• General Overview
  • From Economic Problem to Game Theory
    • Thesis: economic behavior in social exchange must be modeled as games of strategy, not individual maximization.
    • Why calculus failed: economics lacks a universal system; vague formulation and irrelevant tools blocked mathematical progress.
    • Social exchange: each participant maximizes a function he does not fully control — a mixture of conflicting maxima.
    • Utility made numerical: preference axioms yield utility unique up to linear transformation; expectations gain meaning.
    • Solution shape: rational behavior must cover every situation and never let irrationality pay; the imputation is the quantitative core.
  • Formal Description and the Strategic Form
    • Game elements: games have players, moves, choices, information, chance probabilities, and payoff functions.
    • Strategy: a complete plan for every possible information state; strategic form compresses a whole game into one move per player.
    • Chance absorbed: chance moves average out through expectation; only personal-move strategies remain.
    • Zero-sum restriction: zero-sum games make coalitions meaningless in two-person play; total payoff is constant and opposed.
    • Perfect vs imperfect information: Chess is strictly determined; Poker's intransitive knowledge makes bluffing and signaling strategic.
  • Zero-Sum Two-Person Theory
    • Saddle points: a payoff matrix has a value when maximin equals minimax; all saddle points share one value.
    • Minimax theorem: every mixed-strategy zero-sum two-person game has a saddle point; randomization is necessary protection.
    • Mixed strategies: probability distributions over pure strategies, forced by expectation; pure strategies can be found out.
    • Perfect information: backward induction proves strict determinateness of Chess-like games, but optimal play is usually impracticable.
    • Poker analysis: bluffing is inverted signaling; a unique good strategy balances high bids on strong and weak hands.
  • n-Person Games and the Characteristic Function
    • Coalitions become decisive: with three or more players, alliances and compensations determine outcomes.
    • Characteristic function: v(S) gives each coalition its guaranteed value in the two-person game against its complement.
    • Essential vs inessential: inessential games reduce to fixed individual payoffs; essential games make coalition strategy unavoidable.
    • Imputation and domination: payoff vectors are compared by effective coalitions; domination is intransitive and can cycle.
    • Solution as standard of behavior: a set of imputations that is internally stable and dominates all outsiders; multiple standards can coexist.
  • Three, Four, and More Players
    • Three-person solution: the essential game yields a finite three-imputation solution plus discriminatory line solutions.
    • Discriminatory standards: two players may fix the third's payoff within bounds; stable prejudice has quantitative limits.
    • Four-person games: normalized games form a cube; corner types include privileged player, dummy, and fully symmetric center game.
    • Higher-n complexity: essential games need ten parameters at n = 5; six players first show transfer phenomena and unweighted simple games.
    • Composition and decomposition: games split into indecomposable constituents; outside gifts or tribute couple otherwise independent standards.
  • General Non-Zero-Sum Theory and Markets
    • Fictitious player: any general game embeds as a zero-sum game with an extra player receiving the negative total.
    • General characteristic function: drops complementary symmetry; only emptiness and superadditivity remain.
    • Domination restricted: effective coalitions exclude the fictitious player; old theory returns for zero-sum games.
    • Market applications: bilateral monopoly and one-seller–two-buyer games reproduce classical price intervals and add coalition-based solutions.
    • Indivisible utility: integer utility scales reveal that finer scales can dominate bargaining; continuous indeterminacy collapses.
• Deep Dive
  • Introduction
    • Origins and Reception
      • Prehistory: Scattered minimax and game analyses preceded von Neumann's 1928 breakthrough.
      • Collaboration: Morgenstern midwifed von Neumann's theory into Theory of Games and Economic Behavior.
      • Reviewer acclaim: Leading economists praised it; Simon and Stone later won Nobel Prizes.
      • Reviewer inaction: Praise was not followed by research, except for Guilbaud's contributions.
    • Economists' Resistance
      • Princeton isolation: Morgenstern's seminar drew only four students in 1949 economics.
      • Anti-mathematics culture: Department voted against a calculus requirement for undergraduates.
      • Masked curriculum: Mathematical subjects were hidden in titles like "Managerial theory of the firm."
      • No incentive: Graduate students and junior faculty avoided game theory; mathematicians led.
    • Mathematical Development
      • Dantzig–Tucker link: Linear programming proved equivalent to zero-sum games in 1948.
      • Fine Hall seminar: Gale, Kuhn, and Tucker self-studied TGEB and created a weekly forum.
      • Centers at Princeton and RAND: Mathematicians advanced game theory with military sponsorship.
      • New theory: Nash, Shapley, Kuhn, and Tucker repaired TGEB's gaps, often against von Neumann's views.
    • Enduring Legacy
      • Delayed use: Applications emerged after a quarter century; the zero-sum stereotype faded.
      • Nobel 1994: Nash, Harsanyi, and Selten signaled game theory's central place in economics.
      • Foundation: Later expansions rest on von Neumann and Morgenstern's framework.
      • Work in progress: TGEB was incomplete despite its authors' confidence; repairs drove growth.
  • Technical Note
    • Mathematical prerequisites and style
      • Elementary tools: no advanced algebra or calculus; set theory, linear geometry, group theory basics suffice
      • Not truly elementary: deductions intricate, logical possibilities fully exploited
      • Two minor exceptions: simple integrals in an example at 19.7 and a remark in A.3.3
      • Required mindset: mathematical way of reasoning beyond routine phases, especially logic, set theory, functional analysis
    • Presentation approach
      • Not a strict treatise: definitions and deductions deliberately broader than formal mathematics
      • Parallel verbal expositions: every major mathematical deduction gets an unmathematical explanation
      • Model: follows best examples of theoretical physics in method and outlook
      • Moderate readers: can acquire necessary practice while studying the book
    • Reading strategy
      • Skip asterisked sections: those marked in contents are most likely too mathematical for average reader
      • Early omissions harmless: comprehension of early parts survives, though rigorous logical chain breaks
      • Later omissions accumulate: gaps become more significant as the book proceeds
      • Restart after skipping: greater familiarity makes a second reading easier to understand
  • Chapter I Formulation of the Economic Problem
    • Mathematics, Rational Behavior, and Games (Chapter I Formulation of the Economic Problem · I)
      • The Case for a New Mathematical Method
        • Purpose: economic theory needs a treatment different from that found in existing literature.
        • Core problems: individual maximum utility, entrepreneur maximum profit, exchange, monopoly, duopoly, oligopoly, free competition.
        • Game theory: the mathematical theory of games of strategy, developed in 1928 and 1940–1941, is the instrument.
        • Not analogy: typical economic behavior problems become strictly identical with suitable games of strategy.
      • Why Mathematics Has Failed in Economics
        • Universal systems: economics is too difficult for one; even physics has none, so work on mature special problems.
        • Invalid objections: human factors and unmeasurability are no barrier; heat theory shows measurement followed theory.
        • Real causes: vague formulation, inappropriate tools, and inadequate empirical background blocked success.
        • Astronomy precedent: Kepler and Newton needed Tycho’s observations; economics still lacks its empirical base.
        • Overemphasis on calculus: current mathematical economics leans on calculus, but new problems demand new mathematics.
      • Necessary Limitations of Objectives
        • Clear problems first: begin with well-described questions even if they are not practically important.
        • Exact proofs: known results, like indeterminateness of bilateral monopoly, need derivation from an exact theory.
        • Modest procedure: master a limited field, then widen; no shortcuts to burning questions.
        • Heuristic stage: theory evolves from plausibility to formal rigor, then corroboration, extension, and prediction.
        • Practical questions deferred: employment, national income, and distribution have no scientific answers yet.
      • The Problem of Rational Behavior
        • Individual basis: economic theory approaches prices, production, and incomes via behavior of individuals.
        • Traditional motives: consumer maximizes utility or satisfaction; entrepreneur maximizes profit.
        • Utility difficulties: notion and numerical measurement of utility are conceptually hard, treated later in 3.3 and 3.5.
        • Opportunistic stance: the book adopts an opportunistic position on utility, not a primary objective.
    • From Crusoe Maxima to Strategic Interdependence (Chapter I Formulation of the Economic Problem · II)
      • Simplifying Aims and Rationality
        • Monetary utility: assume all participants aim at money, divisible, transferable, identical with satisfaction.
        • Utility not tautological: like physics concepts, it gains empirical meaning through the theories built on it.
        • Rational behavior lacks satisfactory treatment: needs quantitative formulation, not just qualitative.
        • Mathematical methods missing: the maximum problem behind rationality is not unambiguously formulated.
        • Austrian School and Böhm-Bawerk offer qualitative preliminaries; game of strategy gives quantitative form.
      • Crusoe vs. Social Exchange
        • Crusoe’s economy: a single will controls all variables; an ordinary maximum problem, technical only.
        • Social economy: each participant maximizes a function he does not fully control; a mixture of conflicting maxima.
        • Pseudo-maximum: “greatest possible good for the greatest number” is self-contradictory when all maxima are desired at once.
        • No probabilistic shortcut: rivals act rationally, not as statistical data, so their choices cannot be averaged away.
        • Dead data vs living wills: Crusoe faces physical background; social participants face others’ expectations and volitions.
        • Conceptual gap: social exchange is no maximum problem; the theory of games of strategy is devised to meet it.
      • Number of Variables and Participants
        • Partial and total sets: each participant’s variables describe his actions; all sets form the total set.
        • More variables per participant: complicates technically, but preserves the pure maximum character.
        • More participants: changes the nature of the problem; this is an increase in the number of players.
        • Combinatorial explosion: a three-person game differs fundamentally from a two-person game, and so on.
        • Mixed sources: economic growth raises both variables per participant and the number of participants.
      • Free Competition and Coalitions
        • Small vs great numbers: duopoly and oligopoly display strategic interdependence; free competition seems to erase it.
        • Physical analogy misleading: statistical mechanics handles huge numbers, but social theory for 2, 3, 4 participants is lacking.
        • Free competition claims are surmises: the narrowing of indeterminateness with many participants is not established.
        • Coalitions decide: large groups acting together can nullify the effect of great numbers.
        • Real challenge: explain when big coalitions form and when large numbers produce free competition.
      • Utility and the Lausanne School
        • Utility as simplification: strong utility assumptions divide difficulties and focus on strategic interaction.
        • Lausanne equilibrium theory: recognizes interdependence but only under far-reaching restrictions.
        • Free competition assumption: turns participants into independent Crusoe maximizers, avoiding the strategic problem.
        • Petitio principii: excluding coalitions and hidden cooperation assumptions begs the central question.
        • Methodological defense: these investigations are useful, but they do not answer the empirical problem posed.
    • From Preference to Numerical Utility (Chapter I Formulation of the Economic Problem · III)
      • Measurability and Historical Analogy
        • Utility origins: first treated as numerically measurable, but preference alone supports only ordinal comparison.
        • Heat analogy: early thermometry knew only "warmer," yet later produced quantity of heat and temperature.
        • Caution: history warns against final negative assertions; utility's unnumerical look may prove temporary.
      • Probability as Bridge to Numerical Utility
        • Complete preferences: the individual can rank any two imagined events, including probabilistic combinations.
        • 50-50 combinations: comparing A to a mix of B and C reveals whether preference A over B exceeds C over A.
        • Direct scaling: using all probabilities, the α that equals A to a mix of B and C fixes utility ratios.
        • Probability interpretation: frequency in long runs, not subjective estimation, provides the needed numerical foothold.
        • No begged question: only probability combinations are assumed; comparing preference differences is derived, not postulated.
      • Measurement, Models, and Transformation Groups
        • Natural operations: measurable quantities need physically defined operations, e.g., mass admits addition, position admits center of gravity.
        • Mathematical models: correlations must carry natural relations and operations into synonymous numerical concepts.
        • Transformation systems: numerical description is determined only up to a group of transformations, e.g., distance up to positive scaling.
        • Utility as monotone: bare preference makes utility a number up to any monotone transformation—the indifference-curve view.
        • Narrowing transformations: additional observable operations, like utility differences, could reduce utility to linear transformations.
      • Axiomatic Treatment of Numerical Utility
        • Natural operation: combining utilities with probabilities α and 1−α acts as a center of gravity.
        • Representation: order must map to numerical >; combinations must map to weighted averages.
        • Uniqueness: any valid numerical utility is determined up to a linear transformation.
        • Axiom choice: postulates must be few, simple, intuitively meaningful, and guarantee the representation's existence.
    • Axioms, Utility, and Game Solutions (Chapter I Formulation of the Economic Problem · IV)
      • Axioms of Preference and Combination
        • Complete ordering: for any two utilities, exactly one of equality, preference, or inferiority holds, and preference is transitive.
        • Combination operation: any αu+(1–α)v is the prospect of u or v with known probabilities.
        • Continuity: a sufficiently small chance of a preferred v cannot overturn the preferability of w over u.
        • Algebra of combining: order of constituents is irrelevant and compound probabilities reduce to single probabilities.
      • Numerical Utility Derived
        • Representation theorem: axioms guarantee utilities can be mapped to real numbers, unique up to a linear transformation.
        • Expectation legitimated: this construction defines numerical utility as the quantity for which mathematical expectations are legitimate.
        • Justified postulates: each axiom mirrors a plausible trait of preference: completeness, transitivity, continuity, no complementarity.
        • Alternative if incomparability allowed: dropping full comparability leads to a many-dimensional vector utility, less satisfactory but possible.
      • Gambling and Its Utility
        • Self-questioning: do the axioms smuggle in mathematical expectation from the start?
        • Specific gambling utility excluded: a “utility of gambling” cannot be formulated free of contradiction at this level of analysis.
        • Crucial axiom: compounding-probability associativity is the postulate closest to excluding pleasure or pain from taking chances.
        • Interpretation: the theory does not deny deeper psychology; it sets utility so expectations apply under plausible axioms.
      • Marginal Utility and Complete Information
        • Complete-information assumption: all subjects know physical characteristics and can perform needed statistical operations.
        • Divide difficulties: incomplete information is set aside as too vast; it deserves separate treatment.
        • Incomplete-information phenomena reinterpreted: discrimination, incomplete exploitation, and transfer appear within a complete-information theory.
        • Marginal utility in isolation: the maximum utility gain from an added unit measures the effort a Crusoe will expend for it.
        • Social exchange pending: marginal utility’s role remains unclear until a theory of rational behavior in social exchange is built.
      • Shape of a Solution
        • Complete rules: rational behavior must prescribe conduct for every conceivable situation, including others’ irrationality.
        • Irrationality must not pay: a rule set fails if deviation benefits nonconformists and harms conformists.
        • Games as models: precise, exhaustive, simple, and reality-similar constructs fulfill the role of geometric models in economics.
        • Quantitative core: with numerical utility, a solution reduces to how much a rational participant can get, at a minimum.
        • Imputation: the single quantitative statement is the distribution of total proceeds among participants.
      • Beyond One Imputation
        • Single imputation rare: no one imputation generally meets all reasonable optimality conditions.
        • Set of imputations: the solution must broaden to a set, reflecting common-sense features of social organization.
        • Two-person zero-sum games: a unique imputation exists, gives each player at least it, and underlies the general theory.
        • Variable-sum two-person case: bilateral monopoly shows a zone of uncertainty requiring the broader concept.
        • Three-person zero-sum game: any pair can combine against the third; imputations must account for possible coalition shifts.
    • Coalitions, Domination, and Standards of Behavior (Chapter I Formulation of the Economic Problem · V)
      • Coalition Payoffs and Compensation
        • Coalition apportionment: depends not only on game rules but on alternative coalitions available to partners.
        • Compensation payments: a player may transfer gains to a partner to preserve a coalition against better offers.
        • Three-person prototype: solution is a system of three imputations, one per possible alliance, not a single final verdict.
        • Virtual existence: unconsummated alliances shape the actual division, so the whole system of imputations is the significant entity.
        • Stability of the whole: balance and stability belong to the imputation system, not to any single imputation.
      • Domination as Intransitive Superiority
        • Domination defined: x dominates y if a group prefers x and can enforce it, making y unacceptable as a settlement.
        • Effective set: the coalition of participants that enforces a domination.
        • Intransitivity: x over y and y over z does not imply x over z; effective sets may be disjoint.
        • Cyclical dominations: y over x, z over y, and x over z is a characteristic difficulty of social theory.
        • No first element: because domination is intransitive, the idea of a uniquely superior imputation collapses.
      • Precise Definition of a Solution
        • Postulates for solution set S: no element of S dominates another; every imputation outside S is dominated by some element of S.
        • Single imputation case: reduces to the first element—dominates all others and is dominated by none.
        • Implicit definition: the conditions characterize all possible solutions without guaranteeing existence or uniqueness.
        • Open questions: existence unproven in full generality; uniqueness generally fails, yielding multiple solutions.
      • Standards of Behavior and Social Order
        • Solution as standard of behavior: an accepted set of imputations expressing general principles with varied particular details.
        • Inner stability: the standard has no inner contradictions and discredits every non-conforming imputation.
        • Circularity is natural: stability holds only under general acceptance, so different standards can contradict each other.
        • Games as models: game theory examples test and illustrate assertions about social organizations.
        • Equilibrium over a priori purpose: the theory locates the equilibrium of forces, not what ought to happen by fiat.
    • Non-Unique Solutions and the Static Method (Chapter I Formulation of the Economic Problem · VI)
      • The Answer Is a Set of Sets
        • Levels of analysis: single imputations → solution sets → the set of all solutions.
        • Non-uniqueness: complete answer is not one solution but the full set of solutions.
        • Formal justification: game theory's mathematical structure makes this layered object the natural goal.
        • Desirable complexity: multiple standards of behavior require sets of sets, not simple single answers.
      • "Standards of Behavior" and Strategies
        • Older view: solution as explicit rules telling each player how to behave in every situation.
        • Limits: narrow standard works only where coalitions and partner compensations play no role.
        • Typical games: social exchange economies require coalitional compensation, so the narrow view fails.
        • Strategies: those rule-sets are renamed strategies once imputations become the working concept.
      • A Static Theory with Rudimentary Dynamics
        • Static core: the theory deals with equilibria, not with change.
        • Dynamic ideal: a dynamic theory would be more complete, but building one first is futile.
        • Domination talk: dynamic-sounding discussion of imputations is legitimate for characterizing stability.
        • Rudimentary dynamics: equilibrium analysis needs minimal dynamic concepts, not a full theory of motion.
        • Real dynamics: precise motions far from equilibria require deeper knowledge than statics supplies.
      • Departure from Mathematical Physics
        • Different relationship: statics/dynamics structure in social theory may differ generically from classical physics.
        • Anticipated dynamic form: dynamics may use a single current imputation rather than sets of imputations.
        • Narrow solution rejected: a unique number or aggregate proves too narrow for social equilibrium.
        • Mathematical shift: method moves toward combinatorics and set theory, away from differential equations.
  • Chapter II General Formal Description of Games of Strategy
    • From Economics to Formal Game Theory (Chapter II General Formal Description of Games of Strategy · I)
      • Shift in Point of View
        • Independent subject: theory of rational behavior requires studying games of strategy for their own sake.
        • Economic roots remain: most main concepts echo economic literature, but details become alien to it.
        • Guiding aim: exact formulation before returning to economics and social organization.
      • Zero-Sum Versus Non-Zero-Sum
        • Zero-sum games: total payments to all players always zero; entertainment games fit this model.
        • Non-zero-sum games: total social product varies with players' behavior; economically significant schemes are such games.
        • Reduction: every n-person game reduces to a zero-sum (n+1)-person game.
        • Foundations: zero-sum theory rests on the two-person case, where coalitions and compensations play no role.
        • Monetary motive: final transactions are purely monetary, a necessary simplification tied to utility theory.
      • Terminology of Games and Moves
        • Game vs play: game is the totality of rules; a play is one complete instance from beginning to end.
        • Move vs choice: move is an abstract occasion of choice; choice is the alternative selected in a concrete play.
        • Rules vs strategies: rules are absolute commands; strategies are freely chosen principles governing choices.
        • Personal vs chance moves: personal moves are decided by a player; chance moves by a random device with fixed probabilities.
        • Strategy concept: the mathematical device replacing the complicated general scheme by a simpler, rigorously equivalent one.
      • Formal Elements of a Game
        • Fixed sequence: a game has v moves in chronological order, each offering ακ alternatives.
        • Complete play: described by the sequence of chosen alternatives σ1,...,σv.
        • Payoff functions: rules specify each player's payment as a function of the complete play.
        • Choices are variables: fixing σκ values defines a play; it is no part of the game itself.
      • Information, Preliminarity, Anteriority
        • Information state: at each personal move, the rules specify a set of anterior moves whose outcomes the player knows.
        • Preliminarity: λ is preliminary to κ when λ's outcome is known at κ; anteriority alone does not ensure this.
        • Perfect information: Chess and Backgammon make preliminarity coincide with anteriority.
        • Imperfect information: Poker and Bridge separate the two; preliminarity need not be transitive.
        • Bridge partnerships: A and C form one player, B and D another, acting through representatives.
        • Poker intransitivity: player 2 sees player 1's bid but not the original deal, breaking the chain of knowledge.
      • Signaling
        • Direct signaling: spreads information within one's own organization; Bridge's conventions are strategy, not rules.
        • Inverted signaling: irregular, illogical behavior makes signals ambiguous and misleads opponents.
        • Bluffing: Poker's bluff is inverted signaling; the same device appears in almost all games.
    • Information, Variability, and Set Formalism (Chapter II General Formal Description of Games of Strategy · II)
      • Signaling and Information
        • Intransitive preliminarity: arises when information flow among players is not simply ordered.
        • Direct vs. inverted signaling: Bridge splits a player to create direct signaling; Poker avoids it, producing inverted signaling.
        • Costly signaling: deviating from unsophisticated play to signal more or less carries direct losses.
        • Rational balance: players seek an optimum where signaling advantages overbalance its direct costs.
        • Chance moves: typical in intransitive preliminarity, but later analysis shows they barely affect strategy essentials.
      • The Complete Concept of a Game
        • Move characteristics: ακ, kκ, Λκ and probabilities may depend on anterior choices, not just κ.
        • Personal moves: dependence on unknown prior choices creates information conflicts for the player.
        • Special form: no conflict if Λκ is fixed and ακ, kκ depend only on known σλ.
        • Combination knowledge: knowing a function like σμ+σλ without its parts cannot be captured by Λκ.
      • The General Description via Φκ
        • Information functions: replace Λκ with Φκ, the set of functions of anterior choices whose values the player knows.
        • Consistency: ακ and kκ must themselves belong to Φκ when κ is a personal move.
        • Generality: Φκ is needed to exhaust all combinatorial possibilities and meet possible objections.
        • Economic relevance: there are economic models where Φκ is necessary.
        • Ultimate simplicity: formal elements complicate only description, not the final strategic problem.
      • Bounded Plays and Dummy Moves
        • Stop rule: variable play length ends at v depending on choices; must guarantee every play stops.
        • Bounded stop: assume a fixed bound v* so no infinite play is possible.
        • Dummy moves: extend the game to fixed v* moves by adding one-alternative chance moves after the stop.
        • Fixed number justified: this restores the initial assumption that the number and arrangement of moves are given ab initio.
      • Set-Theoretic Formulation
        • Motivation: set theory yields an equivalent but more unified and transparent description of a game.
        • Sets: arbitrary unordered collections; the empty set and one-element sets are legitimate.
        • Operations: subset, sum, product, difference, complement, and disjunctness are defined for sets.
        • Sets of sets: elements can themselves be sets, forming systems or aggregates.
      • Partitions and Trees
        • Partition: nonempty pairwise disjunct subsets of Ω; subpartitions refine them.
        • Subpartition: refines another partition; mutual subpartition implies equality.
        • Superposition: combined partition of all nonempty intersections of two partitions.
        • Local relations: subpartition and equality can be relativized within a subset C.
        • Trees: nested partitions form tree diagrams, representing successive refinements.
    • Set-Theoretic Games, Information, and Axioms (Chapter II General Formal Description of Games of Strategy · III)
      • Sets, Properties, and Information
        • Sets as properties: every property of elements of Ω is the subset of elements possessing it.
        • Logical operations map to set operations: disjunction is union, conjunction is intersection, negation is complement.
        • Information as subset: a body of information narrows the unknown element to the set of still-possible elements.
        • Absurd and incompatible information: the empty set is absurd; disjoint sets are mutually exclusive information.
        • Actual information vs. pattern: a subset gives definite information; a partition announces what information may be given later.
      • Partitions as Information Patterns
        • Partition: pairwise disjoint, nonempty subsets of Ω; a preliminary announcement, i.e. a pattern of information.
        • Subpartition: one partition is a subpartition of another when the latter's announced information includes the former's.
        • Play universe Ω: finite set of all conceivable plays, possibly including absurd sequences for convenience.
        • Pre-move partition ℘κ: umpire's full information up to move κ; its cells Aκ are possible histories.
        • Boundary partitions: ℘₁ is the single set Ω; ℘ᵥ₊₁ consists of one-element sets.
      • The Anatomy of a Move
        • Move owner: who moves at κ is constant within each history Aκ; Bκ(k) groups histories by chance or player.
        • Chance moves: alternatives and probabilities are constant within each history set; each alternative is a cell of chance partition.
        • Personal moves: player's information is a partition κ(k); history partition is a subpartition of it.
        • Choice cells: the number of alternatives is constant within Dκ; choices are cells of a subpartition of the information partition.
        • Possible-choice condition: intersections Aκ∩Cκ inside the same Dκ are nonempty; each Dκ offers at least one choice.
        • Forbidden moves are spurious: apparent impossible choices like double-blind chess are resolved by splitting moves.
      • From Move to Move
        • Successor partition: ℘κ₊₁ adds to ℘κ the outcome of the choice made at move κ.
        • Superposition rule: ℘κ₊₁ is formed by intersecting ℘κ with all choice partitions κ(0),…,κ(n), then discarding empty sets.
        • Chance region: within Bκ(0), ℘κ₊₁ coincides with the chance-choice partition κ(0).
        • Personal region: within Bκ(k), ℘κ₊₁ is built from intersections Aκ∩Cκ inside each information set Dκ.
        • Payoffs: each player's outcome H_k is a function of the actual play π over Ω.
      • Axiomatic Formulation
        • Specification of a game: data are v, finite Ω, payoff functions H_k, partitions ℘κ, Bκ(k), choice partitions, and information partitions.
        • Structural conditions: subpartition relations connect history, information, and choice partitions at each move.
        • Chance probabilities: for each chance history set, probabilities are nonnegative and sum to one.
        • Boundary and recursion axioms: ℘₁ is {Ω}, ℘ᵥ₊₁ is singletons, and each ℘κ₊₁ comes from superposition.
        • Axiomatic method: define concepts without intuitive names; attach intuitive labels only after exact analysis.
    • Axioms, Strategies, and the Game's Reduced Form (Chapter II General Formal Description of Games of Strategy · IV)
      • Meaning of the Axioms
        • Technical vocabulary: length v, plays Ω, outcomes k(π), patterns of information, assignment, and choice, chance probabilities pκ(Cκ).
        • Umpire's information: includes the move's assignment and, for personal moves, the player's information.
        • Choice patterns: chance choices include umpire info; personal choices include the player's actual information.
        • Chance probabilities: behave like disjunct exhaustive alternatives, with the sum of probabilities equal to one.
        • Memory condition: umpire info at move κ+1 is the superposition of prior info and the choice made.
        • End conditions: first move begins with void info; final umpire info determines the play uniquely.
      • Logistic Properties
        • Consistency: games exist, so the axioms are free from contradiction.
        • Categoricity absent: axioms define a class of games, not one unique game.
        • Independence: holds but is not demonstrated.
      • Axiomatization as Method
        • From twilight to precision: axiomatization progressively removes the imprecision of the intuitive game notion.
        • Psychological phenomena: decisions, information, and information interrelations are shown to be axiomatizable.
      • Graphical Representation
        • Tree of partitions: each Aκ+1 is a subpartition of Aκ, so a fixed-length game forms a tree.
        • Full information case: preliminarity equals anteriority when player info coincides with umpire info within personal moves.
        • Bracketing device: elements belonging to the same Bκ(k) are encircled and labeled with the moving player.
        • Single-owner moves: if move ownership is independent of prior play, only move-type labels are needed.
      • Strategies and the Umpire's Choice
        • Strategy: a complete plan specifying a choice for every possible actual information; it does not restrict freedom of action.
        • Formal strategy: a function Σk(κ; Dκ) choosing a Cκ within Dκ; existence rests on postulate (10:1:j).
        • Umpire's choice: predetermined chance outcomes for every possible umpire information, formalized as Σ0(κ; Aκ).
        • Probabilities: an umpire's choice has probability equal to the product of its independent chance-move probabilities.
      • Final Simplified Description
        • Determination: fixed strategies and an umpire's choice uniquely determine the play and all outcomes.
        • Inductive construction: starting from Ā1 = Ω, each move's choice yields the next umpire information, ending in a single play.
        • One-move-per-player form: the game reduces to n+1 simultaneous moves with finite alternatives βk, made without information about other choices.
        • Absorbing chance: the chance move becomes a probability distribution over the outcome-tuples linked to the chosen strategies.
    • From Moves to Strategic Choice (Chapter II General Formal Description of Games of Strategy · V)
      • Final Simplified Form
        • Simplified game: each player (k) makes one move, choosing (\tau_k) from (1) to (\beta_k), in absolute ignorance of the others.
        • Payoff function: an umpire determines player (k)’s outcome as (\mathscr{F}_k(\tau_1,\ldots,\tau_n)).
        • Mathematical expectation: player judgment is directed solely by expectation, because moves are completely isolated from each other.
        • Only personal moves matter: chance moves are averaged out; only the (n) personal moves remain.
      • Strategies as Moves
        • No higher-order strategies: once original moves are replaced by strategies, no room remains for strategies of a higher order.
        • Crystallization: the transition from original moves to strategies compresses the whole game into this rigid, final form.
        • Strategy in this form: with one personal move and nil information, a strategy is simply a definite choice (\tau_k).
        • Ignorance is total: every player chooses without knowing the choices of anyone else.
      • Zero-Sum Restriction
        • Zero-sum condition: the sum of all players’ payoffs is identically zero for every combination of strategies.
        • Coalitions impossible: in a two-person zero-sum game, any understanding or coalition involves all players and zero total payoff.
        • No objectives left: with no opponents outside the coalition, there are no possible objectives for it.
        • Proof deferred: the complete theory of zero-sum two-person games is built in Chapter III, with the decisive result in §17.
      • Rules and Stop Conditions
        • Game data: the rules are fully determined by the numbers (\beta_k) and the payoff functions (\mathscr{F}_k).
        • Stop rule essential: every game must bound the number of moves, or a play could continue indefinitely.
        • Examples: Ecarté has an explicit finite bound; Poker needs overbid limits; Bridge and Chess require added tie or stop rules.
        • Finite plays imply bound: if no infinite play is possible, König’s theorem guarantees a fixed finite upper bound (v^*).
      • Information and Set-Theoretic Structure
        • Partitions model knowledge: moves are described by partitions of choices; information sets specify what a player knows.
        • Chance vs personal moves: chance moves have fixed probabilities; personal moves depend on the player’s free decision.
        • Anteriority matters: the order of moves is essential, and information at a move depends on anterior choices.
        • Axiomatic method: rules are stated abstractly, as in geometry, without identifying moves with their intuitive counterparts.
  • Chapter III Zero-Sum Two-Person Games: Theory
    • From Extensive Games to Saddle Values (Chapter III Zero-Sum Two-Person Games: Theory · I)
      • Game Forms and Research Order
        • Extensive vs. normalized form: strategies compress any game into a fully equivalent strategic form; use whichever suits the task.
        • Division of labor: normalized form proves general theorems; extensive form analyzes special structures and differences.
        • Program of complication: proceed from one-person to zero-sum two-person, then zero-sum three-person, then n-person, then general n-person.
        • General n-person games: theory will reduce to the zero-sum (n+1)-person game as a special case.
        • Classical mathematical games: chance evaluation via probability and expectation is left behind; not games proper.
      • The One-Person Game and Its Limits
        • Rational play: one player maximizes payoff over strategies; the general one-person game is a pure maximum problem.
        • Strategies as plans: the single variable represents an entire policy for all contingencies, not a single move.
        • Solitaire's gap: ordinary patience lacks incomplete information; such cases require splitting one player into communicating agents.
        • Robinson Crusoe economy: a fixed distribution scheme is a one-person game, valid only when the scheme is unquestioned.
        • Maximum approach's limit: appraising or contesting the distribution scheme itself demands n-person games with n ≥ 2.
      • Functional Calculus for Game Theory
        • Functions as dependencies: value determined by variables; domain and number of variables are part of the function's identity.
        • Max and Min operations: denote extreme values; the attained value is unique, though the maximizing choice need not be.
        • Killing variables: Max_x or Min_x removes x, leaving a function of the remaining variables.
        • Repeated operations: applying one extreme operation per variable, in any order, yields a constant.
      • Commutativity and Saddle Points
        • Max-Max, Min-Min commute: order of repeated maxima or repeated minima never matters.
        • Max-Min do not generally commute: Max_x Min_y and Min_y Max_x can differ.
        • Matrix reading: Max_x Max_y is the absolute matrix maximum; Max_x Min_y is the maximum of row minima.
        • Saddle point criterion: equality holds exactly when some matrix field is simultaneously row minimum and column maximum.
        • Decisive for games: this commutativity question underlies the forthcoming theory of zero-sum two-person games.
    • Saddle Points and Strictly Determined Games (Chapter III Zero-Sum Two-Person Games: Theory · II)
      • Saddle Point Concept
        • Definition: x0,y0 is a saddle point if φ(x,y0) peaks at x0 and φ(x0,y) bottoms at y0.
        • Orographic image: row x0 = mountain ridge; column y0 = road crossing the pass.
        • Existence not automatic: a payoff matrix may have no saddle point.
        • Saddle value: all existing saddle points yield the same value, denoted Sa.
      • Minimax Theorems
        • Minimax inequality (13:A*): Max_x Min_y φ(x,y) ≤ Min_y Max_x φ(x,y) always holds.
        • Equivalence (13:B*): MaxMin = MinMax iff a saddle point exists.
        • Key sets: A = x maximizing Min_y φ; B = y minimizing Max_x φ.
        • (13:D*): saddle points are exactly the product A × B.
        • Plateau shape: all saddles together form a rectangular plateau at one altitude.
      • Existence for Function Payoffs
        • Functional payoff: if φ is ψ(x, f(x)) over functions f, a saddle point always exists.
        • Construction: choose f0(x) making ψ(x,u) minimal for each x; then use minimax inequality.
        • Generality: this functional form already covers a wide class of games.
      • The Two-Person Zero-Sum Game
        • Normalized form: zero-sum game; 1 chooses τ1, 2 chooses τ2 in ignorance, payoff φ(τ1,τ2).
        • Opposed aims: player 1 maximizes φ, player 2 minimizes it.
        • Partial control: each player controls only his own variable; the other is a rational adversary, not chance.
        • Matrix representation: rows and columns are strategies; the payoff matrix is unrestricted.
      • Minorant and Majorant Games
        • Minorant Γ1: 1 moves first, 2 sees it; value v1 = Max Min φ — worst for 1.
        • Majorant Γ2: 2 moves first, 1 sees it; value v2 = Min Max φ — best for 1.
        • Good strategies: Γ1's 1 uses A; Γ2's 2 uses B; responding player optimizes against the known choice.
        • Secured value: in each auxiliary game, both players can guarantee their payoff independently.
        • Symmetric characterization: value statements are identical in form for Γ1 and Γ2, despite asymmetric roles.
      • Original Game Value
        • Ordering: v1 ≤ v2 follows from (13:A*); Γ lies between its minorant and majorant.
        • Interpretation: v1/v2 correspond to Γ under suitable assumptions about players' knowledge.
        • Open path: a later device must close the gap to give a complete theory of Γ.
    • Strict Determinateness and Perfect Information (Chapter III Zero-Sum Two-Person Games: Theory · III)
      • Finding Out and the Interval of Value
        • Discovery advantage: a player who “finds out” the rival’s strategy turns Γ into Γ₁ or Γ₂, with value v₁ or v₂.
        • Bounds on value: any value v for Γ itself must satisfy v₁ ≤ v ≤ v₂.
        • Interval length: Δ = v₂ − v₁ measures the advantage of finding out one’s adversary rather than being found out.
        • Strictly determined: Γ is strictly determined when Δ = 0, i.e. v₁ = v₂, equivalent to existence of a saddle point.
      • Guaranteed Value in Strictly Determined Games
        • Value secured: if strictly determined, player 1 can guarantee gain v; player 2 can guarantee gain −v.
        • Optimal strategies: a good way for 1 is any τ₁ in A, the maximin set; for 2, any τ₂ in B, the minimax set.
        • Mutual optimality: if both play well, the payoff equals the value v.
        • Saddle-point link: both play well iff their strategies form a saddle point of K(τ₁, τ₂).
      • Symmetry Under Player Interchange
        • Swapping players: interchange replaces payoff K(τ₁, τ₂) by −K(τ₂, τ₁).
        • Effect on bounds: v₁ and v₂ become −v₂ and −v₁, so the value v is unchanged.
        • Invariance: strict determinateness and all optimality statements survive the interchange.
      • Non-Strictly Determined Games
        • Unsolved interval: when Δ > 0, no unique value is established; only v₁ ≤ v ≤ v₂ remains.
        • Matching Pennies: heads/tails payoff ±1; no strategy is intrinsically better, only guessing the rival matters.
        • Stone, Paper, Scissors: cyclic defeat gives the same in-vitro difficulty: any pure strategy is as good as any other.
        • Promise: a solution for Δ > 0 will be found later along the same lines.
      • Perfect Information and the Induction Step
        • Perfect information: each player at a personal move knows all outcomes of anterior moves; such games are candidates for strict determinateness.
        • Inductive reduction: fix the first move’s choice σ₁ = 1 to form Γ₁ with one move fewer.
        • Condition for reduction: Γ₁ can be formed without altering information only if the first move is preliminary to all later personal moves.
        • Set-theoretic form: dictating the first move restricts play to C₁(λ); partitions are then cut by intersection with C₁(λ).
    • Perfect Information Yields Strict Determinacy (Chapter III Zero-Sum Two-Person Games: Theory · IV)
      • Condition for Inductive Decomposition
        • Preliminarity vs. anteriority: To compose games from subgames, every earlier move must be known before later moves.
        • Perfect information: Exact requirement for full backward induction; preliminarity must coincide with anteriority.
        • Subpartition condition: All κ(k) must be subpartitions of κ whenever κ ≤ λ, keeping player information intact.
        • Sequence of subgames: Inductive decomposition reduces length from v to 0, ending in a vacuous game.
      • Inductive Step by First Move Type
        • Chance move (k=0): Value is the probability-weighted average of the subgames’ values.
        • Personal move of player 1 (k=1): Player 1 chooses the subgame, so v₁ maximizes and v₂ minimizes over σ₁.
        • Personal move of player 2 (k=2): Interchange players: v₁ minimizes and v₂ maximizes over σ₁.
        • Zero-sum symmetry: Sign-flip transforms the player-1 rule into the player-2 rule.
      • Abstract Operators for Value Composition
        • Operator O_k: Average for chance, maximum for player 1, minimum for player 2.
        • Variable elimination: Each operator removes dependence on the first move σ₁.
        • Game dependence: O_k uses k₁, the probabilities, and the range of σ₁.
        • Uniform transformation: For each k₁, the same operation applies to both players’ values.
      • Strict Determinacy and Explicit Value
        • Induction on game length: A vacuous game has fixed payoff; the inductive step preserves v₁ = v₂.
        • Main theorem: Every zero-sum two-person game with perfect information is strictly determined.
        • Explicit formula: Apply the operators backward from the final payoff through all moves to get v.
        • Construction path: Form subgames of decreasing length, then reverse the induction to recover Γ’s value.
      • Consequences for Chess and General Games
        • Chess outcomes: With payoffs 1, 0, –1, the value means white wins, a tie, or black wins.
        • Abstract existence only: Optimal strategies are guaranteed but usually too lengthy to construct effectively.
        • Why Chess remains playable: Without a usable determination, heuristic “good” play keeps struggle and surprise.
        • Verbal backward induction: Starting from the last move, assign expected, maximal, or minimal value at each stage.
        • General n-person limit: The same plausibility argument fails beyond zero-sum two-person games, requiring different methods.
    • Convex Foundations for Mixed Strategies (Chapter III Zero-Sum Two-Person Games: Theory · V)
      • Rationality and Recursive Value
        • Recursive value: fixing later choices reduces the game to v–1 and yields its value before the first move.
        • Rationality presupposed: the recursive argument makes each player rely on the later optimal play of the opponent.
        • Zero-sum immunity: with only two players, an opponent's irrational loss automatically becomes an equal gain for you.
        • Rigorous route: the earlier proof of 15.4–15.6 avoids this objection and is the decisive one.
        • Generalization warning: the rationality issue becomes relevant once the zero-sum two-person restriction is removed.
      • Linearity and Convexity
        • Linear space: L_n consists of all n-tuples of real numbers, treated as vectors or functions.
        • Vector operations: scalar multiplication and addition are componentwise; coordinate vectors span every vector.
        • Hyperplanes and half-spaces: a nontrivial linear equation cuts L_n into two opposing half-spaces.
        • Convex set: contains the whole interval between any two of its points.
        • Convex hull: the set of weighted sums with nonnegative weights totaling one is the smallest convex set containing the given points.
        • Standard sets: P_p is the positive orthant; S_p is the simplex of nonnegative weights summing to one.
      • Supporting Hyperplanes
        • Separation theorem: any point outside a convex set can be separated from it by a hyperplane; this holds for arbitrary convex sets.
        • Nearest-point idea: the closest point of the set to the outside point determines the separating hyperplane.
        • Geometric intuition: if a convex point lay on the near side, some point of the interval to it would be closer still.
        • Dimension-free proof: algebraic componentwise inequalities make the theorem rigorous for all n.
      • Matrix Alternative and Skew-Symmetric Case
        • Alternative theorem: for any matrix, exactly one of two opposite inequality systems has a solution in the appropriate simplex.
        • Dual view: applying the theorem to the negative transpose interchanges rows and columns and reverses the inequality direction.
        • Exclusion: the alternatives are mutually exclusive, so no matrix satisfies both at once.
        • Skew-symmetric matrix: equal to its negative transpose; row and column conditions then coincide.
        • Equilibrium hint: in the skew-symmetric case, some ω in S_n makes all weighted sums Σ_i a(i,j)ω_i nonnegative.
      • Mixed Strategies and Non-Strictly Determined Games
        • Pure-strategy dead end: in Matching Pennies or Stone-Paper-Scissors, no single choice is better than another.
        • Statistical strategy: choose heads and tails each with probability 1/2, by a chance device.
        • Guaranteed expectation: against any opponent strategy, pure or mixed, this procedure yields expectation zero.
        • General program: the convexity results extend the strictly determined solution to all zero-sum two-person games.
    • Mixed Strategies and General Determinateness (Chapter III Zero-Sum Two-Person Games: Theory · VI)
      • Randomization as Defense
        • Statistical strategy: mixing possible plays with chosen probabilities lets a player avoid certain loss.
        • Matching Pennies: 50:50 mixture of heads and tails gives the play value zero and counts as a good strategy.
        • Paper, Stone, Scissors: playing all three alternatives with equal probabilities is the common-sense solution.
        • Protective ignorance: the player cannot reveal a strategy he does not know himself.
      • Generalizing to Mixed Strategies
        • Mixed strategy vector: a point in the probability simplex assigning likelihoods to all pure strategies.
        • Pure strategy: coordinate vector giving one strategy probability 1 and all others probability 0.
        • Full flexibility: probabilities of zero exclude strategies; probability 1 reproduces pure choice.
        • Goal: extend the insight of Matching Pennies to every zero-sum two-person game.
      • Justifying the Static Viewpoint
        • Apparent conflict: static single-play analysis clashes with the idea of a strategy being “found out.”
        • Indirect proof: imagine a complete theory exists; then each player must assume his strategy is known by the opponent.
        • Two extremes: a satisfactory theory must reconcile Γ₁, where player 1 is found out, with Γ₂, where player 2 is found out.
        • Final check: after deriving the theory, justify it directly rather than only heuristically.
      • Majorant and Minorant Games
        • Bilinear payoff: K(ξ,η) is the expected outcome when players choose mixed strategies independently.
        • Γ₁ value: v₁′ = max over ξ of min over η of K(ξ,η), since player 2 chooses after knowing ξ.
        • Γ₂ value: v₂′ = min over η of max over ξ of K(ξ,η), with the order reversed.
        • Always ordered: v₁′ ≤ v₂′; equality is the criterion for a saddle point.
        • No extra assumptions: the earlier rational-behavior justifications apply literally to these mixed-strategy games.
      • General Strict Determinateness
        • Two concepts: special strict determinateness requires v₁ = v₂; general strict determinateness requires v₁′ = v₂′.
        • Implication: v₁ ≤ v₁′ ≤ v₂′ ≤ v₂, so special determinateness implies general determinateness.
        • Main theorem: every zero-sum two-person game has a saddle point for K(ξ,η); the indeterminate case never occurs.
        • Bilinear structure: because K is linear in each variable, the minimax argument succeeds for arbitrary payoff matrices.
        • Secure gain: a player can, by an appropriate mixed strategy, secure the value v′ regardless of the opponent’s play.
      • Why Mixed Strategies Succeed
        • Pure payoff function: arbitrary as a function of τ₁, τ₂; no saddle point can be guaranteed.
        • Mixed extension: K(ξ,η) is a bilinear function on a much larger probability domain.
        • Linear interpolation: extending pure payoffs to mixed strategies is forced by mathematical expectation.
        • Containing the original: every pure strategy appears as a coordinate vector, so mixed theory truly generalizes pure theory.
        • Full protection: mixed strategies protect both players against having their strategies found out; abandoning them restores the disadvantage.
    • Optimal Strategies, Mistakes, and Symmetry (Chapter III Zero-Sum Two-Person Games: Theory · VII)
      • Value and Good Strategies
        • Good strategies: player 1 chooses from Ā, player 2 from B̄, securing v′ and –v′ respectively.
        • Saddle point criterion: a strategy pair is good iff it forms a saddle point of K(ζ, η).
        • Value v′: the unique payoff when both players play well; the value of a play for player 1.
        • No rationality assumptions: the argument does not rely on beliefs about the opponent’s intelligence or choices.
      • Characterizing Good Strategies
        • Mutual optimality: good mixed strategies exclude pure strategies that are not optimal against the opponent’s good strategy.
        • Pure optimality: the game is specially strictly determined iff each player has a good pure strategy.
        • Equality criterion: v₁ = v̄₁ and v₂ = v̄₂ hold exactly when a pure strategy belongs among the good ones.
      • Mistakes and Permanent Optimality
        • Distance from goodness: α(ζ) and β(η) measure the maximum loss risked by using a non-good strategy.
        • Defensive, not offensive: good strategies secure the value but do not exploit the opponent’s mistakes.
        • Permanent optimality: a strategy optimal against every strategy is necessarily good, but such strategies rarely exist.
        • Perfect information: these games possess permanently optimal pure strategies.
        • Limits: permanent optimality implies special strict determinateness, but the converse is false.
      • Interchange and Symmetry
        • Interchanging players: transforms v₁, v₂, v′ into –v₂, –v₁, –v′, with optimal sets swapped.
        • Symmetric games: skew-symmetric payoff matrices make each play’s value zero.
        • Fair games: v′ = 0 can hold without symmetry, as in Matching Pennies.
        • Self-optimality: in symmetric games, every good strategy is optimal against itself.
        • Mixed strategies protect: avoiding loss in symmetric games requires mixing; pure strategies can be found out.
    • Mixed Strategies and Minimax Equilibrium (Chapter III Zero-Sum Two-Person Games: Theory · VIII)
      • Normalized Form and Symmetry
        • Normalized form: a strategy choice in the normalized game corresponds to a strategy in the original extensive form.
        • Advantage Δ: the same for both players; Δ > 0 means they cannot each be cleverer than the other.
        • Simultaneous choice: in the normalized game each player picks τ₁ or τ₂ without knowing the opponent's selection.
        • Role interchange: interchanging players fully exchanges their functions and variable domains, preserving the formal structure.
        • Skew-symmetry: a symmetric game corresponds to a skew-symmetric matrix or bilinear form, not a symmetric one.
      • Inductive Reduction of Games
        • Inductive step: fixing the first move reduces a ν-move game to one with ν−1 moves.
        • Complete solution: repeated reduction reaches zero moves, yielding a fixed, unalterable outcome.
        • Chess illustration: dictating an opening move makes the next move belong to player 2 in the new position.
        • Practice in Bridge: umpires assigning hands, as in Duplicate Bridge, fixes chance moves before play.
        • Dummy moves: variable-length games must be padded with dummy moves to make fixed-length regression possible.
        • Practical limits: only simple games like Tit-tat-toe can be solved this way; larger games like Chess are enormous.
      • Mixed Strategies and Randomization
        • Matching Pennies: no deterministic choice is safe, because the opponent chooses simultaneously and can mirror or counter.
        • Randomization: chance devices like a die randomize pure strategies, making actual play unpredictable.
        • Probabilities as strategy: the essential step is letting strategies be probability distributions; particular values are accidental.
        • Discovered strategies: if the opponent learns frequencies, randomization still prevents predictable losses on each play.
        • Stone-Paper-Scissors: cyclic pure strategies defeat each other; mixing equalizes outcomes and introduces ties.
        • Mathematical expectation: linear expected payoff underlies the connection between mixed strategies and numerical utility.
      • Min-Max Theorem and Saddle Points
        • Min-Max equality: the game's value satisfies max over ζ₁ min over ζ₂ K = min over ζ₂ max over ζ₁ K.
        • Bilinear payoff: K(ζ₁, ζ₂) is a bilinear form because probabilities enter through mathematical expectation.
        • Saddle-point criterion: a pair of mixed strategies is good iff it forms a saddle point of K.
        • Permanent optimality: a permanently optimal strategy combined with an optimal response yields a good strategy pair.
        • Convexity proof: the theorem rests on convex-set separation, with a key connection to Brouwer's fixed-point theorem.
    • Fairness Without Symmetry in Games (Chapter III Zero-Sum Two-Person Games: Theory · IX)
      • Symmetry Forces a Square Matrix
        • Reflection invariance: a game carried into itself by interchanging players must leave its matrix scheme unaltered.
        • Square shape required: a rectangular scheme would change under reflection, so β₁ must equal β₂.
        • Same-role games: when players 1 and 2 have identical roles, the squareness follows automatically.
        • Theoretical basis: this conclusion rests on the general theorem 16:F, not on separate value arguments.
      • Fairness Need Not Imply Symmetry
        • Fairness ≠ symmetry: a grossly unsymmetric game can still be fair if advantages are carefully balanced.
        • Matching Pennies: player 1 matches, player 2 avoids, but the asymmetry is felt to be inessential.
        • Desired example: a truly unsymmetric game whose value is exactly v′ = 0 would demonstrate fairness without symmetry.
      • Rolling Dice: Nearly Fair Asymmetry
        • Player versus house: the player rolls two dice; the house has no influence on play.
        • Immediate results: 7 or 11 win, 2, 3, or 12 lose, other totals force further rolls.
        • Repeated rolls: winning requires repeating the original total; rolling a 7 loses.
        • Near-even chances: the player has 244 winning chances against 251 for the house out of 495.
        • Slight house edge: the value is −7/495 per unit stake, a close approximation to fairness.
  • Chapter IV Zero-Sum Two-Person Games: Examples
    • Simplest Games: Determinateness, Strategies, Limits (Chapter IV Zero-Sum Two-Person Games: Examples · I)
      • The Simplest Games and Their Symmetries
        • Two-choice games: the basic field of study; normalized form with β₁ = β₂ = 2.
        • Equivalent adjustments: row/column interchanges and swapping players yield eight representations of the same game.
        • Explicit computation: concrete examples reveal the common-sense meaning of value, saddle points, and good strategies.
      • Exhaustive Cases for 2×2 Games
        • Case A: max and min lie on opposite diagonals; the remaining field becomes a saddle point, so the game is strictly determined.
        • Case B₁: max and min share a row or column, but the inequality still yields a saddle point.
        • Case B₂: max and min share a row or column without that inequality; the game is not strictly determined.
        • Unique mixed solutions: in B₂, each player has exactly one good strategy, and both components are positive.
      • Qualitative Criteria for Strict Determinateness
        • Diagonal separation: not strictly determined iff one diagonal’s payoffs all exceed the other’s.
        • Majorization: strictly determined iff one row or column is element-wise at least as good as its alternative.
        • Rationale: majorization gives a player an unconditionally safe choice; proving its necessity is the nontrivial insight.
        • Mixed strategies mandatory: in the indeterminate case, no pure strategy is good; both players must randomize.
      • Applications: Matching Pennies Variants
        • Ordinary Matching Pennies: value zero; unique good strategies use heads and tails equally.
        • Premium on matching heads: matcher’s value rises, but he chooses heads less often; the opponent shifts similarly.
        • Asymmetric penalties: further matrix changes move value and probabilities; only the formal formulae guarantee the result.
        • Holmes–Moriarty: the escape episode is Matching Pennies in disguise; Moriarty chooses Dover 60%, Holmes Canterbury 60%, value 40.
      • Limits of the Simple Criteria
        • Call-off game: a 3×3 Matching Pennies variant adds “calling off” with extra payoffs α, β, γ.
        • Diagonal criterion fails: with three alternatives, diagonals no longer exhaust the matrix.
        • Asymmetric mixed case: a non-strictly-determined game may have a pure good strategy for one player and several for the other.
        • General theory irreplaceable: 2×2 short-cuts cannot replace the linearity–convexity analysis and the minimax theorem.
    • Strict Determinateness, Chance, and Bluffing (Chapter IV Zero-Sum Two-Person Games: Examples · II)
      • The Limits of Majorization
        • Majorized row: if one row is no better than a mixed strategy of other rows, player 1 can discard it.
        • Majorizing column: if a column is no worse than a mixed strategy of other columns, player 2 can discard it.
        • Dominance is not required: Figure 34 has a saddle point, yet no row or column satisfies such an average dominance.
        • Narrowing choices: dominance gives an immediate, plausible simplification of a player's strategic options.
      • Chance Moves vs Imperfect Information
        • Key equivalence: any chance move can be replaced by two personal moves with imperfect information, leaving strategic possibilities unchanged.
        • Replacement procedure: each chance move becomes one personal move for each player, both made without knowing the other's choice.
        • Matrix design: the payoff matrix arranges that every row and every column contains each original chance outcome equally often.
        • Strategic equivalence: uniform random choices by the players reproduce the chance move's probabilities, so values stay the same.
        • Real source of probability: mixed strategies come from imperfect information and non-strict determinateness, not from the presence of chance moves.
      • Chance Made Personal: Examples
        • Matching Pennies: a 50–50 chance device is reproduced by two personal moves, yielding the Matching Pennies game.
        • Stone, Paper, Scissors: a three-way chance device becomes the classic three-option game with incomplete information.
        • Cutting the deck: mixing and cutting cards achieves a uniform chance move through two personal moves and ignorance of the deck's order.
        • General condition: any replacement works if each row and each column of the matrix contains every alternative equally often.
      • Poker as an Extensive-Form Game
        • Poker's role: a multi-move game in which passing from extensive to normalized form and to strategies is non-vacuous.
        • Simplified form: two players, stud deal, no draw, hands ordered linearly from weakest to strongest.
        • Dealing as chance: each player draws a hand s with probability 1/S, all hands being equally likely.
        • Restricted bids: bids are only high (a) or low (b), with a > b > 0; the ratio a/b sets the game's risk.
        • Symmetric opening: both players make initial bids without knowing the other's choice, then learn both bids before passing or seeing.
      • Bluffing and Inverted Signaling
        • Posterior inference: a high bid suggests a strong hand, giving the opponent a motive to pass.
        • Motive one: fake strength on a weak hand to win by inducing a pass; success comes when the bluff is not seen.
        • Motive two: fake weakness on a strong hand by bidding low, keeping uncertainty alive; success may come when the bluff is seen.
        • Inverted signaling: both bluffing motives seek to mislead the opponent about the bid–strength connection.
        • Countermeasures: making bluffing costlier restricts its use, but such counter-moves are also indirectly motivated.
        • Exact treatment: the theory can separate these entangled motives and connect bluffing to strategy and initiative.
    • Poker, Bluffing, and Optimal Strategies (Chapter IV Zero-Sum Two-Person Games: Examples · III)
      • Rules and the Strategic Form
        • Chance move: each player draws a hand (s=1,\dots,S) with probability (1/S); hands rank by numerical order.
        • Bid choice: each player, knowing only his own hand, bids high (a) or low (b), then observes the opponent’s bid.
        • Pass or see: if bids differ, the low bidder can pass, paying the low amount, or see, converting to a high-bid comparison.
        • Payoff matrix: hand strength decides payment only when both bids are high or the low bidder sees; no payment occurs on equal hands.
        • Pure strategy: an index (i_s=1,2,3) per hand — high, low-then-see, or low-then-pass — giving (3^S) pure strategies.
      • Mixed Strategies and the Payoff Kernel
        • Mixed strategy: a probability vector over the (3^S) pure strategies defines each player’s randomized choice.
        • Reduced probabilities: (p_{s,i}, q_{s,j}) describe the chance of choosing bid-type (i) with hand (s); the expectation depends only on them.
        • Simplification: full strategies need (3^S-1) constants, but the reduced vectors are (2S) — a vast compression.
        • Symmetric game: a good strategy is characterized by being optimal against itself.
      • The Continuous Model
        • Granular hands: map hand (s) to (z=(s-1)/(S-1)), filling ([0,1]); the continuous limit lets (z) be uniform.
        • Integral equations: sums become integrals and reduced probabilities become functions (p_i(z), q_j(z)).
        • Limit of validity: the continuous solution captures the main strategic features, while discrete “fine structure” is left aside.
      • The Unique Solution
        • No low-then-see: a good strategy never chooses “low with seeing”; the optimality condition eliminates (j=2).
        • Density argument: between any two intermediate hands further intermediate points are dense, forcing constant behavior between thresholds.
        • Threshold structure: threshold points split the hand interval into regions of qualitatively different play.
        • Uniqueness and value: exactly one good strategy exists; the game’s symmetry gives the value (K=0).
      • Bluffing and Optimality
        • Bluff identified: sporadic high bids on weak hands, with probability shrinking as high bids cost more, are bluffs.
        • Defensive role: correct bluffing prevents the opponent from profitably deviating from the good strategy.
        • Mistakes matter: incorrect bluffing costs nothing against a good-strategy opponent, but invites profitable counter-deviations.
        • Immediate exposure: any deviation beyond bluffing errors is exploited immediately by a good-strategy opponent.
        • No permanent optimality: the unique good strategy is not permanently optimal; no strategy is safe against all opponent errors.
    • Generalizing Poker: Bluffing, Bids, Initiative (Chapter IV Zero-Sum Two-Person Games: Examples · IV)
      • Punishing Opponent's Bluffing
        • Punish extra bluffing at z₀ by imitating it on stronger hands, doing the opposite on weaker ones.
        • Correct bluffing protects the game value against opponent bluffing that is too frequent or too rare.
        • Unfinished reflections: the authors note these reflections could extend further, but choose not to pursue them.
      • More General Forms of Poker
        • Original solution assumes three simplifications: continuous hands, two bid levels, simultaneous bidding.
        • General real Poker, removing all three at once, is unsolved; each extension is studied separately.
        • Complete solutions exist for discrete hands and many bids; alternating bids has only limited progress.
      • Discrete Hands: Fine Structure
        • Discrete strategy for S-level hands again splits into a bluffing zone and a high-bid zone.
        • Fine structure: in the bluffing zone exact bid probabilities oscillate with arithmetical peculiarities of S, even as S→∞.
        • Averaging: averages over nearly equal hands match the continuous game; individual hand choices may differ widely.
        • Average strategy: playing the continuous-style average strategy is not good, but its maximal loss tends to zero as S→∞.
        • Cautious continuity: strategic fine detail need not vary continuously, though practical stakes may be tiny.
      • Multiple Bids: Density Without Convergence
        • m-bid solution: strong hands always take the highest bid; weak hands mix irregularly among bids with probabilities set by z.
        • Zone structure: bluffing zone becomes more varied and complicated than in the two-bid case.
        • No unique limit: as m→∞ with bids densely filling the interval, different interpolation rules give different good strategies.
        • Continuous-bid game—all bids between a and b allowed—has many good strategies, not a unique approximation limit.
        • Finite approximations fail: nearby finite-bid games need not converge to a common good strategy as the bid grid refines.
      • Alternating Bids: Sequential Structure
        • Sequential rule: player 1 bids first; if high, player 2 sees it and chooses high or low; if low, play ends.
        • Pure strategies: each player’s strategy is a high/low choice for each hand; there are 2^S strategies.
        • Mixed reduction: mixed-strategy space collapses from 2^S−1 to S parameters; only per-hand bid probabilities matter.
        • Continuous analogue: the same probability-vector method carries over with integrals, preserving the solution criteria.
        • Complete classification: player 1 has a unique good strategy; player 2 has an extensive family of good strategies.
      • Interpretation: Initiative and Bluffing Varieties
        • Three zones: three behavioral regions replace the original two zones of simultaneous Poker.
        • Player 1’s aggression: bids high on weakest hands and low on medium hands—aggressive bluffing in pure form.
        • Player 2’s defense: irregularly “sees” on medium hands to deter bluffing; on weakest hands he always bids low.
        • Initiative advantage: player 1’s positive expectation plausibly comes from possessing the initiative.
        • Two bluffing varieties: aggression by the initiating player, defensive “seeing” by the last bidder; original simultaneous poker mixed both.
        • Real Poker: longer alternating-bid sequences fits the same framework, but remains mathematically unexplored.
    • Mixed Strategies and Bluffing in Poker (Chapter IV Zero-Sum Two-Person Games: Examples · V)
      • Dominance and the Role of the Opponent
        • Dominance: in some cases one column or row majorizes another, settling the choice directly.
        • Exceptional condition: normally the relative merits of two choices depend on what the opponent does.
        • Case (B2): (18:10) excludes all four possibilities, so no simple dominance solution applies.
      • The Holmes–Moriarty Paradigm
        • Conan Doyle's story: classical pursuit example, with mixed strategies misrepresented as pure choices.
        • Pure-strategy narrative: the "best possible under limitations" is to pick the more probable route: Holmes leaves at the intermediate station.
        • True odds: the value favors Moriarty; on leaving Victoria, Holmes is "as good as 48% dead".
        • Morgenstern's doubt: if fully appraised beforehand, the whole trip might be unnecessary—loser determined before the start.
      • Chance Moves Reduced to Personal Moves
        • Normalized form: every game is equivalent to one without chance moves, only personal moves.
        • Detailed elimination: replace each chance move by two personal moves, preserving the move's structural role.
        • Rational probabilities: arbitrary rational odds can be reproduced with equal-probability alternatives, so no generality is lost.
      • Poker: Model and Strategies
        • Poker as zero-sum game: deals are chance moves, bids are personal moves, final "Passing" or "Seeing" closes the play.
        • Simplifications: ignore ante, deck depletion, and minor complications; reduce hands and bids to discrete levels.
        • Symmetry: in the symmetric variant, the payoff matrix is skew-symmetric.
        • Strategy redundancy: different mixed strategies can coincide behaviorally, e.g. 50–50 "Bold"/"Cautious" equals 50–50 "Normal"/"Bluff".
        • Static strategy: repeated-play and observational language is a verbal convenience; the analysis is for one isolated play.
        • Continuous limit: with billions of possible hands, replace discrete probabilities by continuous distributions.
      • Bluffing and the Value of the Game
        • Mixed strategies required: no pure strategy is good; optimal play randomizes over several bids.
        • Good strategy: bid "high" on very strong and very weak hands, "low" with a view to "Seeing" on middling hands.
        • Bluffing: high bids on weak hands are deliberate, with probabilities inversely proportional to the bid's cost.
        • Strict determinacy: Borel's simplified variant is strictly determined; its pure max-min equals the mixed value, obscuring bluffing.
        • von Neumann's variants: the symmetric and unsymmetric treatments bring out bluffing explicitly; overbidding rules change the solution.
        • Value: the game's value and optimal strategies emerge only from the full mixed analysis.
  • Chapter V Zero-Sum Three-Person Games
    • Coalitions, Compensations, and Symmetry (Chapter V Zero-Sum Three-Person Games · I)
      • From Pure Opposition to Alliance Choice
        • Transition: the three-person zero-sum game removes the pure opposition of interest of two-person play.
        • Partial parallelism: one player’s gain may hurt both rivals, or help one while hurting the other.
        • Choosing allies: a player can decide with whom to align and possibly how far the alignment goes.
        • Agreements arise: parallel interests make cooperation and pre-play understandings meaningful; two-person games required none.
      • Coalitions as the Decisive New Element
        • Core question: coalitions—with whom, and against whom—are the new strategic subject.
        • Only two candidates: each player can seek a coalition with either of the two other players.
        • Forced vs. chosen: if only one coalition is possible, it is closer to strategy than to cooperation.
        • Pure isolation: the chapter constructs games where coalitions are the only thing that matters.
      • The Simple Majority Game
        • Game rules: each player secretly names one of the other two; two mutual choices form a couple.
        • Payoffs: a couple’s members win ½ unit each, the excluded player loses 1, no couple yields zero.
        • Secret moves: collaboration cannot arise during play; agreements must be concluded outside the game.
        • Enforcement gap: the rules need not sanctify agreements, yet without them rational conduct is undefined.
        • Bridge analogy: coalition understandings resemble Bridge conventions, but bind two separate players.
        • Fair rules, unfair play: the game is symmetric, but any play is unsymmetric—one couple necessarily excludes one player.
      • Unequal Splits and Compensations
        • Privileged split: in couple 1,2, rules give player 1 an extra ε; other couples split evenly.
        • Illusory advantage: insisting on ε lets couple 2,3 alone form, so player 1 is excluded and loses 1.
        • Compensation requirement: player 1 must return the full ε to make couple 1,2 competitive for player 2.
        • Bidding for partner: if two players are favored against the third, they bid for the third; equilibrium restores the split.
        • General principle: what a coalition yields depends on competing alternatives, so compensations are the norm.
      • General Coalition Values and Feasible Claims
        • General model: coalitions 1,2, 1,3, and 2,3 can extract at most c, b, and a from the outsider.
        • Justified ceiling: player 1 cannot rationally claim more than α=(b+c−a)/2; analogous bounds hold for 2,3.
        • Coalition consistency: partners’ justified claims exactly exhaust their coalition’s total: α+β=c, β+γ=a, γ+α=b.
        • Excluded player: only coalition members collect their justified claims; the outsider gets −a, −b, or −c.
        • Rationality condition: coalitions are worth entering only if claims beat isolation, i.e. Δ=a+b+c≥0.
        • Solo floor: each player can secure the negative of what the other two could extract alone, making Δ≥0 necessary.
    • Chapter V Zero-Sum Three-Person Games · II
      • From Special Case to General Case
        • General reduction: any two-player coalition can be treated as a composite player, making Γ a zero-sum two-person game.
        • Coalition values: a, b, c denote what coalitions {2,3}, {1,3}, {1,2} can enforce against the excluded player.
        • Computational only: a, b, c are defined by the minimax theorem, not by assuming any coalition actually forms.
        • Forward path: the three-person solution directs n-person theory to coalitions, their competition, and compensations.
        • Minimum scale: coalitions first matter when three players exist; a two-person coalition requires a third to oppose.
      • Inessential and Essential Games
        • Δ = 0: no coalition has a raison d’être; each player can secure alone what any coalition would yield.
        • Inessential games: unique basic valuations a′, b′, c′ exist, with a′ = α = −a, etc.
        • Δ > 0: every player’s inducement to join a coalition is the same Δ/2; α > a′, β > b′, γ > c′.
        • Essential games: coalition members gain premium Δ/6; the excluded player loses Δ/3; coalition formation is unavoidable.
        • Central distinction: inessential versus essential extends beyond three-person games as a fundamental classification.
        • Symmetry trap: formally fair rules do not ensure fair play; coalitions drive players into unsymmetric arrangements.
      • Complete Formulae
        • Normalized form: coalition {1,2} controls τ₁,τ₂; player 3 controls τ₃; payoff is a bilinear form.
        • Mixed strategies: coalition {1,2} chooses a joint distribution over τ₁,τ₂; player 3 over τ₃.
        • Cyclical symmetry: b and a are obtained by cyclic permutations of players 1,2,3.
        • Derived parameters: α, β, γ, a′, b′, c′, and Δ are all functions of a, b, c.
      • Objection: Perfect Information
        • Tempting alternative: backward induction for perfect-information games seems to work for any number of players, without coalitions.
        • Stakes: even a special-case objection would not supply a universal alternative, but it would undermine the claim of general validity.
        • Objection fails: perfect-information games can be essential; sequential majority play preserves a, b, c.
        • Burden of proof: claiming general validity for any theory obliges answering even special-case objections.
      • Backward Induction’s Fatal Gap
        • Last-move ambiguity: player 2’s maximizing move may be non-unique, and choices can matter differently to player 1.
        • Two-person special case: with zero-sum two-person play, maximizing one player’s payoff automatically fixes the other’s.
        • Three-person break: player 2’s payoff no longer determines player 1’s, so “indifferent” choices become contested.
        • Compensations: players can pay others to choose favorable alternatives, even at a loss to the chooser.
        • Coalitions return: comparing gains and losses across players leads back to the coalition and compensation analysis.
  • Chapter VI Formulation of the General Theory: Zero-Sum n-Person Games
    • Coalition Values and Strategic Equivalence (Chapter VI Formulation of the General Theory: Zero-Sum n-Person Games · I)
      • Defining the Characteristic Function
        • Coalition value: A subset S playing against complement –S forms a zero-sum two-person game with definite value v(S).
        • Characteristic function: v(S) is a numerical set function on all subsets of players, the proposed foundation for the whole theory.
        • Composite strategies: Coalition S controls joint strategy τ_S; complement controls τ_–S; the value comes from the resulting bilinear form.
        • Program: v(S) alone must determine coalitions, compensations, mergers, and fights, despite ignoring how proceeds split within S.
        • Surmise to prove: The three-person analysis suggests v(S) determines even the imputation; proving this for all n is the theory's main objective.
      • Fundamental Properties
        • Zero and complement: v(∅)=0 and v(–S)=–v(S), hence v(I)=0; proved conceptually from the two-person value.
        • Superadditivity: For disjoint S and T, v(S∪T) ≥ v(S)+v(T); merging coalitions cannot reduce their joint guarantee.
        • Decomposition bound: For pairwise disjoint coalitions, sum v(S_i) ≤ v(S_1∪...∪S_p); partitioning I gives sum ≤ 0.
        • Equivalent axioms: The three properties are equivalent to one decomposition inequality for p=3 plus equalities for p=1,2.
      • Constructing Games from Characteristic Functions
        • Converse theorem: Any set function satisfying the three characteristic properties is the characteristic function of some zero-sum n-person game.
        • Rings: Players choose subsets containing themselves; a set all of whose members chose exactly it forms a ring.
        • Partition and payoffs: Rings and solo sets decompose I; each member of a ring C_q receives an equal share of v0(C_q).
        • Guarantee: A coalition can force its set to be a ring and secure at least v0(S); complementarity turns this into exact equality.
      • Strategic Equivalence and Reduced Form
        • Fixed side payments: Adding constants α_k to players' payoffs, with sum α_k=0, leaves the zero-sum property and all strategic possibilities intact.
        • Equivalence formula: v′(S)=v(S)+Σ_{k∈S} α_k identifies strategically equivalent families of games.
        • Reduced form: Requiring all one-player coalitions to have value –γ gives a unique reduced representative for each equivalence class.
        • Reduced bounds: In reduced form γ≥0 and –pγ ≤ v̄(S) ≤ (n–p)γ for every p-element S.
      • Inessential versus Essential Games
        • Inessential games: γ=0 makes the reduced form identically zero; each player can secure a fixed amount alone and no coalition improves it.
        • Essential games: γ>0 means a lone player gets –γ while any n–1 player coalition gets γ, so coalition strategy becomes decisive.
        • Normalization: For essential games one may rescale units to set γ=1 without affecting strategic structure.
    • Essentiality, Symmetry, and Coalition Stability (Chapter VI Formulation of the General Theory: Zero-Sum n-Person Games · II)
      • Inessentiality and Additivity
        • Inessential criterion: Γ is inessential exactly when its reduced form has γ = 0; essential when γ > 0.
        • Additive decomposability: Inessential games have v(S) = v((k₁)) + ··· + v((k_p)) for every coalition S.
        • Strategic equivalence to zero: Such v(S) is strategically equivalent to the identically zero characteristic function.
        • Economic value: Additivity occurs only in the uninteresting case; essential games have non-additive value and generate complementarity effects.
        • Measure analogy: Additive v(S) resembles a measure on players with total measure zero; general v(S) generalizes measure.
      • The Shape of Essential Games
        • Range constraints: Essential γ > 0 restricts v̄(S) for each set size p to the interval −pγ ≤ v̄(S) ≤ (n − p)γ.
        • Minimum participants: Essential games are impossible for n = 1 or 2; necessarily n ≥ 3.
        • n = 3 uniqueness: For n = 3 the inequalities determine the characteristic function completely, giving one essential type.
        • n ≥ 4 freedom: For 2 ≤ p ≤ n−2, v̄(S) can vary through an interval of length nγ, so many distinct essential games exist.
        • Endpoints attainable: For every n ≥ 4, games exist in which p-element sets achieve both interval endpoints.
        • Real ramifications: The genuinely complex behavior of the theory begins only at n ≥ 4.
      • Operations on Characteristic Functions
        • Scalar multiplication: For t ≥ 0, t v(S) is again a characteristic function; for t > 0 it changes the utility unit.
        • Vector addition: v(S) + w(S) is a characteristic function; it superposes two games played independently by the same players.
        • Center of gravity: t v + (1 − t) w forms a weighted superposition of games; strategic effects can be very involved.
        • Strategic equivalence: Passing to a strategically equivalent game is superposing an inessential game, leaving strategic structure unaffected.
        • No strategic consequence: For t > 0, scalar multiplication is a pure change of utility unit, so it does not alter strategy.
      • Symmetry and Fairness
        • Player permutations: A permutation P turns Γ into Γ^P, with characteristic function v^P(SP) = v(S).
        • Invariance group: The permutations leaving Γ unchanged form a group G_Γ; its size measures the game's symmetry.
        • Size-only dependence: If G_Γ moves any subset to any other of equal size, then v(S) depends only on the size of S.
        • Fair game: An n-person game is fair when v(S) depends only on the size of S, extending the two-person notion.
        • Three-person fairness: All reduced zero-sum three-person games are fair; for n ≥ 4 reduced games need not be fair.
        • Unfairness beyond payments: For n ≥ 4, strategic differences between players can be fundamental and cannot be reduced to fixed side payments.
      • The Three-Person Solution Pattern
        • Reduced essential game: Normalize so γ = 1; the characteristic function is fixed: singles get −1 and pairs get +1.
        • Coalition outcomes: Three two-person coalitions can form, producing three payoff distributions; partners split 1/2 each, outsider gets −1.
        • Solution is a set: No single distribution solves the game; stability lies in the system of all three distributions.
        • Stable rejection: Any outside distribution is rejected because two players can be convinced to prefer one of the solution distributions.
        • Internal stability: In each solution distribution only one player is dissatisfied, and no partner can improve by changing allies.
        • Why outside offers fail: A prospective partner fears later vulnerability in further negotiations after accepting an offer outside the solution.
    • Solution, Saturation, and Domination Criteria (Chapter VI Formulation of the General Theory: Zero-Sum n-Person Games · III)
      • Exact Definitions and Solution
        • Imputation: a vector of n payoffs, one per player, satisfying the distribution conditions.
        • Effective set: coalition that can realize the payoffs it is offered by an imputation.
        • Domination: β is dominated by α if a nonempty effective S gives every i in S α_i > β_i.
        • Solution: imputations exactly undominated by the set; every non-member dominated by some member.
      • Standard of Behavior
        • Solution as standard: conditions express inner stability expected of accepted norms.
        • Unsound outside imputations: preferable payoffs fail to attract because players reject unsound distributions.
        • Circularity: soundness is defined by the accepted solution, familiar in everyday standards.
        • Multiplicity: several consistent, mutually conflicting solutions can exist for one game.
      • Saturation Formalism
        • Symmetrized relation: new relation s requires both x≺y and y≺x.
        • Satisfactory set: all internal pairs satisfy the relation; unaffected by symmetrization.
        • Saturated set: consists exactly of all elements compatible with it.
        • Symmetric result: saturated iff maximal satisfactory; every satisfactory subset extends to one.
        • Asymmetric barrier: the relevant relation is asymmetric, so maximal satisfactoriness is necessary but not sufficient for saturation.
      • Three Immediate Objectives
        • One-element solutions: determine for which games a single imputation can be a solution.
        • Three-person essential case: verify earlier solutions and find all additional solutions.
        • Strategic equivalence: prove rigorously that strategically equivalent games have identical solution structure.
        • Scope: these cover all games with n ≤ 3; new difficulties begin at n ≥ 4.
      • First Consequences: Domination Criteria
        • Preliminary conditions: for domination, S must be nonempty and effective; payoff superiority is the main condition.
        • Certainly necessary sets: criteria guarantee a candidate coalition meets the preliminary conditions.
        • Certainly unnecessary sets: criteria rule out a coalition for all relevant comparisons.
        • Auxiliary principles: convexity, flatness, and criteria (31:A)–(31:H) constrain which coalitions matter.
    • Solution Criteria and Discriminatory Standards (Chapter VI Formulation of the General Theory: Zero-Sum n-Person Games · IV)
      • Coalition Necessity and Flatness
        • Necessary coalition: must promise every member definitely more than solo payoff, otherwise ignore.
        • Redundant coalitions: drop S if a superset or same set with better payoff is already considered.
        • Convexity: total advantage from forming S; zero convexity means S is flat and negligible.
        • Complement rule: nonempty coalition is certainly necessary if its complement is flat.
        • Doubtful sizes: only coalitions of size 2 to n−2 ever need consideration; for n=3, only pairs matter.
      • Imputations and One-Element Solutions
        • Imputation set: inessential games have exactly one imputation; essential games have an (n−1)-dimensional continuum.
        • Solution never empty: every solution contains at least one imputation.
        • Undominated imputation: exists only in inessential games; essential games always have dominated alternatives.
        • One-element solution: exists iff the game is inessential; then no other solutions exist.
      • Strategic Equivalence as Isomorphism
        • Isomorphic correspondence: strategically equivalent games have imputations mapped by fixed side payments.
        • Invariants preserved: mapping preserves effectivity, domination, and solutions.
        • Reduced form justified: strategic-equivalence classes allow canonical normalization.
      • Three-Person Solutions: Graphical Classification
        • Graphical model: imputations fill a triangle in the plane with 60° axes.
        • Domination by pairs: in 3-person games only two-player coalitions matter; domination uses sextant regions.
        • Non-dominating pairs: line between two undominated imputations is parallel to a side of the fundamental triangle.
        • Three-point solution: middle points of triangle sides form the original desired solution, (32:B).
        • Line solutions: for each c with -1 ≤ c < 1/2, all imputations with one fixed component c form a solution.
      • Stability and Discriminatory Standards
        • Multiplicity accepted: rigorous theory yields infinitely many extra solutions beyond the intended one.
        • Discriminatory solution: two players fix the third's payoff, then split the remainder arbitrarily by bargaining.
        • Quantitative bounds: discriminated player's fixed payoff c satisfies -1 ≤ c < 1/2; c=-1 is full exploitation.
        • Stability imposes bounds: arbitrary discrimination can be stable only within quantitative limits.
        • Broader import: for n≥4 expect richer schemes of prejudice, privilege, and standards.
    • Coalition Values and Stable Standards (Chapter VI Formulation of the General Theory: Zero-Sum n-Person Games · V)
      • Characteristic Function and Coalitions
        • Coalition values: every subset of players is assigned a guaranteed worth v(S), including the empty set and the all-player set.
        • Zero-sum normalization: for zero-sum games, v(I)=0 and complementary coalitions satisfy v(S)+v(-S)=0.
        • Superadditivity: for disjoint S, T, v(S∪T) ≥ v(S)+v(T); merging coalitions never lowers guaranteed value.
        • Fictitious two-person game: v(S) is derived from treating S against -S as a two-person zero-sum game.
        • Rings: a coalition forms when every member chooses the same ring; rings are disjoint by definition.
        • Virtual actors: the empty set and I are formally treated as coalitions; they have no moves or gains but simplify set theory.
      • Strategic Equivalence and Inessential Games
        • Strategic equivalence: v and v′ represent the same game if v′ = kv plus additive individual constants, with k>0.
        • Reduction: normalizing by individual values simplifies analysis without changing strategic content.
        • Inessential games: v(S) equals the sum of the individual v({i}) in S; coalitions add nothing, so no domination occurs.
        • Value of a coalition: v(S) measures only concerted acts of behavior; goods themselves represent acts of exchange.
        • Never-dominated imputation: an imputation undominated by any other exists only in inessential games.
      • Symmetry and Permutation Groups
        • Permuting players: a permutation P sends strategies and payoff functions to a transformed game Γ^P.
        • Superscript order: applying P then Q must match multiplication PQ; correct placement of superscripts is essential.
        • Group-theoretic language: symmetry of games is expressed through permutation groups, though no advanced group theory is required.
        • Set-transitive groups: only special groups, like the alternating group for some n, preserve equal treatment of player sets.
      • Domination and Standards of Behavior
        • Domination: α is dominated by β if a coalition S can enforce β and all members prefer it.
        • Standards of behavior: a solution is a set of imputations that are mutually compatible and discredit all non-approved alternatives.
        • Satisfactoriness and saturation: two dual properties that characterize stable subsets of imputations.
        • Parameter count: solution sets are defined by n−1 equations rather than n, forming one-dimensional families.
        • Quasi-dynamics: negotiations and fears before coalition formation are quasi-dynamic but motivate the static solution concept.
      • Discriminatory Solutions and Geometry
        • Majority-game analogy: general n-person games generalize the three-person simple majority game; rings replace couples.
        • Discriminatory solutions: "inobjective" solutions are the bridge to general non-zero-sum games and economic application.
        • Geometric form: solutions appear as line segments and shaded areas in the fundamental triangle, with boundary cases excluded.
        • Variable spoils: the coalition's "spoils" can be positive or negative, yet v(S) still determines solutions.
        • Stability by indifference: in some bargaining moves one player gains, one loses, and the third blocks change by indifference.
  • Chapter VII Zero-Sum Four-Person Games
    • Cube Q and Four-Person Coalitions (Chapter VII Zero-Sum Four-Person Games · I)
      • General Viewpoints and Formalism
        • State of theory: general n-person theory exists, but only n ≤ 3 cases are solved; four-person games first expose full coalitional complexity.
        • Reduced normalization: with γ = 1, one-player coalitions have value –1, so only the six two-player coalitions remain variable.
        • Coordinate triplets: complementary two-player coalitions force three parameters x₁, x₂, x₃ with –1 ≤ xᵢ ≤ 1.
        • Cube Q representation: each essential zero-sum four-person game corresponds uniquely to a point in the cube Q, centered at the origin with edge length 2.
      • Permutations and the Symmetries of Q
        • Broken symmetry: choosing coordinates singles out player 4; only the six permutations of players 1,2,3 act as plain coordinate permutations.
        • Generating all 24: permutations A, B, C interchange player 4 with 1,2,3 and flip two coordinates each, leaving the third invariant.
        • Cube motions: only the 24 sign-change-even motions of Q correspond to player permutations; they preserve the tetrahedron of the winning-corner vertices.
      • The Corner I: A Privileged Player
        • Winning coalitions: (1,4), (2,4), (3,4), and (1,2,3) win; player 4 needs one ally, while the others must all three combine.
        • Winner payoffs: heuristic symmetry gives players 1,2,3 one-third each and player 4 five-thirds; defeated players all get –1.
        • New four-person trait: advantage over another player may differ depending on whether both win or both lose—impossible when n = 3.
        • Extreme advantage degenerates: if every coalition without player 4 loses, the game becomes inessential; the advantage reduces to a fixed payment, not strategy.
      • The Corner VIII: A Dummy Player
        • Additive dummy: adding player 4 to any coalition leaves its value unchanged, so he contributes nothing strategically.
        • Inflated three-person game: players 1,2,3 play an essential three-person game; winners get 1, losers get –1, and the dummy always gets –1.
        • Objective vs. prejudicial exclusion: the dummy is excluded by the characteristic function; a discriminatory solution excludes by a chosen standard of behavior.
      • Interior of Q: The Symmetric Game
        • Center game: at (0,0,0) the game is totally symmetric; any three-player coalition wins, while a two-versus-two tie yields no payments.
        • Bewildering variety: the center game has numerous solutions not yet systematically ordered; notable specimens are examined in later sections.
    • Main Diagonals and Solution Domains (Chapter VII Zero-Sum Four-Person Games · II)
      • Geometry of Q and Main Diagonals
        • Main diagonal (x_1=x_2=x_3): exactly the games symmetric in players 1–3, leaving only player 4 special.
        • Four main diagonals of Q: each encodes a game in which one player alone may hold a special role.
        • One-parameter family on the diagonal: (x_1\in[-1,1]) connects corner VIII, center, and corner I.
        • Known solutions: complete only at the eight corners; elsewhere known regions form a scattered net of lines and areas.
      • Heuristic Route from Corner VIII
        • Near (x_1=-1): a two-person coalition among players 1–3 remains the strongest aim, but coalitions involving player 4 turn slightly profitable.
        • Stable first coalition: once formed, it acts as a single player in a three-person game with the remaining two.
        • Predicted payoffs: successful first-coalition partners split (1/2); defeated first coalition leaves both with (x_1).
        • Guessed solution: collects outcome patterns for all possible first coalitions and final allies, with player-1–3 permutations.
      • Exact Verification of the Guess
        • Heuristic guesses must be justified: exact domination conditions from §30.1 decide whether a guessed set is a solution.
        • Criterion (36:A): the guessed set is a solution if and only if (-1\le x_1\le -1/2).
        • Proof is mechanical: it only classifies certain and unnecessary coalitions, raising no new point of principle.
        • Reader may skip the details: retaining criterion (36:A) suffices; exact proof anchors heuristic guessing to rigorous theory.
      • Extension across the Diagonal
        • Second interval: adding further imputations gives a solution for (-1/2\le x_1\le 0), adjoining the first at the border.
        • Negative half covered: two solution families exhaust the VIII–center half of the diagonal.
        • Positive half: three intervals from center to I show similar qualitative changes, with solutions switching at thresholds.
        • Behavior at thresholds: (x_1=0) and (3/5) admit both neighboring solution types continuously; (1/2) remains unresolved.
      • Center and Its Environs
        • Center is the unique fully symmetric game in Q; the characteristic function makes three-person coalitions the aim.
        • Strategic question at center: whether a preliminary two-person coalition admits the third player on equal or unequal terms.
        • Equal alternative: all three participants divide the total 1 equally, (\frac13) each, as at corner I.
        • Unequal alternative: the first two partners each receive more than (\frac13), as in the interval near VIII.
        • Three-dimensional part around center: the same qualitative solution type extends beyond the diagonal, but complete mapping of Q remains open.
    • Central Four-Person Solutions and Their Families (Chapter VII Zero-Sum Four-Person Games · III)
      • Two Alternatives and the Role of Symmetry
        • Flexibility: the first pair can recruit either outsider; a hostile two-person coalition only ties.
        • Two principles: the final trio may form around a two-person core or with all three members on equal terms.
        • Two-stage logic: the first coalition separates defeat from tie; the final coalition separates tie from victory.
        • Symmetry assumption: only fully symmetric solutions are examined; asymmetric ones are too recondite for now.
        • Small-number caveat: with four players, organizational principles may hold for all or none.
      • First Alternative: Stratified Coalition at the Center
        • Core-plus-ally payoff: the winning pair splits 1/2 each; the ally gets 0; the excluded player gets –1.
        • Tie imputation: if no ally is found, all four receive 0, stabilizing the winning imputations.
        • Permutation closure: all permutations of that payoff pattern, plus the tie, form a symmetric solution.
        • Deferred proof: exact verification is omitted because a later general proof covers it.
      • Second Alternative: Equal-Terms Coalition
        • Equal-trio payoff: each member of the winning trio gets 1/3, while the excluded player gets –1.
        • Incomplete set: these imputations do not dominate one another, yet leave other imputations undominated.
        • Tie is dominated: the all-zero imputation cannot stabilize them because it is itself dominated.
        • Unique extension: exactly one symmetric extension exists, adding compromise imputations rather than a tie.
      • Comparison and Unsymmetrical Central Solutions
        • Two normal solutions: both central solutions are finite, fully symmetric, and express distinct social principles.
        • Transparent extra: the first solution’s tie is obvious, yet the solution is surrounded by unexplained phenomena.
        • Asymmetric diagonal solution: near-center games yield a solution symmetric only in players 1,2,3, with player 4 special.
        • Privileged group: players 1,2,3 form an exclusive class; no outsider joins the first coalition.
        • Structural discrimination: unlike three-person discriminatory solutions, asymmetry comes from the solution, not the game.
      • A Family of Scaled Solutions
        • Scaled central solution: multiplying the first central solution’s imputations by z gives a solution for 2/3 < z ≤ 1.
        • Reversed viewpoint: fix an imputation set, then characterize the games for which it is a solution.
        • Exact conditions: with two-element coalitions worth 0 and v((k)) = –y_k, solutionhood holds iff 1 ≤ y_k < 3/2.
        • Neighborhood of the center: restoring normalization extends this family around the center of cube Q.
    • Normalized Solutions and Strategic Restraint (Chapter VII Zero-Sum Four-Person Games · IV)
      • Normalized Reduced Form
        • Normalization: divide the characteristic function and all shares by γ to reach the reduced form.
        • Two-player coalitions: for S = (i,j), v(S) is further adjusted to complete the reduction.
        • Coordinate space: normalized games form the three-dimensional region Q with coordinates x1, x2, x3.
      • Region Where the Solution Exists
        • Implicit definition: equations (38:10)–(38:11) exhaustively delimit the games for which the transformed solution (37:2) applies.
        • Explicit criterion: given x1, x2, x3, a z works iff the four u_k satisfy min > 2/3 of max.
        • Region Z: the points satisfying this condition form a region Z inside Q, with the center in its interior.
        • Central check: at the center all u_k equal 1, so the earlier symmetric formulas of 38.1.1 are reproduced.
      • The Transformed Solution Family
        • Multiplicity: every game in Z admits infinitely many solutions, each a finite set of imputations.
        • Symmetry: each solution carries the full symmetry of the game, with no player singled out.
        • Imputation values: the old values 1, 0, –1 become z/2 + u_k – 1, u_k – 1, and –z + u_k – 1.
        • Organizational principle: none of the earlier distinctions applies, yet a simple qualitative principle distinguishes these solutions.
      • Partial Exploitation of Defeated Players
        • Worst outcome: a defeated player's worst result is generally better than his solo value v((k)) = –1.
        • Restraint: victorious coalitions do not fully exploit defeated players, reducing them to the lowest possible level.
        • Limit case: full exploitation occurs only at maximum z, and usually only for the player whose u_k is minimal.
        • Equal treatment: in contrast to three-person mild discrimination, this restraint applies to all players; at the center the solution is fully symmetric.
        • Social insight: non-exploitation is a possible but not necessary feature of solution-defined social organizations.
      • Place in the General Theory
        • Precedent: the three-person mild discriminatory solutions already showed restraint, but only toward one excluded player.
        • Related families: some solutions in region C share this feature, yet are distinct from the present family.
        • Outlook: the phenomenon is likely to play a larger role in the general theory.
  • Chapter VIII Some Remarks Concerning n ≧ 5 Participants
    • Counting Parameters for Essential Games
      • Essential games: reduced form with γ = 1; the zero-sum three-person game is unique, the four-person games form a 3-dimensional manifold.
      • General formula: essential zero-sum n-person games need 2^(n−1) − n − 1 parameters; already 10 for n = 5.
      • Symmetric formula: free parameters are (n−3)/2 for odd n and n/2−2 for even n; only 1 at n = 5, 2 at n = 7, 8.
      • Combinatorics: complement symmetry halves the 2^n coalition values; normalization fixes v(∅) and the n single-player values.
      • Low n: no essential zero-sum games for n = 1, 2; the first symmetric variety appears at n = 5.
      • Complexity: the parameter explosion shows how sharply game complexity grows with participants.
    • The Symmetric Five-Person Game
      • Direct attack: the general zero-sum five-person game has 10 parameters; only the symmetric case is now feasible.
      • One-parameter family: v₀ = 0, v₁ = −1, v₄ = 1, v₅ = 0, with v₂ = −η and v₃ = η.
      • Domain: superadditivity restricts η to −1/2 ≤ η ≤ 2.
      • New phenomenon: n = 5 is the first case where symmetric games form a genuine one-parameter family.
    • Extreme Ends of the Family
      • η = 2: every pair is defeated, every triple wins; the transition from defeat to victory occurs between size 2 and 3.
      • η = −1/2: pairs and quartets win; the transition occurs between size 1 and 2, making pairs the effective objective.
      • Extreme solutions: heuristic guessing and exact proof are easy at both endpoints.
      • General construction: the extremes illustrate a broader way to define games by declaring every p-player coalition winning.
      • Arithmetic restriction: no symmetric game makes all p-player coalitions winning if p divides n, because n/p coalitions leave no loser.
    • From Five Players to Four: The Merge
      • Composite player: merging players 4 and 5 into one player 4′ turns Γ into a 1,2,3-symmetric four-person game Γ′.
      • Forced merger: Γ′ changes the rules; nothing in Γ compels players 4 and 5 to act together.
      • Parameter map: normalization gives (3 − x₁)(3 + η) = 10, a monotone hyperbola from η in −1/2..2 onto x₁ in −1..1.
      • Zone transfer: four-person classes A–E map to candidate zones for the five-person game; detailed analysis replaces some with new zones.
      • Strategic comparison: matching solutions reveals the effect of forced unity — divergence means possible separation dislocates the joint bargaining position of players 4 and 5.
  • Chapter IX Composition and Decomposition of Games
    • Game Decomposition and Constant-Sum Extension (Chapter IX Composition and Decomposition of Games · I)
      • Why Special Classes Are Needed
        • Complexity explosion: exhaustive casuistic treatment hopeless beyond five participants.
        • Qualitative novelty: every new n brings new phenomena; crucial ones first appear at n=6.
        • Pacemaker strategy: fully analyze manageable special cases embodying essential principles.
        • Two prototypes: families generalize the two strategic types among cube Q's corners; corner VIII anchors this chapter.
      • Composition of Games
        • Composition: Γ from constituents Δ, H with disjoint player sets J, K; no interaction.
        • Additive characteristic function: value of a coalition in Γ equals sum of values in Δ and H.
        • Solution correlations: separate rules do not prevent stable standards from correlating the two societies.
        • Scientific analogy: combining noninteracting mechanical or economic systems is a standard preparatory step.
      • Exact Decomposability Criteria
        • Constituent formulas: recover parts from the whole via vΔ(S)=vΓ(S), vH(T)=vΓ(T).
        • Core additivity: vΓ(S∪T)=vΓ(S)+vΓ(T) for S⊆J, T⊆K; no cross-boundary attraction.
        • Measurability analogy: this condition is exactly Carathéodory's definition of measurability.
        • Full criterion: decomposability iff additivity plus vΓ(J)+vΓ(K)=vΓ(I).
      • From Zero-Sum to Constant-Sum
        • Lacuna: recovered constituents may violate zero-sum normalization; corner VIII shows v(J) not zero.
        • Invariance problem: decomposability should be invariant under strategic equivalence.
        • Widening: discard (27:1), keep transformation (27:2), passing to constant-sum games.
        • Constant-sum equivalence: every constant-sum game is strategically equivalent to a zero-sum game.
        • Old games unaffected: for zero-sum games, the new equivalence coincides with the old one.
      • Characteristic Functions in the New Theory
        • Equivalent definition: v′(S) is same whether recomputed by 25.1.3 or transformed from a zero-sum game.
        • Conditions preserved: (42:6:a)–(42:6:c) characterize constant-sum characteristic functions.
        • v(I) vanishes: the zero-sum normalization condition is obliterated; v(I) need not be zero.
        • Deferred revision: full abandonment of the zero-sum condition postponed to Chapter XI.
    • Constant-Sum Decomposition and Its Partition (Chapter IX Composition and Decomposition of Games · II)
      • From Zero-Sum to Constant-Sum
        • Imputations: distribute the fixed sum s of the game, not necessarily zero.
        • Strategic equivalence: maps imputations one-to-one between equivalent constant-sum games, preserving domination and solutions.
        • Inessentiality: defined via strategic equivalence to the game with v(S) ≡ 0; criterion uses v(I) instead of 0.
        • Constant-sum domain: every constant-sum game is strategically equivalent to a zero-sum game, so the old theory carries over.
      • Decomposability in the Wider Domain
        • Decomposability condition: in constant-sum games, condition (41:6) alone suffices; the extra zero-sum restrictions disappear.
        • Splitting set: a self-contained group of players who neither influence nor are influenced by the others.
        • Complement symmetry: J is splitting iff its complement is; the empty set and I are always splitting.
        • Closure properties: unions and intersections of splitting sets are again splitting sets.
      • The Decomposition Partition
        • Minimal splitting sets: nonempty splitting sets with no smaller nonempty splitting subset; they are pairwise disjoint and cover all players.
        • Decomposition partition: the system of all minimal splitting sets; every splitting set is a union of its elements.
        • Indecomposable constituents: each partition element corresponds to a constituent game that is indecomposable.
        • Characterization: a set is splitting exactly when each partition element lies wholly inside or wholly outside it.
      • Extreme Cases and Player Count
        • Fine partition: one-element splitting sets for every player iff the game is inessential.
        • Coarse partition: the trivial partition {I} iff the game is indecomposable.
        • Player count: n=1 games are both indecomposable and inessential; n=2 games are always inessential/decomposable; n≥3 decomposability is exceptional.
        • Dummies: one-element partition classes are self-contained players; the remaining classes have at least three players.
        • Social principle: non-dummy players are grouped into indecomposable constituents of at least three — a general principle of social organization.
      • Solutions of Decomposable Games
        • Composition of imputations: any imputation for J and any for K compose to an imputation for the whole game.
        • Decomposable imputation: decomposition is possible only when each constituent group receives exactly its "just dues" (zero).
        • Composed solution set: composing a solution for each constituent yields a set of imputations expressing a standard of behavior.
        • Expected result: this composed set should be a solution of the whole game, with each constituent's internal behavior governed by its own solution.
    • Composition, Solutions, and the Excess (Chapter IX Composition and Decomposition of Games · III)
      • Common Sense vs. Mathematical Proof
        • Common sense and rigour: intuitive claims are invalid without proof; failure of proof may demand theory change.
        • (44:C) Composition of solutions: composing solutions J, K of constituents Δ, H yields a solution I of Γ.
        • (44:D) Converse fails: a decomposable game Γ need not have only decomposable solutions.
        • Split failure: (44:D:b) holds—constituents of a decomposable solution are solutions; (44:D:a) fails—not every solution decomposes.
        • Source of trouble: indecomposable solution violates the primary decomposability condition (44:B:a).
        • Remedy: relax decomposability so condition (44:B:a) can be discarded and the theorem restored.
      • Extended Imputations and Outside Sources
        • Extended imputations: vectors with only individual lower bounds, no requirement that components sum to game value.
        • Outside sources: proposals can involve transfers—contributions or withdrawals—not just suggestions.
        • Non-isolated games: the composite game and its constituents are viewed as coexisting with outside transfers.
        • Deferred construction: including the outside source as a player in a larger game Γ′ is postponed until final results.
        • Restored decomposability: with extended imputations, composition and decomposition can always be carried out.
      • The Excess
        • Excess defined: transfer's size equals the extended imputation's total minus v(I), the game's assigned value.
        • Negative excess: split into hostile coalitions can make players accept less than v(I), stabilizing inferior totals.
        • Positive excess: free gift must be distributed; coalitional claims adjust to the available total.
        • Too large a gift: excess dissolves organizational mechanisms because no coalition can exhaust it.
        • Lower bound |Γ|₁: for e < |Γ|₁, E(e0) and F(e0) are empty and the empty set is the only solution.
      • Setups E(e0) and F(e0)
        • Set E(e0): extended imputations whose excess is exactly prescribed as e0.
        • Set F(e0): extended imputations whose excess is only bounded above by e0; technically useful auxiliary case.
        • Solution definition: taken from 30.1.1: the set of all undominated imputations within E(e0) or F(e0).
        • E(0): the old zero-sum and constant-sum theories reappear as the special case e0 = 0.
        • Invariance: excess conditions and domain definitions are invariant under the isomorphism of games.
      • Detachment and Domination
        • Detached imputation: no non-empty coalition is effective for it; every coalition secures at least its coalitional value.
        • Fully detached: strict form of detachment; detached allows the limiting equality cases.
        • Threshold |Γ|₂: detached imputations exist for excess e ≥ e*; fully detached for e > e*.
        • Fully detached dominates none: no effective coalition exists, so it cannot dominate any other imputation.
        • Detached iff undominated: an extended imputation is detached exactly when no other imputation dominates it.
        • Antithetic roles: the strict and limiting notions stand in opposite relations to domination but complement each other.
    • Excess Bounds and Decomposable Solutions (Chapter IX Composition and Decomposition of Games · IV)
      • Quantifying Essentiality
        • |Γ|₁ and |Γ|₂: two quantitative measures of a game's essentiality, expressed through excess bounds.
        • Bounding inequality: |Γ|₁ and |Γ|₂ are tied by inequalities that tighten as n changes.
        • n = 1, 2: games are inessential, so both measures vanish despite apparent contradictions.
        • n = 3: the two measures coincide, merging the inequalities into an equation.
        • n ≥ 4: the lower bound is attained; the upper bound remains undetermined.
        • Qualitative reading: the two measures capture different, partly independent aspects of essentiality.
      • Detached Imputations and the E/F Distinction
        • Detachment mandate: every solution for E(e₀) or F(e₀) must contain every detached extended imputation in that set.
        • Gift semantics: E(e₀) fixes the outside gift at exactly e₀; F(e₀) only caps it at e₀.
        • D*(e₀): detached imputations lying in F(e₀) but not E(e₀); they are exactly what separates the two families.
        • Critical threshold: e₀ > |Γ|₂ makes D*(e₀) nonempty, forcing F- and E-solutions to differ.
        • “Too large” excess: an outside contribution beyond |Γ|₂ disorganizes normal stable behavior.
        • Only difference: the E- and F-solutions differ solely by the presence of D*(e₀).
      • Proof of the E–F Equivalence
        • Base lemma (45:J): a general inferiority result that works despite the intransitivity of domination.
        • Construction lemma (45:K): turns a detached imputation in F but not E into one in E for the solution.
        • Unique intersection (45:L): every F-solution restricts to a unique E-solution; the remainder is exactly D*(e₀).
        • Transfer lemmas (45:M–N): restriction and extension preserve the solution property in both directions.
        • Main equivalence (45:1): combining the lemmas yields a one-to-one correspondence between F- and E-solutions.
      • Decomposition: Additivity and Local Domination
        • Additive measures: for a decomposed game, |Γ|₁ and |Γ|₂ are sums of the constituent games' measures.
        • Decomposition of detachment: an imputation is detached in Γ exactly when both constituents are detached in Δ and H.
        • Excess decomposition: the excess of an imputation is the sum of the excesses of its J- and K-constituents.
        • Local domination (46:B): a dominating set in a decomposable game can always be chosen inside J or inside K.
      • Solutions of Decomposable Games
        • Projection sets: a solution I for F(e₀) determines constituent sets U_J and U_K of imputations in Δ and H.
        • Absorption condition: an imputation belongs to I when its components lie in U_J and U_K and its total excess is ≤ e₀.
        • Constituent solutions: U_J and U_K are solutions of their games with suitable excess limits f₀ and g₀.
        • Composition theorem: arbitrary solutions of Δ and H, tied by matching excess ranges, build a solution of Γ.
        • Only true restraint: the constituent solutions are coupled solely through their excess ranges relative to e₀.
    • Composition, Normal Zone, and Imbedding (Chapter IX Composition and Decomposition of Games · V)
      • Complete Result in F(e0)
        • Implicit conditions (46:16)–(46:17) are replaced by a transparent two-case analysis.
        • Case (a): both sides of the key equation vanish, giving explicit inequalities and equalities.
        • Case (b): nonzero sides impose stricter equations that fully describe the solution.
        • Summary (46:H): the original conditions reduce to exactly one of these two explicit cases.
      • Complete Result in E(e0)
        • Cases (a) and (b) align with (45:O:b) and (45:O:c), linking the F- and E-theories.
        • Too-small excess: the empty set is the only solution for both E(e0) and F(e0).
        • Normal excess (46:I:b): E and F solutions of Γ coincide; choose constituent excesses in bounds, then compose any E-solutions of Δ and H.
        • Too-large excess: solutions come only from E-solutions of constituents, with excesses fixed by equations.
      • Graphical Representation and Normal Zone
        • Figure 69 maps excess pairs to e0 values: rectangle for normal case, line for excessive case.
        • Normal zone lies between -|Γ|₁ and |Γ|₂; constituent excesses remain in their normal zones.
        • Heredity holds for normal position, not for vanishing excess: e0 = 0 need not imply constituent excesses zero.
        • At zone endpoints the possible excess interval constricts to a point, making the transition continuous.
        • Width of theory: even original e0 = 0 solutions force constituent games to be treated with nonzero excesses.
      • Dummies and Inessential Compositions
        • Adding dummies: compose with an inessential game; its unique solution components are merely appended.
        • Heredity criterion (46:J): e0 = 0 implies zero constituent excesses iff Δ or H is inessential.
        • Old theory stays hereditary only for dummy-addition imbeddings, not general composition.
      • Imbedding of a Game
        • Imbedding: Γ is an imbedding of Δ if Γ is the composition of Δ with another game H.
        • Non-isolated analysis: enumerate J-constituents of solutions of all imbeddings of Δ.
        • Main result (46:L): these constituents are precisely Δ's normal-zone solutions, in both old and new theory.
        • Restoring old heredity requires Δ inessential or the added game inessential—just dummies.
        • Outside source cannot be one player; an essential three-person added game suffices.
        • Methodological payoff: imbedding without modification justifies the new theory, like closed systems in mechanics.
      • Significance of the Normal Zone
        • Stable order in the composite game requires a fixed transfer/tribute from one constituent group to the other.
        • Internal standards remain stable once that transfer is accepted as beyond dispute.
        • Possible transfers lie within the bounds (46:35); zero transfer is always possible.
        • Zero-only transfer occurs exactly when one of the two constituent games is inessential.
    • Tributes, Transfer, and Solution Curves (Chapter IX Composition and Decomposition of Games · VI)
      • Tributary Limits of Composite Games
        • Tribute direction: in mixed essential games, either constituent may be the tribute-paying group.
        • Physical bounds: each group’s tribute is confined between limits set by both games.
        • Minima and maxima: –|Δ|₁ is the worst endured; |Δ|₂ is the maximum externally accepted claim.
        • Vanishing together: the two bounds vanish only when the game is inessential.
      • Transfer Appears at Six Players
        • Transfer condition: the new element emerges as a nonzero tribute when both constituents are essential.
        • Minimal size: each essential constituent needs three players, so six is the smallest case.
        • Construction: two essential three-person games with γ=1 allow tribute between their bounds.
        • Novelty threshold: six players, like two, three, and four, introduce a qualitatively new phenomenon.
      • Need and Graphical Setup for the Three-Person Game
        • Motivation: the essential three-person game is central to decomposition and earlier existence arguments.
        • Normalization: set γ=1; the fundamental triangle represents all imputations with excess e₀.
        • Quasi-components: transform imputations to zero-sum coordinates so the old graphical figure applies.
        • Domination rule: only two-player coalitions dominate; conditions split the triangle into effective regions.
      • Six Cases for Solutions
        • Cases I–II: too-negative excess gives no imputations; e₀ = –3 gives exactly one point.
        • Case III: for nonpositive excess, the old-theory solutions transfer with a proportionality factor.
        • Case IV: small positive excess adds arbitrary curves in the fringe around old-type solutions.
        • Case V: larger positive excess forces the undominated central triangle into the solution.
        • Case VI: high excess makes the whole fundamental triangle the solution.
        • Variety: freely chosen curves make the family of solutions vastly larger than before.
      • Interpreting Curves and Areas
        • Coalition claim: with positive excess, a coalition can obtain more than its effective maximum.
        • Standard of behavior: the extra fraction of excess is fixed by convention, not physical possibilities.
        • Curves as rules: each curve prescribes how the coalition divides its acquired excess among members.
        • Bounded arbitrariness: threats between partners restrict the curve, but many stable standards remain.
        • Social order: solutions leave some distributional adjustments open while binding others by definite conventions.
    • Excess, Decomposition, and Solution Structure (Chapter IX Composition and Decomposition of Games · VII)
      • Excess and Disorganization
        • Two-dimensional solution areas: in cases (V) and (VI), solutions contain two-dimensional areas, not just curves.
        • Stages of breakdown: at moderate excessive gift, solution is an area plus coalition curves; at larger gift, area alone remains.
        • Standard of behavior: disorganization sets distribution limits where it no longer sanctions coalitions.
        • Critical thresholds: the successive stages appear at two excess values; quantitative values apply only to the three-person case.
      • Extending Decomposition
        • Dummy variant: one set plays a game while the other has no influence—an inessential constituent.
        • Essential constituents: when both component games are essential, new phenomena appear that dummies hide.
        • Multi-part composition: composition/decomposition concepts extend naturally to more than two constituents.
        • Symmetry caveat: a symmetric game need not possess symmetric solutions.
        • Superposition differs: simultaneous play of several games is not composition; its strategic effects are more complex.
      • Formal Structure
        • General games: characteristic functions extend to non-zero-sum games by constants; zero-sum is a special case.
        • Empty coalition: treating the empty set as a coalition simplifies decomposition proofs.
        • Self-contained groups: intersections and sums of self-contained sets remain self-contained; decomposition rests on them.
        • Indecomposability: most games are indecomposable; decomposition requires restrictive equations.
      • Excess, Detachment, and Imbedding
        • Detached imputations: detachment requires a minimum excess; |Γ|₂ is the lower limit.
        • Bounded solutions: every solution for F(e0) or E(e0) is bounded and closed in the imputation space.
        • Excess splitting: in a decomposable game, e0 distributes between the two constituents, with exhaustive alternatives.
        • Imbedding transitivity: an imbedding of an imbedding is an imbedding, so only direct imbedding relationships matter.
      • Three-Person Solutions
        • Normal excess range: between the two critical bounds, solutions resemble the discriminatory solution of the old theory.
        • Degenerate extremes: outside that range solutions degenerate or vanish; too-small excess makes E(e0) empty.
        • Large-excess uniqueness: high excess yields the unique solution consisting only of the two-dimensional area.
    • Detached Imputations and Solution Geometry (Chapter IX Composition and Decomposition of Games · VIII)
      • Geometry of the Fundamental Triangle
        • Fundamental triangle: the space of imputations; the inner triangle lies within it as the core region of the decomposition.
        • Critical scale: ε0 measures the inner triangle's linear size relative to the outer one; limiting case ε0 = 1.
        • Apex: on non-lower triangles it lies on the inner border; on the lower triangle it lies below.
        • Degenerate cases: the lower triangle may shrink to a point or vanish without affecting the argument.
        • Symmetry: configurations are handled up to rotations of 0°, 60°, or 120°.
      • Detached Imputations
        • Detached imputations: points dominated by no imputation at all; they qualify as in (45:D).
        • Fully detached imputations: interior points that also dominate no imputation at all, as in (45:C).
        • One-sided detachment: being undominated does not by itself prevent an imputation from dominating others.
        • Interior passivity: fully detached points are the passive extreme—no one dominates them and they dominate no one.
      • Domination and Boundaries
        • Boundary coverage: the remainder of the fundamental triangle is dominated by the inner boundary belonging to the solution.
        • Closedness: the dominating boundary is part of a closed set, which makes the coverage argument possible.
        • Threshold regimes: inequalities such as ε0 < 3 and ε0 > 3 separate different domination behaviors.
        • Active boundary: the boundary does the domineering; interior points do not.
      • Decomposition and Six-Person Games
        • Decomposable structure: the inner-and-outer-triangle geometry represents a game split into independent components.
        • Six-person connection: when |Γ|2 = 2, the same decomposition yields a six-person game in the earlier theory.
        • Uniform method: the same reasoning applies to all relevant triangles, not only to the lower one.
        • Composition viewpoint: decomposing a game isolates the detached regions that make up its solution.
  • Chapter X Simple Games
    • Winning Coalitions and Simple Games (Chapter X Simple Games · I)
      • From Corner I to Simple Games
        • Starting point: generalize the corner I game, where coalitions either win decisively or lose completely.
        • Decisive coalitions: winning sets include their proper supersets; losing sets are their complements.
        • Simple games defined: essential games in which every coalition is classified as winning or losing, with no quantitative middle ground.
        • Strategic invariance: the winning/losing classification is unchanged under strategic equivalence.
      • Axioms for Winning and Losing Coalitions
        • Complement rule: of any two complementary coalitions, exactly one is winning.
        • Superset closure: any superset of a winning coalition is winning; any subset of a losing one is losing.
        • One-element condition: losing coalitions include the empty set and all singletons.
        • Characterization theorem: these axioms are necessary and sufficient for W and L to come from an actual essential game.
        • Excluded case: a fixed victor whose mere presence wins is not a real game; it is inessential.
      • Simplicity via the Characteristic Function
        • Flatness criterion: losing coalitions are flat; winning coalitions are those whose complement is flat.
        • Boundary values: in simple games, v(S) reaches the upper bound for winning sets and the lower bound for losing sets.
        • Geometric picture: simple games are the vertices of the convex domain Qn; for n = 4 they are the cube's eight corners.
        • Uniqueness: for simple games, W and L determine the game up to strategic equivalence.
      • Minimal Winning Coalitions
        • W or L suffices: either class alone determines the other through complementation.
        • Minimal winning sets Wm: winning coalitions containing no proper winning subset.
        • Real decisiveness: Wm lists the coalitions in which no participant can be spared.
      • Solutions of Simple Games
        • Certainty classification: winning coalitions are certainly necessary; losing ones are certainly unnecessary.
        • Simplified analysis: every coalition being one or the other makes solution theory more tractable than in general games.
    • Weighted Majorities and Imputation Solutions (Chapter X Simple Games · II)
      • Necessary and Unnecessary Coalitions
        • Necessity criterion: every winning set in W is certainly necessary; all other coalitions are certainly unnecessary.
        • Refined criterion: treat only minimal winning coalitions as necessary; every winning set contains one.
        • Preference: the minimal-coalition criterion is more useful for determining solutions.
      • Weighted Majority Games
        • Direct majority game: for odd n, winning means holding more than n/2 players; the unique essential 3-person game is this.
        • Weighted majority symbol: assign weights w_i; S wins if its weight exceeds half the total; write [w₁,…,wₙ].
        • Weight constraints: each weight is nonnegative, never half or more of total; no coalition has exactly half weight.
        • Even n: weighted majorities extend majority rule to even n; four-person corner game is [1,1,1,2].
      • Homogeneity and the Direct Imputation Approach
        • Homogeneous weights: require the surplus a_S to be constant over minimal winning coalitions; normalize a = 1.
        • Known homogeneous games: direct majorities [1,…,1]_h and corner [1,1,1,2]_h satisfy homogeneity.
        • Imputation link: ordinary economic imputation offers a direct route to solutions for a class of simple games.
        • Minimal coalitions in play: players form minimal winning coalitions; outsiders receive -1, insiders receive -1 + x_i.
        • Imputation vector: the payoff vector is -1 + x_i for members, -1 otherwise; summing to zero makes it an imputation.
        • Profitable set U: choose which minimal winning coalitions are profitable; equations hold on U, inequalities outside.
      • Solution Criterion and Verbal Interpretation
        • Solution condition: the imputations from profitable coalitions form a solution iff undominated ones correspond exactly to the set.
        • U* and U+: U* contains coalitions with a profitable subset; U+ contains those whose complement is not in U*.
        • Behavior of U*, U+: at U = W_m both equal W; as U shrinks, the gap between certainly winning and certainly losing widens.
        • Indifferent players: zero x_i players can be added or removed from a coalition without affecting solution status.
        • Verbal construction: pick profitable minimal winning coalitions, solve the constraints, then verify all inequalities.
    • Main Simple Solutions and Enumeration (Chapter X Simple Games · III)
      • Main Simple Solutions and Homogeneous Weights
        • Proper U choice: decreasing U adds inequalities, increasing U adds equations; U = W_m is the natural maximal case.
        • Main simple solution: if U = W_m and its equations are solvable, the minimal winning coalitions themselves form a solution.
        • Homogeneous majority games: every homogeneous weighted majority game possesses a main simple solution.
        • Converse condition: a main simple solution yields homogeneous weights only if the key positivity condition (50:20) holds.
        • Limitations: other solutions generally exist, and this approach covers only simple games.
      • Enumeration by Winning Coalitions: 1-Saturation
        • Enumeration goal: all simple games are enumerated by listing all possible W or W_m; W_m is more compact.
        • 1-saturation: the basic axioms on W are equivalent to S belonging to W iff S meets every member of W.
        • Criterion for W: W fits the simple-game axioms iff it is 1-saturated and contains neither empty nor one-element sets.
        • Embedding test: a candidate V extends to some W iff no two V-sets are disjoint and V excludes empty sets and singletons.
        • Construction: grow V gradually until maximal; rigorous but cumbersome even for small n.
      • Direct Characterization of Minimal Coalitions: 2-Saturation
        • Minimal sets determine game: W is exactly all supersets of W_m, so enumerating W_m suffices.
        • Asymmetric criterion: V equals some W_m iff it is 2-saturated and contains neither empty nor one-element sets.
        • Blocked construction: because 2 is asymmetric, this criterion cannot drive a gradual build-up like the W method.
      • Enumeration by Minimal Coalitions: 3-Saturation
        • Membership condition: V is contained in some W_m iff each S of V is winning and each (–S)∪(i) is winning.
        • Four explicit conditions: V avoids disjoint pairs, inclusions, complements summing to I unless intersection is a singleton, and forbidden extremes.
        • Symmetric saturation: combined conditions form a symmetric relation 3, so maximal 3-satisfactory sets in the allowed domain are exactly all W_m.
        • Key contrast: mere symmetrization of 2 gives only part of 3; the extra condition is the essential technical advance.
      • Simplicity and Decomposition
        • Component-wise membership: in a decomposition, R∪T is winning in Γ iff R is winning in Δ and T is winning in H; same for losing.
        • Simple decompositions: Γ is simple iff one constituent is simple and the other is inessential.
        • Proof mechanism: choosing empty and full complementary sets forces one constituent inessential; an inessential constituent makes all sets both winning and losing.
    • Simplicity, Decomposition, and Small-n Enumeration (Chapter X Simple Games · IV)
      • Composition and Inessential Constituents
        • Composition rule: a simple game stays simple only when composed with an inessential game of dummies; two simple games do not compose to a simple game.
        • Hereditary solutions: solutions of the composite come directly from the simple constituent's solutions.
        • Zero excess only: a nonzero-excess theory of simple games would also embrace nonsimple games, so zero excess is forced.
        • Imbedding failure: simplicity does not survive general imbedding, so the methodological principle of 46.10.5 has limits.
      • Significant Players and the Kernel
        • Significant players: exactly those in some minimal winning coalition; I₀ collects them.
        • Same W_m: a decomposable simple game and its simple constituent have identical minimal winning coalitions.
        • I₀ splits: I₀ is a splitting set, its constituent is simple, and the complement is inessential.
        • Decomposition partition: I₀ plus one-element sets for each remaining dummy; the kernel is the unique simple indecomposable constituent.
      • Classification by Two-Element Coalitions
        • Enumeration plan: enumerate simple games by W_m; n = 3 is already [1,1,1]_h, so start at n ≥ 4.
        • Two-element criterion: a two-element coalition lies in W_m exactly when it lies in W.
        • Cases C_k: W_m contains exactly (1,n), …, (k,n) for k = 0,…,n−1; C* is the exceptional (1,2),(1,3),(2,3) case.
        • Disposable cases: C*, C_{n−2}, C_{n−1} are exactly the games [1,…,1,l−2]_h with dummies.
        • Remaining cases: any other simple n-person game must fall in C_0,…,C_{n−3}.
      • Small Cases Completed
        • n = 4: complement symmetry rules out C_0,C_1, leaving exactly [1,1,1,2]_h.
        • n = 5 list: exactly four essential games (up to permutation): [1,1,1,1,1]_h, [1,1,1,2,2]_h, [1,1,2,2,3]_h, [1,1,1,1,3]_h.
        • C0 for n = 5: no two-element winning coalitions; all three-element coalitions win, yielding direct majority.
        • C1 for n = 5: unique winning pair (1,2); yields [2,2,1,1,1]_h.
        • C2 for n = 5: winning pairs (1,2),(1,3); yields [3,2,2,1,1]_h.
      • New Regularities at n = 6
        • Homogeneity fails: first simple game without any homogeneous weighted-majority symbol appears at n = 6.
        • Classes multiply: from n = 6 onward, some C_k classes contain more than one game.
        • Weights are rigid: exhaustive small-n lists give one symbol for n = 3, one for n = 4, and only four for n = 5.
        • W_m varies: total winning coalitions are fixed at 2^{n−1}, but minimal winning coalitions vary (for n = 5: 10, 7, 5, 5).
    • Beyond Main Simple Solutions (Chapter X Simple Games · V)
      • Six Counter-Examples for n = 6 and 7
        • Case C_k nonunique: [1,1,1,2,2,4]_h and [1,1,1,3,3,4]_h are distinct games in the same C_2.
        • No main solution: weighted majority game [w,w,w,1,1,1] with 1<w<3 has inconsistent main equations; substitution rates cannot be defined.
        • No symbol, but solution: a game with no [w_i] representation yet has main simple solution x_1=...=x_6=2.
        • Neither exists: another n=6 game lacks both weighted-majority symbol and main simple solution; (E_3) is contradictory.
        • Criterion gap: examples give equality and strict inequality in (50:21), so homogeneous weighted majority and main-solution existence differ.
        • Relational order: n=7 cyclic minority triples are unweighted yet have a main solution; no player has advantage, relations decide.
      • Reasons to Determine All Solutions
        • Main solution insufficient: homogeneous weighted majority games have many solutions; extraordinary ones may be natural.
        • General n: find all solutions for at least one simple game at every n, exposing structural possibilities.
        • Simple-game advantage: preliminary conditions of 30.1.1 cause no difficulties, since every coalition is necessary or unnecessary.
        • Known full-solution cases: inessential games, the essential three-person game, and decomposable games built from known constituents.
        • Composite games are polymers: decomposing essential three-person games yields 3k players but no rule-based linkage among triples.
        • Only general n known: the homogeneous weighted majority game [1,...,1,n-2]_h is the sole n-person game with all solutions determined.
      • The Chief-Player Game
        • Rule: player n wins with any ally; players 1..n-1 win only as a complete bloc of n-1.
        • Privilege: for n≥4 player n is privileged; for n=3 the game is symmetric [1,1,1]_h.
        • Monopolist analogy: chief player resembles a monopolist who must find at least one ally; only all-others coalition can beat him.
        • Winning-set restriction: making the one-element set (n) winning is impossible; it would destroy the game’s essentiality.
      • Domination and Solution Cases
        • Domination condition (55:A): α dominates β iff chief and an ally improve, or all ordinary players improve.
        • Direct vs cooperative domination: (55:1) is chief player plus one ally; (55:2) is general cooperation of the other n-1 against him.
        • Chief always affected: in any domination the chief gains in (55:1) and loses in (55:2); other players may be unaffected.
        • Solution cases: within a solution, chief payoff ranges from a to b; Case I means a=b, fixed chief payoff.
        • Solution variety: the complete list is a complex array of classes; the proof itself is a vehicle for interpretation.
    • Chief Player Solutions and Discrimination (Chapter X Simple Games · VI)
      • Case I: Segregation of the Chief Player
        • Chief player segregation: the solution isolates the chief player from the rest of the game.
        • Solution set: every imputation giving the chief player the assigned amount belongs; other allocations are arbitrary.
        • Indefinite distribution: excluding the chief player leaves the other players’ shares unspecified.
        • Upper limit: the assigned amount must stay within a bound set by the game.
      • Case II: Two Extremes
        • Decomposition: the solution splits into chief-player defeat, chief-player success, and intermediate imputations.
        • Complete defeat: a unique imputation gives the chief player −1 and all others their best possible shares.
        • Complete cooperation: this unique imputation is the state of full cooperation against the chief player.
        • Complete success: the chief player reaches his maximum value but needs exactly one ally.
        • Ally choice: the ally can be selected from a least favored set, not from natural supporters.
      • The Least Favored Set S*
        • Set S*: players least favored in the coalition that fully defeats the chief player.
        • Intermediate optimum: each S* member reaches his maximum in every apportionment where the chief player is not fully successful.
        • Overlap condition: success and intermediate sets meet only if S* is a one-element set or α* = −1.
        • Strategic interaction: internal ranking of the opposition determines the chief player’s choice of ally.
      • Case II′: Nondiscriminatory Coalition
        • Case II′: S* contains all other players, so no discrimination exists within the general coalition.
        • Finite solution: this is the only finite solution, with all coalition members treated equally.
        • Successful alliance: the chief player offers an ally the same share used in the general coalition.
        • Non-discriminatory result: reasonable and fair, yet not the only possible solution.
      • Case II″: Discriminatory Coalition
        • Case II″: S* is a proper non-empty subset, so discrimination appears within the opposition.
        • Reduced domain: after fixing S* components, the solution space has dimension n − p − 1.
        • Unique path: each chief-player payoff y in [−1, ω*] determines exactly one imputation in the solution.
        • Complete system: all solutions in this case come from the characterized solution V and satisfy the same conditions.
        • Degenerate exception: when S* is one-element or α* = −1, extreme imputations may coincide; otherwise they are disjoint.
    • Classifying Simple-Game Solutions (Chapter X Simple Games · VII)
      • Reformulating the Complete Result
        • Complete enumeration: the three cases (I), (II′), and (II″) exhaust all solutions of the game.
        • Proof closure: internal stability and external domination are verified by exhaustive case splits, leaving no unaccounted imputations.
        • Role of S*: in the unified enumeration, S* is any nonempty subset of players 1 to n−1; only the empty set is excluded.
        • Unifying Case (II′): including S* = all other players lets one formula cover both Case (II′) and Case (II″).
        • Excluding the empty set: forcing S* = ∅ into that formula yields a set that is not a solution.
      • Finite Symmetric Solution
        • Finite solution: Case (II′) is a finite set of imputations, the only finite solution in the classification.
        • Full symmetry: it is invariant under all permutations of players 1,…,n−1, matching the game’s own symmetry.
        • Intuitive appeal: this is the solution suggested by heuristic study of the three- and four-person cases.
        • Non-uniqueness: the formal theory reveals additional infinite solutions that cannot be ignored.
        • Special contingency: existence of a unique finite solution is a favorable accident of this game, not a general rule.
      • Segregation of the Chief Player
        • Case (I) solution: the set of all imputations giving the chief player a fixed amount, with arbitrary shares for the others.
        • No internal standard: among the remaining players, no quantitative rules govern how they divide their gains.
        • Unanimity only: with the chief player excluded, the others can only combine unanimously, so bargaining outcomes are indeterminate.
        • Perfect symmetry: Case (I) shares full symmetry with Case (II′) and is the other extremal solution.
      • Partial Segregation of the Remaining Players
        • Two payoff values: in Case (II″) each player in S* receives either α* or −1 in every solution imputation.
        • Sole-player segregation: a single segregated player can receive more than −1, up to his share in the finite solution.
        • Group segregation: if more than one player is segregated, they all receive the absolute minimum −1.
        • Segregated sets: the sets that can be segregated are precisely the defeated sets L.
        • Escape clause: under the exceptional condition, players in S* may join coalitions and improve their status, ending segregation.
      • Bargaining Correlations and Partition Principle
        • Causal linkage: in Case (II″), the shares of players outside S* are uniquely determined by the chief player’s amount through chosen functions.
        • Regulated bargaining: bargaining runs between the chief player and some outsiders, while the standard fixes correlations among those outsiders.
        • Multiplicity of standards: different choices of the defining functions generate many distinct solutions.
        • Complementary partition: S* and its complement partition the players into two sets, and this pair decides the solution type.
        • General principle: in a simple game a two-part partition seems decisive; general games may require partitions into more sets.
    • Simple Games, Classification, and Chief-Player Solutions (Chapter X Simple Games · VIII)
      • Minimal Winning Coalitions and Indifferent Players
        • Minimal winning coalitions: every player belongs to at least one; membership marks significant service.
        • Indifferent players: belong to minimal winning coalitions but never receive a share, so they must be provided for.
        • Open problem: no simple game is known with a solution derived from U W_m, and none is ruled out.
        • Weight zero: players outside all minimal winning sets get w_i = 0, treating them as dummies.
      • Classification and Exhaustive Cases
        • Simple games exist only for n ≥ 3, with W and L disjoint and equinumerous at 2^(n−1).
        • Case exhaustion: most classes are void or contain exactly one game; n = 5 first yields counter-examples.
        • First counter-examples: [1,1,1,1,1]_h and [1,1,1,2,2]_h break the void-or-unique pattern.
        • Significant alternatives: non-main solutions are essential, not mere appendages.
      • Exceptional Six- and Seven-Player Games
        • Six-player example: has 13 minimal winning coalitions; no finite solution is known, and none is suspected.
        • Privileged-player variant: an exception to tie-breaking yields [2,1,1,1,1,1]_h rather than a simple numerical privilege.
        • Seven-player geometry: minimal winning sets are the seven lines of the 7-point projective plane, W_m having 7 elements.
        • Symmetry without fairness: the 7-point game is pairwise transitive, yet two three-element sets can act differently.
      • Decomposition and Indecomposable Constituents
        • Decomposition principle: a game is a composite of indecomposable constituents; participants split into one- or three-element sets.
        • Known solutions: all solutions are known exactly for games formed from one- and three-element constituents.
        • Progress follows classification: obtaining all solutions tracks the sizes of indecomposable constituents.
        • Typical games are indecomposable, so these complete results remain very special.
      • Chief-Player Discriminatory Solutions
        • Chief-player role: unique for n ≥ 4; for n = 3 any player can fill it.
        • Two cases: case (I) segregates the chief player; case (II) covers intermediate and non-discriminatory possibilities.
        • n = 3 aggregation: case (II) combines two discriminatory possibilities with the non-discriminatory solution.
        • Extreme discrimination: β = –1 yields an immediate solution; intermediate β > –1 needs the full analysis.
        • Solution as standard: within a standard, a player's pessimum can become his optimum, illustrating imputation influence.
      • Analytic Proofs and Continuity
        • Dimensional reduction: graphical methods would need (n – p – 1) dimensions, so proofs become analytic.
        • Lipschitz continuity: the functions α_i(y) satisfy a Lipschitz condition; continuity is proved, not assumed.
        • Analogy to graphical deduction: lemmas (55:D′)–(55:P′) mirror the earlier two-dimensional proof step by step.
        • Verbal arguments are inconclusive: only mathematical proof fixes which standard of behavior is valid for this game.
    • Chief Player and Degenerate Cases (Chapter X Simple Games · IX)
      • Chief Player's Bargaining Limit
        • Chief player: needs any one ally to win; the chief-plus-ally coalition commands n − 2.
        • Concession to an ally: keeping ω means offering each possible ally n − 2 − ω.
        • Limit to exaction: the total amount the n − 1 potential allies can make together sets the ceiling on ω.
        • Upper bound: hence ω is constrained by n − 2 − 1/(n−1); beyond that no ally is available.
        • Heuristic solution: chief gets n − 2 − 1/(n−1) on success, −1 on failure; chosen ally gets 1/(n−1).
      • Exact Cases and Deductions
        • Continuous α_i(y): continuity is the hinge linking equation (a) to equation (b).
        • Construction: setting y = −1 yields α* and S* from (55:L′:c).
        • Case (II′): the heuristic chief-player solution coincides with the exact solution in (55:V).
        • Case (II″): every other solution has S* a nonempty proper subset of {1,…,n−1}.
      • Degenerate S* = ∅ Case
        • Symbolic case (I): S* = ∅ is used to denote the degenerate case (I).
        • Contradictory result: p = 0 gives ω* = α* + 1, so ω* can exceed the required bound.
        • Violated definition: with S* = ∅ the equations no longer satisfy the defining condition for α*.
        • Not a solution: the resulting object is absent from the solution list and is no solution at all.
      • Uniqueness, Permutations, and n = 4
        • Permutation argument: any proper S* can be changed by permuting players 1,…,n−1, altering the solution.
        • Uniqueness failure: a counterexample to uniqueness is analyzed in 38.3.1.
        • Two-element S*: when p = 2, constraint p ≤ n−2 forces n ≥ 4.
        • Why n = 3 differs: the unusual phenomena (i) and (j) first appear only at n = 4.
        • Chief player's segregation: apart from the chief, α_i(y) varies for players outside S*.
  • Chapter XI General Non-Zero-Sum Games
    • General Games Beyond Zero-Sum (Chapter XI General Non-Zero-Sum Games · I)
      • Dropping the Zero-Sum Restriction
        • General games: payoffs are entirely unrestricted; zero-sum and constant-sum games become special cases.
        • Economic relevance: zero-sum theory overemphasized apportionment; general games restore questions of productivity.
        • Conceptual break: unrestricted payoffs invalidate characteristic functions, domination, and solutions as previously defined.
        • Not merely technical: extending the theory is a foundational problem, not a routine adjustment.
      • The Fictitious Player Device
        • Zero-sum extension: any n-person general game can be re-read as an n+1-person zero-sum game.
        • Fictitious player: an extra player receives the negative of the real players' total payoff and controls no variables.
        • Formal constraint: he must have no direct influence on the course of the play.
        • Coalition danger: in a coalition game he can still pay real players to break alliances, gaining indirect influence.
      • Limits of the Zero-Sum Extension
        • Illustrative game: two players each get −1 alone, but 1 together; its extension is the essential three-person simple majority game.
        • Symmetry trap: the characteristic function treats real and fictitious players alike, inviting fictitious coalition competition.
        • Equivalence fails: applying zero-sum theory literally lets the fictitious player act as a real bargainer, contaminating Γ.
        • No escape by reverting: abandoning characteristic functions sacrifices the only workable zero-sum theory without removing the objection.
      • Discriminatory Solutions
        • Right remedy: restrict the zero-sum theory to solutions that discriminate against the fictitious player.
        • Non-discriminatory solution: natural for symmetric three-person games, but wrong because it lets player 3 compete for coalitions.
        • Discriminatory solutions: exclude player 3 from coalition competition and assign him a fixed amount in every imputation.
        • Worst exclusion c = −1: real players divide freely, with no threat of defection to a fictitious partner.
      • The Fixed Payment Question
        • Variable c: the excluded player's fixed amount can range over −1 ≤ c < 1/2.
        • Payments from fictitious player: absurd, since he does not exist and cannot pay.
        • Payments to fictitious player: not inherently absurd, leaving open whether c > −1 should be permitted.
    • Pruning Solutions and Ruling Domination (Chapter XI General Non-Zero-Sum Games · II)
      • Setup and Candidate Solutions
        • Extension: general n-person games are studied through their zero-sum (n+1)-person extension with a fictitious player.
        • Self-denial: stable standards of behavior may require forgoing a possible collective advantage.
        • Ω: all solutions of the zero-sum extension.
        • Ω_c, Ω′, Ω″: fix fictitious player’s payoff at c; Ω′ is the union of nonempty families, Ω″ takes the minimum c.
        • Empty families: Ω_c is empty for inadmissible c, so only certain fictitious-player payoffs can matter.
      • New Imputation Restrictions
        • Real notation: imputations can be written by real players’ components alone; the fictitious component is implied.
        • Individual rationality: no real player accepts less than he can guarantee alone against all others.
        • Collective ceiling: all real players together cannot be promised more than full cooperation and best strategy can yield.
        • Rational refusal: imputations violating either restriction are manifestly absurd and will not be accepted.
      • Agreement with the Zero-Sum Case
        • Dummy addition: extending a zero-sum Γ adds a dummy, so solutions correspond with the dummy getting v((n+1)).
        • Old theory preserved: for zero-sum Γ, the new solutions coincide with old solutions and the Ω′/Ω″ choice becomes irrelevant.
        • Normal zone: the two imputation sets E(0) and F(0) have the same solutions, confirming agreement.
      • Domination: Two Cases
        • Ordinary case: a real coalition prefers one imputation and can enforce that preference as an alliance.
        • Paradoxical case: a group prefers one imputation, outsiders cannot block it, yet society as a whole is worse off.
        • Fictitious treatment: treating the fictitious player as real produces this strange, anti-social form of domination.
        • Key contrast: ordinary domination requires enforcement; the paradoxical case relies on outsiders’ inability to block and an anti-social effect.
      • Redefinition and Rigorous Consequences
        • Rejection: keep only ordinary domination; drop the paradoxical alternative from the concept of domination.
        • Effectivity restriction: in the extension, a coalition is effective only if it excludes the fictitious player.
        • Lemma 1: every solution in the new sense assigns the fictitious player exactly v((n+1)).
        • Lemma 2: every solution in the new sense is also a solution in the old sense.
    • New Solutions and Characteristic Functions (Chapter XI General Non-Zero-Sum Games · III)
      • Choosing the Solution Class
        • Ω″ wins: restricting solutions by a modified domination in the zero-sum extension yields exactly the desired set.
        • Domination change: require effective sets to exclude the fictitious player n+1.
        • Concurrent results: modification of domination and restriction of solutions lead to the same class.
      • New Definition of Solution
        • General solution: a solution of Γ is any old solution of its zero-sum extension whose imputations give the dummy player v((n+1)).
        • Equivalent formulation: apply the original zero-sum solution concept directly to Γ, using imputation form (56:7).
        • Strengthened imputation condition: (56:25) is a stronger form of (56:10), and may be added to the characterization.
        • Zero-sum games: for a zero-sum Γ the new solution concept coincides with the old one.
      • Characteristic Functions: Restricted and Extended
        • Extended function: defined on all subsets of players including fictitious n+1; inherits old zero-sum properties.
        • Restricted function: defined on original players only; matches old characteristic function when Γ is zero-sum.
        • Axioms for extended: v(∅)=0, v(┴S)=−v(S), and superadditivity for disjoint S,T.
        • Axioms for restricted: v(∅)=0 and superadditivity; complement equality is replaced by v(−S) ≥ v(I)−v(S).
      • Characterization Theorems
        • Sufficiency for restricted: any set-function obeying restricted axioms is the restricted characteristic function of some general game.
        • Construction: players choose subsets containing themselves; agreeing choices form rings; payments are assigned from the given set-function.
        • Sufficiency for extended: any function obeying extended axioms arises as an extended characteristic function.
        • Basis of theory: all strategic coalition behavior in general games depends only on the characteristic function.
      • Removable Sets of Players
        • Removable set: a set S is removable if another game with the same characteristic function gives no player in S direct influence.
        • Every singleton is removable: one dummy player's indirect bargaining role can duplicate any single player's strategic influence.
        • Full set removable iff inessential: if all players are removable, the characteristic function is additive over singletons.
        • Extreme games: all (n−1)-element sets removable—only one player directly influences play; others offer compensation.
        • Low dimensions: all essential 3-person games are extreme; 4-person games are extreme or leave exactly one non-removable triple.
    • General Games: Equivalence, Threats, Normal Forms (Chapter XI General Non-Zero-Sum Games · IV)
      • Strategic Equivalence in General Games
        • Restricted characteristic function: after abandoning the zero-sum extension, the "restricted" qualifier is dropped and Γ's own characteristic function is used.
        • Imputation: a vector α whose components satisfy the conditions from (56:I:b,e).
        • Strategic equivalence: Γ and Γ′ are equivalent when v′(S)=v(S)+Σ_{i∈S} α_i; the α_i are unrestricted.
        • Isomorphism of imputations: strategic equivalence induces a one-to-one correspondence between imputations of Γ and Γ′.
      • Zero-Sum and Constant-Sum Classes
        • Domain: general games satisfy v(∅)=0 and superadditivity for disjoint coalitions.
        • Zero-sum functions: add v(−S)=−v(S) to the general conditions.
        • Constant-sum functions: add v(S)+v(−S)=v(I); merging complementary coalitions yields no extra profit.
        • Zero-sum decomposition: zero-sum property equals constant-sum property plus v(I)=0.
        • Strategic mimicry: a non-zero-sum game can share a zero-sum or constant-sum characteristic function and behave like one.
        • Fictitious player: the fictitious player is a dummy iff v(S)+v(−S)=v(I), i.e. constant-sum.
      • Interpreting the Characteristic Function
        • Value of a coalition: v(S) is the maximin of expected payoff in the zero-sum extension; S maximizes, −S minimizes.
        • Gain vs. loss: in general games, a coalition seeks its own gain, not merely the opponent's loss.
        • Mutual gains: general games allow strategy changes advantageous to both sides — genuine productivity increases.
        • Threats as pressure: inflicting losses can be indirectly profitable by inducing compensation or favorable adjustment.
        • Collective maximum: solutions aim at maximum total profit; after that, one group's gain costs others at least as much.
        • Complete information: net-loss threats are avoided via negotiation and compensation, so one side's gain equals the other's loss.
      • Reduced Forms and Normalization
        • Reduced form: unique strategically equivalent form with one-player sets equal to −γ and total value 0.
        • Zero-reduced form: unique alternative normalization requiring each one-player set to value 0.
        • Unrestricted constants: strategic equivalence has n independent parameters, enabling two normalizations.
        • Nonnegative γ: superadditivity over singletons gives γ ≥ 0.
        • Bound inequalities: for |S|=p, −pγ ≤ ṽ(S) ≤ (n−p)γ, with fixed endpoints at p=0,1,n.
      • Essentiality and the Program
        • Essentiality split: γ=0 yields inessential, vacuous games; γ>0 yields essential games, rescalable to γ=1.
        • Essentiality criteria: chapter 27 tests remain valid, with v((k)) modified and (27:C),(27:D) unchanged.
        • Deferred theory: removable sets and extreme games are postponed; the general theory is unlikely to be simple.
        • Domination geometry: convexity and flatness carry over; doubtful necessary-set cases reduce to 2≤p≤n−1.
        • Classification program: classify all general games for n≤3 completely; for n≥4 treat only special cases.
        • Economic applications: small-n games connect with bilateral monopoly and duopoly; common-sense results hold until coalitions and compensations dominate.
    • General Solutions and Market Interpretation (Chapter XI General Non-Zero-Sum Games · V)
      • Extending Chapter IX Concepts
        • |Γ| asymmetry: the second half of (45:F) survives; the first fails without (25:3:b).
        • |Γ|₁ vs |Γ|₂: |Γ|₂ can be evaluated from |Γ|₁, but not conversely; |Γ|₁ may be positive while |Γ|₂ is zero.
        • Weakened essentiality: inessential gives |Γ|₁ = 0, |Γ|₂ = 0; essential gives |Γ|₁ > 0, |Γ|₂ ≥ 0.
        • Composition theory: extends with minor changes; detailed analysis is deferred since interpretation matches the zero-sum case.
      • Solving n = 1 and n = 2
        • n = 1 trivial: necessarily inessential; solution is the pure maximum problem—Robinson Crusoe or planned economy.
        • n = 2 essentiality: general games need not be inessential; one normalized essential type exists with γ = 1.
        • |Γ|₂ = 0: even essential two-person games have zero excess, while |Γ|₁ = 2γ can exceed zero.
        • Unique solution: all imputations with αᵢ ≥ v((i)) and α₁ + α₂ = v((1,2)).
        • Essential case: infinitely many imputations arise when v((1)) + v((2)) < v((1,2)).
      • Solving n = 3
        • Characteristic function: normalized γ = 1 with parameters aᵢ in [−2, 1]; zero-sum occurs only at aᵢ = 1.
        • Imputation domain: same fundamental triangle as the essential zero-sum three-person game.
        • Domination: only two-person coalitions matter; αᵢ + αⱼ ≥ aₖ and both players must gain.
        • Case (a): a₁ + a₂ + a₃ > 0 gives inner-triangle solutions akin to Figures 82–83, possibly asymmetric.
        • Case (b): a₁ + a₂ + a₃ ≤ 0 yields Figure 86-type solutions; inner triangle may protrude or degenerate.
      • General Games vs Zero-Sum Games
        • Analogy: general n-person games behave like zero-sum n+1-person games via zero-sum extension.
        • Zero-sum extension: every zero-sum n+1-person game comes from some general n-person game.
        • Solutions not exhausted: general game solutions form only a subset; all four-person zero-sum solutions remain incomplete.
        • Existence payoff: general three-person results prove every zero-sum four-person game has at least one solution.
      • Economic Interpretation: Two-Person Market
        • Bilateral monopoly: sale of one unit by seller 1 to buyer 2; values satisfy u < v.
        • Price game: seller offers p, buyer accepts or declines; payoffs p and v − p, or u and 0.
        • Common-sense interval: theory should leave all prices from u to v possible.
        • Robust to bargaining: multi-bid variants and compensation mechanisms yield the same characteristic function.
        • Characteristic function: v((1)) = u, v((2)) = 0, v((1,2)) = v.
        • Adversarial assumption loaded: if the buyer can prefer gain over harming the seller, the seller can force p₀ > u, altering the solution.
    • Markets, Marginal Pairs, and Coalitions (Chapter XI General Non-Zero-Sum Games · VI)
      • Divisible-Goods Two-Person Market
        • Corroboration: general two-person game validates forming the characteristic function; all such games reduce to one form.
        • s divisible units: utility of holding t units is u_t for seller, v_t for buyer.
        • Characteristic function: v((1,2)) = max_t(u_{s−t}+v_t); single players can block all sales.
        • Decreasing utility: marginal-pair inequalities become necessary and sufficient for optimal transfer t_0.
        • Böhm-Bawerk criterion: number of units transferred matches the marginal pairs of ordinary theory.
        • Inessential game: occurs exactly when no transfer takes place, t_0 = 0.
      • Price Determination with Compensation
        • Unique solution: all imputations restrict the per-unit price p to a definite interval.
        • Wider interval: our theory allows a larger price range than Böhm-Bawerk's marginal-pair interval.
        • Compensations: premiums and rebates among players destroy the unique price of ordinary analysis.
        • Average price: p is only a mean over transfers, not a uniform transaction price.
        • Bilateral monopoly: non-unique price structure is consistent with this market form.
      • Three-Person Market: Setup and Solutions
        • One seller, two buyers: valuations u, v, w; stronger buyer 3 with u < v < w in the typical subcase.
        • Characteristic function: v((1,2,3)) = w; both buyers and seller can block sales.
        • Essential, non-zero-sum: three-person essential games form a three-parameter family; one typical class suffices.
        • Solution cases: buyer utilities υ <, =, > w yield distinct geometric and algebraic solution forms.
        • Solution structure: upper side of the fundamental triangle plus a monotone curve of imputations.
        • Arbitrary division rule: curve embodies a definite standard of behavior for the buyer coalition.
      • Comparison with Common-Sense Theory
        • Ordinary marginal-pair result: price lies between the limits of the stronger and weaker buyers; seller's limit is irrelevant.
        • Agreement: common-sense price range matches the upper side of the mathematical solution.
        • Extra imputations: the monotone curve has no counterpart in ordinary marginal-pair analysis.
        • Coalition interpretation: buyers form a coalition with a fixed division rule and bargain jointly with the seller.
        • No conflict: theory extends, rather than contradicts, common-sense results by admitting coalitions.
    • Markets, Coalitions, and Solution Structure (Chapter XI General Non-Zero-Sum Games · VII)
      • Coalitions in the Three-Person Market
        • Coalitions welded: theory unifies competition and coalition in one solution, unlike the classical argument.
        • Divisible supply: s substitutable units with variable utilities for seller and both buyers.
        • Characteristic function: seller alone holds initial utility; buyers alone get zero; mixed coalitions maximize over transfers.
        • Buyer-only coalition: v((2,3)) = 0, since buyers cannot transact among themselves.
      • Buyer Strength and Assumptions
        • Weakness test: equality in the key inequalities marks a buyer unable to activate or influence the market.
        • Active-buyer assumptions: strict inequalities exclude buyer weakness and make divisible supply effective.
        • One-sided dominance: if one buyer can exclude the other, the solution reduces to the earlier single-unit monopoly case.
        • Decreasing utility: yields the key inequality v + w > z + u, essential for the geometric solution.
      • Solution Structure and Interpretation
        • Imputations: α1 ≥ u, α2, α3 ≥ 0, with α1 + α2 + α3 = z; reduced form simplifies the analysis.
        • Two-part solution: an area plus a curve, intermediate between earlier degenerate market solutions.
        • Competition vs coalition: the area represents buyers competing; the curve represents buyers united against the seller.
        • Marginal transfers: transferred unit numbers follow Böhm-Bawerk’s marginal-pair criterion.
      • Prices and Market Form
        • Two prices: the two buyers may pay different unit prices p and q.
        • Price intervals: wider than Böhm-Bawerk’s limits, though the difference tends to shrink with more buyers.
        • Average prices: price differences reflect premiums and rebates even under complete information.
        • Monopoly vs duopoly: the observed price abnormalities belong to this three-person market structure.
      • General Market with l Sellers and m Buyers
        • General characteristic function: pure seller or buyer coalitions receive initial utilities; mixed coalitions maximize transfer utility.
        • Flat coalitions: seller-only and buyer-only sets are flat, so they cannot be needed for domination.
        • Constant-sum equals inessential: the market is constant-sum only when no transactions take place.
        • Effective markets nonconstant-sum: genuine market activity makes the game intrinsically non-zero-sum.
        • Domination mix: any dominating coalition must contain both sellers and buyers; l = 1 or m = 1 gives monopoly or monopsony.
    • General Non-Zero-Sum Markets and Monopoly (Chapter XI General Non-Zero-Sum Games · VIII)
      • From Zero-Sum to General Games
        • Zero-sum vs. non-zero-sum: the distinction reflects purely social vs. social-economic relationships.
        • Fictitious player: fix strategy for player n+1 to embed any non-zero-sum game in a zero-sum framework.
        • Pure maximum only for n=1: with multiple players, coalitional bargaining replaces straightforward maximization.
        • Constant-sum games: a technical generalization; more participants always enrich structural possibilities.
      • The Extended Solution and Its Restriction
        • Characteristic function carries the theory: rules need not be symmetric; v(S) alone determines solutions.
        • Fictitious player is not a dummy: unless the solution is restricted, his role remains active in the rules.
        • Imputation dimension: eliminating the fictitious player’s component αₙ₊₁ is essential for deduction.
        • Zero-sum equivalence: the new theory agrees with the old for every zero-sum game.
        • Enforce vs. block: a coalition’s enforcement and outsiders’ blocking are independent, except in zero-sum games.
      • Domination, Stability, and Collective Benefit
        • Transitive monopoly order: unlike general domination, (64:12)–(64:13) are transitive in monopolistic situations.
        • Stable inefficiency: accepted standards can make fully informed players accept self-denial, producing inefficient but stable social forms.
        • Collective maximum: the fictitious player receives only his solo amount, real players jointly maximize benefits.
        • Compensation without dictation: the solution prevents losses to the totality without predetermining the compromise among players.
      • The Market as a Game
        • Bilateral monopoly: monopolistic/monopsonistic structure emerges when a unique critical seller or buyer satisfies i=1 or j*=1*.
        • Money as transferable utility: sales are modeled with an unrestrictedly transferable numerical utility; barter is set aside until Chapter XII.
        • Classical price range: the solution reproduces the Böhm-Bawerkian bargaining interval and allows both forced and optional transfers.
        • Poker-like complexity: no simple uniform policy works; market rules may build in analogous constraints, e.g. options.
      • Geometry and Interpretation of Solutions
        • Inner triangle rule: in three-person markets, solutions are inner triangles meeting 0–3 sides of the fundamental triangle.
        • Zero-sum limit: equal valuations eliminate inner curves; only symmetric types (Figures 92, 95) survive.
        • Malevolent vs. cooperative: opposition fixes lower individual limits; joint outcome is described by the cooperative ideal.
        • Theory outranks interpretation: verbal market descriptions are illustrative, but only the complete rigorous theory is reliable.
    • Market games and monopoly power (Chapter XI General Non-Zero-Sum Games · IX)
      • From Zero-Sum to General Non-Zero-Sum
        • Grand coalition value: after player interchange, the characteristic function becomes v((1,2,3)) = υ.
        • Allocation sum: individual amounts obey α₁ + α₂ + α₃ = υ, fixing the total utility.
        • Redundant formula: definition (62:20) is unnecessary yet kept for interpretive convenience.
      • Indivisible Goods and Seller Constraints
        • Single unit of value: all amounts are measured in utility; the good itself is one indivisible unit.
        • Seller’s limit: the seller’s reservation value is his best alternative use of the good.
        • Inventory bound: total units sold, t + r, cannot exceed the seller’s original stock, s.
      • Coalitions and Market Transactions
        • Coalition composition: an effective coalition must contain both sellers and buyers.
        • Identity irrelevance: no need to specify which seller serves which buyer, since only utilities enter v(S).
        • Automatic negotiation: bargaining, compensation, and coalition agreements are absorbed by the theory.
      • Monopoly, Monopsony, and Domination
        • Monopolist’s necessity: no effective domination is possible without the monopolist or monopsonist.
        • Shared boundary: the solution’s lowest point coincides with the curve’s highest point at α₁ = υ + w − z.
        • Redundant variable: once α₂ and α₃ are fixed, α₁ follows from their sum and needs no separate analysis.
  • Chapter XII Extension of the Concepts of Domination and Solution
    • Generalizing Domination and Solution (Chapter XII Extension of the Concepts of Domination and Solution · I)
      • Motivation and General Framework
        • Three variations: concepts shifted during deductions, decomposition to constant-sum games, and final general-game generalization
        • Unification: redefinitions for decomposability are subordinate to the original theory via imbedding
        • Generalized setup: arbitrary domain D, arbitrary relation x ≺ y; solution is exactly the set of undominated elements
        • Saturation link: generalized solutions mirror saturation, but asymmetric relations raise the same difficulties
      • Orderings and Acyclicity
        • Complete ordering: trichotomy plus transitivity (65:A), as with usual “greater” on real numbers
        • Partial ordering: at most one of x=y, x ≺ y, y ≺ x, retaining transitivity; incomparable elements allowed
        • Acyclicity: no chains x1 ≺ x0, …, xm ≺ x_{m−1} with x0=xm; weaker than partial ordering
        • Example: immediate succession on positive integers is acyclic yet not an ordering
      • Solutions under Complete Ordering
        • Symmetric relation: exploit the saturation connection to obtain full information about solutions
        • Absolute maximum: unique if exists; solution iff one-element set containing it (65:E)
        • Finite domain: complete ordering guarantees a maximum, since otherwise an infinite ascending chain arises
      • Solutions under Partial Ordering
        • Relative maxima: need not be unique, and existence alone does not ensure a solution
        • Condition (65:G): every non-maximum is dominated by some maximum; necessary and sufficient for solution
        • Unique solution: when (65:G) holds, solution is precisely the set of all relative maxima
        • Finite domains: fulfill (65:G), so a unique solution exists
      • Acyclic Case and Limits
        • Subset maxima: define Em(E) for subsets; decisive property is every non-empty subset possesses maxima
        • Finite D settled: acyclic case gives exhaustive results for finite domains, with deeper insights
        • Limited applicability: original game relation has no simple distinguishing properties for useful specialization
    • Acyclicity and Unique Solutions (Chapter XII Extension of the Concepts of Domination and Solution · II)
      • Strict Acyclicity Defined
        • Strict acyclicity: no infinite sequence x₀, x₁, … with x₁ ≻ x₀, x₂ ≻ x₁, …
        • Core equivalence: strict acyclicity is exactly the property (65:K) that every nonempty subset has maxima.
        • Expected fundamentality: strict acyclicity, not merely acyclicity, is the precise condition behind (65:K).
      • Acyclicity vs. Strict Acyclicity
        • Strict ⇒ acyclic: every infinite domination chain would yield a finite cycle, so strict acyclicity implies acyclicity.
        • Acyclic but not strict: requires an infinite chain with x_p ≻ x_q only when p > q; D must be infinite.
        • Minimal cycles: non-acyclicity forces a cycle where x_p ≻ x_q exactly when p = q + 1.
        • Finite coincidence: for finite D, acyclicity and strict acyclicity are indistinguishable.
      • Unique Solution Construction
        • Inductive decomposition: A₁ = D; Bᵢ are maxima in Aᵢ; Cᵢ are elements dominated by Bᵢ; Aᵢ₊₁ = Aᵢ − Bᵢ − Cᵢ.
        • Termination: finiteness plus (65:K) guarantees every nonempty Aᵢ has nonempty Bᵢ, so the layers exhaust D.
        • Unique solution: the only solution is the union of all Bᵢ layers; existence and uniqueness are proved together.
      • Unique Solution for All Subsets
        • Refined question: demand a unique solution not just in D, but in every subset E ⊆ D.
        • Strict acyclicity sufficient: every subset inherits strict acyclicity, hence has a unique solution.
        • Acyclicity necessary: a violating cycle E yields two solutions when m is even, none when m is odd.
        • Finite characterization: for finite D, unique solutions in all E are equivalent to acyclicity.
        • Infinite gap: acyclic-but-not-strictly-acyclic infinite sets need a full analysis of (A∞).
      • Bearing on Game Theory
        • No direct application: the imputation space D has no maxima, so strict acyclicity always fails.
        • Ordinary acyclicity fails too: already in the essential three-person game.
        • Acyclic pockets: domination is acyclic in the triangle sets T of 47.5.1 and in the set a of 55.8.2.
    • Utility Generalization, Acyclicity, and Bargaining (Chapter XII Extension of the Concepts of Domination and Solution · III)
      • The Paradox of Acyclic Domination
        • Monopoly and monopsony: these market forms produce an acyclic concept of domination.
        • Extensive solutions: acyclic cases yield families of solutions with parameters, curves, or functions entering them.
        • Uniqueness divide: acyclicity implies uniqueness only for finite D; infinite D needs strict acyclicity, and the market examples fail it.
        • Approximation paradox: integer-utility finite modifications have unique solutions yet approximate original acyclic games with many solutions.
        • Deferred interpretation: the discontinuity is analyzed later through the indivisible-utility example.
      • Utility Generalization: Two Phases
        • Narrow utility: current theory assumes utility numerical, substitutable, and unrestrictedly transferable.
        • Technical roots: numerical utilities handle expectation values; transferability makes imputations vectors and characteristic functions numerical.
        • Two phases: the value of zero-sum two-person games comes first; the n-person characteristic function comes second.
        • Phase-dependence: each phase leans on different utility properties, so modifications must be tested separately.
      • First-Phase Obstacles, Second-Phase Latitude
        • Numerical necessity: complete preference ordering plus combination with probabilities forces numerical utility, as established in 3.5.
        • No value otherwise: without numerical utilities, no value can yet be assigned to a zero-sum two-person game.
        • Characteristic function dependency: v(S) rests on two-person values and transferability, so it inherits the numerical requirement.
        • Directly determined cases: synthesized games, simple games, and markets provide characteristic functions without elaborate two-person theory.
        • Second-phase flexibility: once the characteristic function is given, imputations may have non-numerical, non-transferable components, each with its own utility domain.
        • Composite utility: coalition T needs its own utility domain (T); sums of components presuppose unrestricted transferability.
      • Unifying the Two Phases
        • Same principles: good-strategy analysis in two-person games parallels domination of imputations.
        • Intermediary halt: two-stage procedure divides difficulties but later becomes a handicap.
        • Merged reductions: successive reductions merged all steps except zero-sum two-person and general n-person games.
        • Sharper-than-needed result: the characteristic function forces a sharp value for two-person games, beyond what n-person theory requires.
        • Misplaced insistence: demanding that sharp intermediary result may be the true source of n-person difficulty.
        • Unified remedy: a direct theory of general n-person games without the artificial halt is the likely cure.
      • Indivisible Utility and Bargaining
        • Indivisible units: utility is numerical but indivisible; transfers, and the characteristic function, are integer-valued.
        • Small games unchanged: one- and two-person games behave essentially as before; general three-person case is postponed.
        • One seller, two buyers: this three-person market applies acyclicity results and bears directly on bargaining.
        • Two-part solution: competition and buyer-coalition parts jointly make up the general solution.
        • Competition part: reflects buyers competing; uniquely determined and agrees with ordinary economic ideas.
        • Coalition part: reflects a buyer coalition against the seller; rests on highly arbitrary functional connections.
    • Discrete Utility and Bargaining Advantage (Chapter XII Extension of the Concepts of Domination and Solution · IV)
      • The Discrete Setup
        • Market model: one seller and two equally strong buyers, with v=w and u=0; only one indivisible transaction is possible.
        • Integer utilities: all imputations are integral, making the imputation domain finite.
        • Effective coalitions: only (1,2) and (1,3) can dominate; (2,3) and all one-person sets are unnecessary.
        • Unique solution: finiteness plus acyclicity guarantees exactly one solution by the (65:X) criterion.
      • The Solution and Its Interpretation
        • Solution layers: imputations are removed in stages; the final set consists of alternating families (67:A), (67:B), (67:G), (67:H), ….
        • Total payoff: α1+α2+α3 equals w or w−1, so the maximum social gain need not be attained.
        • Indivisibility cause: failure to reach w comes directly from the existence of an indivisible unit of utility.
        • Buyer symmetry: α2 and α3 differ by at most 1; in the continuous limit they are treated exactly alike.
        • Determinate division: the continuous case's arbitrary rule of division collapses to a uniquely determined equal split.
      • Generalization: Different Discrete Utility Scales
        • Asymmetric setup: α1 and α2 kept integer, while α3 must be even; buyer 3 has a coarser utility scale.
        • Solution pattern: α3=0 throughout the solution; buyer 2 receives the entire advantage.
        • Necessary steps: same inductive removal applies, treating w as w−2 each time because α2 is incremented by 2.
        • Large-w limit: as w→∞ the solution approaches the continuous shape, but the advantage is not equalized.
        • General pattern: varying density of utility ranges can reproduce any continuous division rule, recovering the continuous case's arbitrariness.
      • Conclusions Concerning Bargaining
        • Equal fineness: if both buyers have equally fine utility scales, the division rule treats them equally.
        • Unequal fineness: if one buyer has a finer scale, the entire advantage goes to that buyer.
        • Reversal: reversing the fineness relation reverses the advantage.
        • Continuous indeterminacy: in continuous utility "fineness" is undefined, so many division rules are possible.
        • Discernment matters: this is the first observed case where a player's fineness of subjective utility scale determines bargaining position.
        • Psychological frontier: complete settlement waits for a systematic treatment of the psychological conditions behind utility scales.
  • Appendix: The Axiomatic Treatment of Utility
    • From Utility Axioms to Numerical Representation (Appendix: The Axiomatic Treatment of Utility · I)
      • Goal and Stance
        • Theorem to prove: utility axioms make utility a number up to a linear transformation.
        • Proof strategy: build local interval coordinates, then fit overlapping maps into one global utility function.
        • Complete ordering: axiomatized via > and <, with = interpreted as identity.
        • Notational conflation: same symbols for relations/operations on utilities and numbers; safe if type is kept clear.
        • Deduction style: lengthy but elementary; simple ideas behind voluminous technical execution.
      • Local Interval Mapping
        • Basic monotonicity (A:A): for u<v, raising α strictly increases the mix (1−α)u+αv.
        • Parameterization (A:B): α → (1−α)u₀+αv₀ maps 0<α<1 monotonically into u₀<w<v₀.
        • Surjectivity (A:C): every utility between u₀ and v₀ is hit by exactly one α.
        • Endpoint normalization (A:D): f(u₀)=0, f(v₀)=1, f(w)=α for intermediate w.
        • Mixture coordinates (A:E): f((1−β)u₀+βw)=βf(w); f((1−β)v₀+βw)=1−β+βf(w).
        • Interval uniqueness (A:F): endpoint values plus either mixture law force f uniquely.
      • Limits and Generalized Scales
        • Incompleteness of local map: only covers an interval; fitting across pairs and the mixture property remain open.
        • Arbitrary endpoints (A:G): g(w)=(β₀−α₀)f(w)+α₀ sends u₀→α₀ and v₀→β₀.
        • Affine mixture property (A:H): g preserves convex combinations in its interval.
        • Uniqueness theorem (A:I): endpoint normalization plus mixture preservation fixes g uniquely.
        • Equivalence lemma (A:J): the endpoint affine equation is equivalent to the mixture-property conditions.
      • Fitting Local Maps
        • Interval glueing (A:K): maps on overlapping intervals agree on their common domain.
        • Nested intervals (A:L): the larger interval map agrees with the induced map on any subinterval.
        • Fixed anchors (A:M): with chosen u*,v*, each large interval has a unique map sending them to 0 and 1.
        • Consistency (A:N): larger-interval h coincides with smaller-interval h on the smaller domain.
        • Well-defined h(w) (A:O): h_{u₀,v₀}(w) depends only on w, not on the covering interval.
      • Global Utility Function
        • Covering any pair (A:P): for u<v, choose anchors and an outer interval containing both.
        • Order preservation (A:Q): h(u)<h(v), and h matches interval coordinates on u≤w≤v.
        • Normed representation (A:R): h(u*)=0, h(v*)=1, monotone, and h of a mixture equals the mixture of h values.
        • Global uniqueness (A:S): any utility-to-number map with anchors 0/1 and the mixture equation is exactly h.
    • Completing Utility and Its Limits (Appendix: The Axiomatic Treatment of Utility · II)
      • Extending the Utility Theorem
        • Extension to all u, v: the equality case forces (1 – γ)u + γu = u, so linearity holds universally.
        • A:U: h((1 – γ)u + γv) = (1 – γ)h(u) + γh(v) for all u, v, not just ordered ones.
        • Existence (A:V): monotone numerical utility satisfying linearity for probability combinations exists.
        • Uniqueness (A:W): any two such utilities are affine transforms v′ = ω₀v + ω₁ with ω₀ > 0.
      • Complementarity Clarified
        • No complementarity assumption: u, v are alternative imagined events, never coexisting, so additive combination cannot deny ordinary complementarity.
        • Coalition value v(S): in n-person games, it captures possible complementarity among members’ services.
        • Superadditivity: value of S∪T may exceed the sum of values, formally expressing complementarity between coalitions.
      • Bernoulli Utility and Gambling
        • Bernoulli’s utility: utility of gain dx proportional to dx/x; equal losses weigh more than gains, making fair gambles disadvantageous.
        • Logarithmic utility: satisfying the axioms with utility proportional to ln x, not to money x.
        • No forced indifference: numerical utilities need not make 50–50 equal-risk gambles indifferent.
        • Gambling-specific utility: requires abandoning some axiom; then no numerical utility satisfying expectation calculus exists.
      • Axioms in Question
        • Completeness (3:A:a) is dubious; incomparability u || v may be needed, and indifference curves merely broaden equality.
        • Archimedean property (3:B:c,d) is hard to abandon; rejecting it means allowing infinite utility differences.
        • Combination rule (3:C:b) is the critical group; abandoning it could yield a calculus with utility of gambling.
        • Non-Archimedean ordering of probability mixes is possible but conflicts with normal ideas of preference.
  • Afterword
    • The book's legacy
      • Foundational status: first major publication in game theory, setting the tone for the next half century
      • Mainstream triumph: game theory moved from economics' fringe into its core; 1994 Nobel honored the field
      • Cultural impact: terms like "zero-sum game" and "Prisoner's Dilemma" entered everyday language
      • Mathematical landmark: this book helped transform economics into a mathematical discipline
    • Skepticism about usefulness
      • Prediction critique: strategic situations can be modeled many ways, yielding contradictory predictions
      • Forecasters as players: prediction belongs to the game itself, complicating any social-science forecast
      • Performance doubt: little solid evidence that game theory improves real-life strategic performance
      • Reasoning focus: game theory studies patterns of reasoning in interactive situations, more like logic than prediction
      • Burden of proof: rests on those applying game theory to policy, not on its skeptics
    • Language and interpretation
      • Misleading terms: "strategy" and "solution" invite expectations beyond what formal concepts deliver
      • Ambiguity replaced: formal definitions may merely trade natural-language vagueness for confusing interpretations
      • Mathematical cost: heavy formalism limits access and can hide assumptions behind technical mastery
      • Common-sense evaluation: links between formal models and interpretations rely on ordinary judgment
    • Future challenges
      • Dry well: few major new game-theory ideas have emerged recently, despite the field's success
      • Unconventional minds: the profession needs to cultivate broad thinkers who can generate innovative ideas
      • Missing abilities: memory, information processing, and association quality remain poorly captured by game theory
      • World's complexity: today's real-world challenges are far too complex to be contained in any matrix game
  • Reviews
    • Game Theory for the Social Sciences (Reviews · I)
      • A New Mathematical Foundation for Social Science
        • Mathematical maturity: game theory demands rigorous reasoning, not advanced algebra or calculus.
        • Past attempts: most social-science mathematization was sterile; mathematical economics is the notable exception.
        • Wrong tools: calculus tied to physics failed; social theory needs new mathematics comparable to calculus.
        • Sociological promise: schema general enough to reframe competition, cooperation, and organization.
        • Applications: bilateral monopoly, duopoly, revolutions, and party systems; revolutions require a dynamic theory.
      • Formalizing Games and Utility
        • Game postulates: formal description covers moves, information, and chance, with payoffs as functions of moves.
        • Numerical utility: preference ordering among probabilistic alternatives yields utilities, unique up to linear transformation.
        • Ends-and-means schema: replaced by alternatives, consequences, and values for analyzing rational behavior.
        • Interdependence: each player's outcomes depend on others' choices, enabling rigorous definitions of competition and cooperation.
      • Normalized Play and Chance
        • Strategy catalog: every player picks one complete strategy; secretaries could determine outcomes without playing.
        • Chance as player: randomization enters through expected payoffs computed from probabilities of chance moves.
        • One-player game: Robinson Crusoe economy; simple maximization decides the game.
        • Zero-sum conversion: economics is not zero-sum, but a fictitious player can absorb net gains or losses.
      • Two-Person Zero-Sum and Mixed Strategies
        • Diametric opposition: pure maximin and minimax give bounds; games may be undetermined.
        • Concealment: players randomize to prevent opponents from exploiting known strategies.
        • Determination: with probabilities, bounds converge and every two-person zero-sum game is determined.
        • Convexity proof: expected-payoff vectors form a convex set, the key to existence of optimal mixed strategies.
      • Coalitions, Imputations, and Solutions
        • Characteristic function: v(S) assigns each coalition its guaranteed gain; superadditivity makes cooperation worthwhile.
        • Coalition formation: games with more than two persons generally produce organizations of players who coordinate behavior.
        • Additive case: when v(S) is additive, coalitions are ineffective and the game is determined.
        • Complexity: for n≥6, a game can split into distinct yet interdependent subgames.
        • Imputations: vectors dividing stakes; stable sets arise as trusted patterns of play.
        • Solution: maximal set of mutually undominated imputations; dominance is nontransitive, so discovery is hard.
    • Rationality, Strategy, and Coalition Dynamics (Reviews · II)
      • Three-Person Games and Fictitious Players
        • Geometric constraints: the imputation α must lie within an equilateral triangle centered on the coordinate traces.
        • Dominance regions: α dominates imputations inside three parallelograms sharing the triangle's sides.
        • Discrete solution V: three imputations—(1/2, 1/2, 0), (1/2, 0, 1/2), (0, 1/2, 1/2).
        • Linear solution V_c: a continuum of imputations along a line α₃ = c.
        • Fictitious player: reduces non-zero-sum two-player games to zero-sum three-player form; V_c preserves the original character.
        • Market applications: theory applies to one-buyer–one-seller and two-buyers–one-seller markets.
      • The Rationality Gap in Economics
        • Fundamental gap: oligopoly lacks an adequate definition of rational economic behavior.
        • A priori assumptions: Cournot, Bertrand, and Bowley assume known behavior of others.
        • Logical impasse: others' rationality cannot be known a priori if they are also rational.
        • Novel rejection: rational behavior need not mean narrowly interpreted maximization.
      • Duopoly and Stable Strategy Choice
        • Payoff matrices: each strategy pair determines profits for both duopolists.
        • Leakage-proof choice: A chooses A₃, securing at least 5 even if B learns his strategy.
        • Stable equilibrium: (A₃, B₁) leaves neither player wanting to switch after discovery.
        • Coalition gap: joint profits under cooperation could be higher than the stable solution.
      • Saddle Points, Indeterminacy, and Mixed Strategies
        • Saddle point: highest row minimum equals lowest column maximum.
        • Residual indeterminacy: when absent, a profit surplus depends on outguessing the opponent.
        • Mixed strategies: dice-determined chance coefficients replace pure strategies.
        • Existence theorem: expectation tables always possess a saddle point (von Neumann, 1928).
        • Price of determinacy: expectation criteria and mixed strategies must be accepted.
      • Coalitions and Constant-Sum Reduction
        • Multi-player coalitions: subcoalitions pay even when total profits are constant.
        • Constant-sum convenience: two-person games with constant sum preclude collusion.
        • Fictitious player: non-constant-sum games become constant-sum by adding a dummy participant.
        • Three-person case: the simplest constant-sum setting that admits coalition formation.
        • Derived vs. postulated: game theory derives coalitions; traditional theory assumes them.
    • Game solutions beyond classical economics (Reviews · III)
      • From classical gaps to game theory
        • Traditional economics: ignores coalition formation, collusion, bribery; game theory tackles them directly.
        • Gain is payoff after bribes/deals; an imputation is the set of all players’ gains, not a single number.
        • Exchange economy resembles a game of strategy: each player maximizes against interdependent actions of others.
        • New technique: combinatorics and set theory replace infinitesimal calculus for economic equilibrium.
      • Imputations and domination
        • In three-person constant-sum cases, fallback to acting alone is the saddle point of a two-person game.
        • A coalition forms when it can guarantee members more than their individual fallback imputations.
        • Domination means a coalition could obtain the gains promised by one imputation over another.
        • Domination is not transitive: i₁ can dominate i₂, i₂ dominate i₃, yet i₁ not dominate i₃.
        • An imputation is unrealistic if the coalition cannot actually secure the total gains it promises.
      • Solutions and their structure
        • A solution is a set of imputations with two properties: no internal domination; every outside imputation dominated internally.
        • Removing or adding one imputation destroys the solution; outside imputations may dominate solution members.
        • Multiple solutions exist; geometric circle examples show infinitely many possible sets.
        • Existence of at least one solution is not proved; examples without solution can be constructed.
        • Economically, solutions act as accepted standards of behavior; multiplicity maps to alternative institutional setups.
      • Achievements
        • Discriminatory solutions and collusive deals discovered by pure analysis go beyond classical results.
        • A dummy player converts production gains into a zero-sum game, generalizing from bilateral monopoly toward competition.
        • Results for chess, poker bluffing, and military/diplomatic strategy show broad reach beyond economics.
        • The book is lucid and constructive; a rare event despite complexity.
      • Limitations and critique
        • No general model yet for m sellers and n buyers; monopoly, monopsony, and perfect competition remain unresolved.
        • Harsh opening attacks on existing economic theory are regrettable; imperfect tools still serve social needs.
        • Sparse references ignore Cournot, Chamberlin, Robinson, Frisch, and Stackelberg; economics seems reduced to Böhm-Bawerk plus Pareto.
        • Literary economics is implicitly mathematical; the attack targets elements shared by both forms.
        • Potential is tremendous but largely promise; complex cases lack concrete results.
    • Rethinking Economics via Game Theory (Reviews · IV)
      • The New Approach
        • Participant count changes theory: quality shifts with number of players, not merely complexity.
        • Traditional cases stand: bilateral monopoly and perfect competition need no new results.
        • Rigorous enumeration: new approach exhaustively catalogs alternative outcomes and scrutinizes assumptions.
        • Discrimination clarified: exact definition emerges, with indeterminacy when discrimination is present.
        • Calculus inadequate: social phenomena need mathematical discoveries of comparable stature to calculus.
      • Scientific Method and Economics
        • Limited scope: theory accepts profit motive and treats economic statics only.
        • Formal viewpoint: economics is approached as a formal mathematical problem.
        • Experimental gap: social sciences rarely allow controlled experiments, blocking empirical testing.
        • Dynamic constants absent: no constants in economic dynamics, so natural-science perfection is unattainable.
        • Neutrality doubted: economic facts are interwoven with behavior; burning questions cannot be walled off.
        • Gradual strategy: master a limited field rigorously before extending to wider ones.
      • Utility and Rationality
        • Profit motive accepted: consumers maximize satisfaction; entrepreneurs maximize profits.
        • Complete ordering: preferences are all-embracing, complete, and transitive.
        • No interpersonal utility: results concern one person's utilities, not social welfare comparisons.
        • Social maximum contradiction: greatest good for greatest number is self-contradictory as a guide.
      • Game Taxonomy
        • One, two, many: Crusoe is a maximum problem; two-person adds uncontrolled variables; many adds coalitions.
        • Coalitions emerge: beyond two players, coalitions reduce complex exchanges and dominate three-person analysis.
        • Large-number analogy risky: statistical simplification cannot be assumed without known multiperson mechanics.
        • Zero-sum two-person: simplest nontrivial game; minimax yields rational strategies and strict determination with perfect information.
        • Mixed strategies: probability mixtures protect intentions when perfect information is absent.
        • Beyond zero-sum: non-zero-sum n-person games reduce to zero-sum (n+1)-person via a fictitious player.
      • Solution and Equilibrium
        • Solution as rulebook: rules tell each player how to act in every situation, allowing for others' irrationality.
        • Single imputation: in simple games, one distribution tells each player's rational minimum share.
        • Imputation sets: complex games replace a single imputation with an unordered set of imputations.
        • Cyclical dominance: B over A, C over B, A over C yields multiple possible equilibria.
        • Static theory: dynamic theory is postponed until equilibrium analysis is thoroughly understood.
        • Catalyst not medicine: the work may catalyze social-science thinking rather than cure quantitative problems.
    • Reception of the Game-Theory Classic (Reviews · V)
      • Technical Notes and Foundations
        • Measurability caveat: core theory survives without measurable or transferable utility; substitute "profits" for "utility" if bothered.
        • Mixed strategies: randomizing choices prevents information leakage, since the decider himself doesn't know his strategy.
        • Constant-sum determinateness: in zero-sum play B minimizes A's profits; even without a saddle point, some strategy combinations are excluded.
        • Dominance can cycle: imputations may dominate one another cyclically, so solution requires more than pairwise comparison.
        • Utility via probability: events in a preference map can be equated by assigning probabilities, making utility numerically measurable.
        • Broader applications: the theory may illuminate politics, party systems, monopolies, and international power struggles.
      • Heads, I Win, and Tails, You Lose — Paul Samuelson
        • Landmark judgment: Theory of Games and Economic Behavior is genius that fed research but failed to revolutionize economics.
        • Math literacy: the book's symbols are Greek to many; mathematical ability is not rare and schools wrongly destroy it.
        • Cultural test: game theory, like entropy, could define 20th-century literacy.
        • Perfect information trivializes: chess and tic-tac-toe are determinate; random mixed strategies maximize your weakest point against a rational adversary.
        • Randomize to conceal: flipping a coin prevents information leakage; poker bluffing, quiz design, and bargaining rules all shift.
        • Limits of game theory: with three players, coalitions are unpredictable; mathematics cannot settle philosophical life-death choices.
      • Big D — Paul Crume
        • Satirical rejection: von Neumann's math contradicts "Great Natural Laws"; adding numbers or taking square roots never gives reliable answers.
        • Logical Repulsion: every force creates an equal force in an unforeseen direction, so causes are inevitably misidentified.
        • Imponderables rule: a dinner forecast collapses under a cat, neighbor, and lawyer; prediction's web of accidents defeats science.
        • Lesser Natural Laws: "things that never happen usually do"; "things equal to the same thing usually mess up."
      • Mathematics of Games and Economics — E. Rowland
        • Economic man as player: individual wealth maximization matches game winnings, but some variables are uncontrolled or unknown.
        • Past errors and method: vague formulations and pseudo-proofs doomed early attempts; proceed from a mastered modest field.
        • Formal equivalence: games can be described extensively or normalized; ten axioms reduce to three.
        • Coalitions emerge: two-person games replace maximization with opposition; three-person games add shifting alliances.
        • Special classes needed: systematic n-person theory stalls beyond five; qualitative novelty at n = 6 forces narrow solvable families.
        • Dropping zero-sum: contact with familiar economics begins when the zero-sum restriction is dropped, but notation excludes non-mathematicians.
      • Theory of Games — Claude Chevalley
        • Frequent attempts: mathematical methods have been applied to economics many times, but without decisive success.
        • Source of diagnosis: the review credits von Neumann and Morgenstern with the explanation; the passage breaks off mid-sentence.
    • The Mathematical Theory of Strategy (Reviews · VI)
      • The Axiomatic Method
        • Physics analogy fails: differential equations predict the future without strategic interdependence.
        • Economics as a game: finite players follow rules and seek maximum advantage.
        • Axioms first: mathematics requires a precise axiomatic description of the situation.
        • Pure deduction then interpretation: logic yields theorems; concrete meaning returns only at the end.
        • Strategy defined: a complete rule telling a player what to do in every circumstance.
      • Two-Player Games and Value
        • Zero-sum solution: one player's gain is the other's loss; the best-strategy question is answered.
        • Perfect information: chess guarantees each player a best strategy—forced win or ensured tie.
        • Imperfect information: solved with mixed strategies, choosing pure strategies by explicit probabilities.
        • Value of the game: player 1 can secure at least v; player 2 can ensure no more than v.
        • Limited control: each player chooses only their own strategy, not opponents' moves.
      • Coalitions and Stable Standards
        • Coalitions reduce to duels: groups split into two coalitions that play as two-player games.
        • Coalition value: minimum a group can force even against all other players united.
        • Characteristic function: the game rests on values assigned to every possible coalition.
        • Imputations: outcomes include rule-based returns and side payments within coalitions.
        • Solutions as stable standards: no coalition prefers another solution imputation; outsiders are worse off.
        • Open problem: existence of a solution for every game is unproved; several can coexist.
      • The Book and Its Reception
        • Accessible to beginners: requires only elementary algebra; advanced notions are defined in the text.
        • Examples enrich theory: special examples and verbal explanations keep meaning clear.
        • Professional breakthrough: the American Economic Review called the work "indeed a rare event."
        • New analytical tools: logic, set theory, and functional analysis replace the traditional probability calculus.
        • Broad scope: techniques apply to political science, sociology, and military strategy, not just economics.
        • Sociological split: reviewers called it either a model of clear exposition or an interesting but useless development.
      • Economic and Military Implications
        • Economic applications: duopoly, oligopoly, coalitions, cartels; rational and irrational behavior included.
        • Duopoly determinacy: seller A picks A₃, seller B picks B₁; neither would change after learning the other's choice.
        • Identity, not analogy: economic problems become strictly identical to games of strategy.
        • Military adoption: Project Rand, Navy, and the Army's Advanced Study Group pursue game theory.
        • Poker as strategy: von Neumann's poker analysis was the essence of strategy, with security implications.
        • Bomb-like hope: a Pentagon scientist hoped "Games" would work as they hoped the atomic bomb would work.
    • Strategy, Games, and Economic Theory (Reviews · VII)
      • A New Foundation for Economics?
        • Origins: von Neumann and Morgenstern developed game theory for economics; its effect is stimulating if not revolutionary.
        • Critics' verdict: it raises new problems and challenges existing market theory, even if it cannot rebuild the science.
        • Accessibility: the mathematical core is closed to laymen, but clear outlines reward the intrepid reader.
      • Strategy in Duel and Market
        • Strategy defined: interaction of persons whose acts rest on expectations about uncontrolled others.
        • Skill and chance: business is a mixed game of hunch and calculation; game theory narrows the gamble to an optimum policy.
        • Duel dilemma: first shot versus better shot are conflicting maximums; Project Rand solved it mathematically.
        • Silent guns: a combatant may not know when the opponent fired, adding uncertainty to the duel.
        • Buyer-seller price: each maximizes only by anticipating the other; price resolves conflicting maximum hopes.
      • Oligopoly, Macro, and Utility
        • Imperfect competition: modern industries are oligopolies where sellers must account for rivals' market influence.
        • Seagram vs. Schenley: each faced irreconcilable brand-sales and profit goals; strategic moves defied classical formulas.
        • Combination omitted: classical pure-competition theory misses the combinations typical of corporate, union, and association life.
        • Keynesian aggregates: General Theory cannot explain strategic manipulation of aggregates, e.g. tax relief amid inflation.
        • Utility measurable: game theory holds that consistent preferences can be expressed numerically, challenging marginalists.
      • Games as Economic Models
        • Why games: strategies are simple, observable, and abstractable; games align with economic life's essential character.
        • Three situations: one man against nature, two-person exchange, and coalitions of two against a third.
        • Rules uncertainty: markets would be games if their rules were fully known; unknown rules make theory harder.
        • Strategical vs. nonstrategical: probability suffices for solitaire, craps, roulette, and chess's perfect information; bridge and poker require strategy.
        • Information central: strategy turns on seeking, signaling, or obscuring information.
      • Minimax and Poker
        • Found-out premise: assume your strategy is already known to your opponent; this removes fear of surprise.
        • Randomization: conceal choices by randomizing—fifty-fifty pennies or irregular bluffs on a controlled basis.
        • Minimax theorem: in two-person zero-sum games, best-play floor equals minimax; proofs are uncontested and irrational opponents surrender gains.
        • Poker's structure: stripped to two players, two bets, and one raise, poker models bluffing and conflicting intents.
    • Bluffing, Coalitions, and Military Games (Reviews · VIII)
      • Bluffing as Inverted Signaling
        • Inverted signaling: bluffing fakes strength in weakness or weakness in strength.
        • Two motives: aggressive bluff succeeds when opponent passes; deceptive bluff succeeds when opponent calls.
        • Correct strategy: bet high on high hands, mostly low on weak hands, with occasional irregular bluffs.
        • Defensive value: bluffing mainly protects against opponents’ deviations, not wins against good players.
        • Countermeasures: penalize incorrect bluffing by irregular “seeing” with medium hands.
        • Two varieties: aggressive initiative and defensive enforcing of minimax.
      • From Two-Man to Three-Man Games
        • Pure opposition breaks: three-person games add one-man, two-against-one, and pure three-person situations.
        • Coalition imperative: rational players must combine when rules allow; refusal risks smaller gains.
        • Two-man prelude: the old two-person game sets the value of any coalition’s internal bargain.
        • Coalition price: staying costs no more than the opponent could pay to break the coalition.
        • Economic inference: not combining invites loss by competitors’ combinations, as antitrust pressures show.
      • The Set Back Model
        • Model game: Set Back, a three-player auction card game, shows coalitions forming and dissolving.
        • First coalition: trailing players must give tricks to stop a leader; enforcement is the threat of giving him the game.
        • Negative coalition: when two strong players threaten, each feeds the weak to deny the other victory.
        • Weakness as strength: under some conditions weakness survives; “fittest” need not always win.
        • Standards of behavior: custom, prejudice, and ethics shape coalition play and economic competition.
      • Large Numbers and Static Limits
        • Combinatorial explosion: ten players can form 511 opposing coalitions; four-person play becomes two-vs-two or three-vs-one.
        • Reduction by organization: trade unions compress many economic individuals into few strategic units.
        • Static and small-number: theory is avowedly static and lacks large numbers; development limits, not objections.
        • Gas analogy: statistical physics suggests ideal free competition might one day be calculable.
        • Open economics questions: which competition gives fuller employment; do profit, production, and welfare coincide?
        • Decision focus: game theory studies man as decision-maker, not inanimate nature.
      • Military Games and Minimax
        • Wartime origin: ASWORG applied von Neumann’s 1928 poker paper to antisubmarine warfare before the full theory appeared.
        • Canonical forms: Duel, Deployment, and Search are military problems translatable into games.
        • Colonel Blotto: deployment problem; mixed strategy guarantees a minimum score against any enemy deployment.
        • Mixed strategy wins: randomizing between “3 and 1” and “4–1’s” gives minimax better than pure deployments.
        • Search problem: plane vs submarine is a two-person pattern-vs-counterpattern game; rational play enforces minimax.
        • Mechanization: optimal play may need chance devices and precomputed calculations, perhaps no pilot.
    • Minimax, Bluff, and Game Theory's Birth (Reviews · IX)
      • Minimax and War
        • War as experiment: opposing forces rarely measure strength accurately unless great disparity.
        • Minimax philosophy: war is chance; optimal play makes the enemy’s attack too costly.
        • Strategy illuminated: game theory clarifies the meaning of strategy even before practical applications.
        • Social promise: game theory may someday apply in many social fields.
      • Military Optimization
        • Air Force problem: maximize bomb tonnage subject to matériel, training, and supply constraints.
        • Linear programming: linking maximum military desires to resources is only at early stages.
        • Two-person game: interacting activities and constraints can be cast as a game.
        • Computation: ~500 factors and billions of multiplications demand new electronic calculators.
      • Rocket Bluff and Counter-Bluff
        • Rocket defense: no maximum continent protection; interception must be far out in tiny time.
        • Bluffing: many rockets are decoys; destroying all may cost more than the damage inflicted.
        • Optimum mix: detection and interception levels should make enemy attack too costly to chance.
        • Poker rule: punish over-bluffing by bluffing less on weak hands, more on strong ones.
      • Morgenstern’s Road
        • Prediction paradox: predictions affect events; perfect foresight leads to inadmissible equilibria.
        • Dead vs. live variables: isolated economies maximize; live variables are other wills interfering or enhancing.
        • Holmes-Moriarty: out-thinking cannot settle pursuit; only an arbitrary strategic decision can.
        • Čech’s hint: von Neumann’s 1928 game paper matched Morgenstern’s prediction problems, but circumstances delayed study.
      • The Collaboration
        • Princeton meeting: after-luncheon talk with Bohr and Veblen led to games-and-experiments discussions.
        • Joint project: von Neumann proposed co-authorship; a planned pamphlet became Theory of Games and Economic Behavior.
        • Expected utility: axioms produced numerical utility up to a linear transformation.
        • Probability choice: classical frequency basis allowed a later subjective-probability extension.
        • Expanding economy: von Neumann’s 1937 model impressed Morgenstern but fell flat on the economics seminar.
    • Forging Game Theory: A Collaboration (Reviews · X)
      • Intellectual Context
        • Pioneering isolation: no mathematical economists at Princeton were ready for von Neumann's new ideas.
        • First notice: Morgenstern's Professor Hicks on Value and Capital introduced von Neumann's expanding-economy model.
        • Inequalities, not equations: economics is inherently about inequalities; von Neumann fully agreed.
        • Primitive mathematics: von Neumann called pre-game economic writing Newton-era, "a million miles" from physics.
      • Game Theory Foundations
        • Expanding-economy restriction: assumed every good enters every other's production; Morgenstern found it too narrow.
        • KMT generalization: Kemeny, Thompson, and Morgenstern removed the restriction, winning von Neumann's approval.
        • Minimax theorem at core: KMT showed game theory can serve as a scientific calculus, not only a model.
        • Borel's book accident: spotting Jean Ville's proof in Borel's treatise redirected their proof strategy.
      • Collaborative Working Style
        • Endless joint sessions: apartment meetings, handwritten pages, and typed duplicates — no secretarial help.
        • Klari's elephant joke: they promised to hide an elephant diagram, and the book contains one.
        • Breakfast planning: mornings at the Nassau Club shaped the day's afternoon work.
        • Elementary proofs preferred: von Neumann and Morgenstern avoided advanced mathematics whenever possible.
      • Publication Journey
        • Press shock: Princeton University Press saw 1,200 pages when expecting a 100-page pamphlet.
        • Subsidies and retyping: $500 grants funded retyping; a Japanese mathematician filled in formulas.
        • Title settled: rejected General Theory of Rational Behavior for Theory of Games and Economic Behavior.
        • Anonymous donation: a friend's gift overcame the press's wartime risk; publication came September 1944.
      • Reception and Aftermath
        • Early reviews: Hurwicz, Marschak, and Wald gave immediate, substantial expositions.
        • New York Times sensation: a 1946 front-page article sold out the printing, forcing a second edition.
        • Utility appendix: second edition proved the numerical-utility axioms and added gambling/ordering notes.
        • Global diffusion: translations, conferences, journal, and 6,200-plus bibliography items by 1970.
        • Last conversations: von Neumann worked on automata and computers before his death.
• Core Conclusion and Practical Takeaways
  • Core Insight: Economics as Games of Strategy
    • Strategic interdependence: each participant maximizes against other living wills, not passive nature.
    • Characteristic function: v(S) assigns every coalition its guaranteed worth, the foundation of solution theory.
    • Coalitions decide outcomes: compensation and competing alliances shape how any partnership splits gains.
    • Solution as standard of behavior: a stable set of imputations expresses social norms, not a single prediction.
    • Non-zero-sum reality: dropping the zero-sum restriction restores productivity and mutual gain to the model.
  • Practical Framework: Model and Solve
    • Normalize the game: compress extensive rules into complete strategies and a payoff matrix.
    • Use mixed strategies: randomize whenever pure choices can be found out or no saddle point exists.
    • Compute minimax bounds: value lies between maximin and minimax; equality marks a saddle point.
    • Build coalition values: compute v(S) from two-person subgames against the complement coalition.
    • Test solution stability: verify no internal domination and that every outside imputation is dominated.
  • Daily Strategic Practices
    • Assume your strategy is known: choose defensively, then conceal remaining uncertainty by randomization.
    • Negotiate compensation: side payments and outside alternatives determine partners' final split.
    • Signal deliberately: bluffing is inverted signaling; its cost and frequency must be rationally balanced.
    • Restrain exploitation: stable discriminatory settlements often leave the excluded party above his floor.
    • Respect initiative: the first mover can bluff aggressively; the last mover defends by selective seeing.
  • Mindset Shifts
    • Abandon unique prediction: accept sets of stable outcomes rather than one final verdict.
    • Expect intransitive domination: x over y and y over z need not imply x over z.
    • Separate essential from inessential: only essential games make coalition strategy decisive.
    • Measure utility operationally: preferences plus probability fix utility up to a linear transformation.
    • Embrace multiplicity: several valid standards of behavior can coexist for the same objective game.