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A Chronology of Game Theory by Paul Walker May 2001 Chronology | Ancient | 1700 | 1800 | 1900 | 1950 | 1960 | 1970 | 1980 | 1990 | Nobel Prize |

0-500AD The Babylonian Talmud is the compilation of ancient law and tradition set down during the first five centuries A.D. which serves as the basis of Jewish religious, criminal and civil law. One problem discussed in the Talmud is the so called marriage contract problem: a man has three wives whose marriage contracts specify that in the case of this death they receive 100, 200 and 300 respectively. The Talmud gives apparently contradictory recommendations. Where the man dies leaving an estate of only 100, the Talmud recommends equal division. However, if the estate is worth 300 it recommends proportional division (50,100,150), while for an estate of 200, its recommendation of (50,75,75) is a complete mystery. This particular Mishna has baffled Talmudic scholars for two millennia. In 1985, it was recognised that the Talmud anticipates the modern theory of cooperative games. Each solution corresponds to the nucleolus of an appropriately defined game. 1713 In a letter dated 13 November 1713 James Waldegrave provided the first, known, minimax mixed strategy solution to a two-person game. Waldegrave wrote the letter, about a two-person version of the card game le Her, to Pierre-Remond de Montmort who in turn wrote to Nicolas Bernoulli, including in his letter a discussion of the Waldegrave solution. Waldegrave''s solution is a minimax mixed strategy equilibrium, but he made no extension of his result to other games, and expressed concern that a mixed strategy "does not seem to be in the usual rules of play" of games of chance 1838 Publication of Augustin Cournot''s Researches into the Mathematical Principles of the Theory of Wealth. In chapter 7, On the Competition of Producers, Cournot discusses the special case of duopoly and utilises a solution concept that is a restricted version of the Nash equilibrium 1871 In the first edition of his book The Descent of Man, and Selection in Relation to Sex Charles Darwin gives the first (implicitly) game theoretic argument in evolutionary biology. Darwin argued that natural section will act to equalize the sex ratio. If, for example, births of females are less common than males, then a newborn female will have better mating prospects than a newborn male and therefore can expect to have more offspring. Thus parents genetically disposed to produce females tend to have more than the average numbers of grandchildren and thus the genes for female-producing tendencies spread, and female births become commoner. As the 1:1 sex ratio is approached, the advantage associated with producing females dies away. The same reasoning holds if males are substituted for females throughout. Therefore 1:1 is the equilibrium ratio. 1881 Publication of Francis Ysidro Edgeworth''s Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. Edgeworth proposed the contract curve as a solution to the problem of determining the outcome of trading between individuals. In a world of two commodities and two types of consumers he demonstrated that the contract curve shrinks to the set of competitive equilibria as the number of consumers of each type becomes infinite. The concept of the core is a generalisation of Edgeworth''s contract curve. 1913 The first theorem of game theory asserts that in chess either white can force a win, or black can force a win, or both sides can force at least a draw. This ''theorem'' was published by Ernst Zermelo in his paper Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels and hence is referred to as Zermelo''s Theorem. Zermelo''s results were extended and generalised in two papers by Denes Konig and Laszlo Kalmar. The Kalmar paper contains the first proof of Zermelo''s theorem since Zermelo''s own paper did not give one. An English translation of the Zermelo paper, along with a discussion its significance and its relationship to the work of Konig and Kalmar is contained in Zermelo and the Early History of Game Theory by U. Schwalbe and P. Walker. 1921-27 Emile Borel published four notes on strategic games and an erratum to one of them. Borel gave the first modern formulation of a mixed strategy along with finding the minimax solution for two-person games with three or five possible strategies. Initially he maintained that games with more possible strategies would not have minimax solutions, but by 1927, he considered this an open question as he had been unable to find a counterexample. 1928 John von Neumann proved the minimax theorem in his article Zur Theorie der Gesellschaftsspiele. It states that every two- person zero-sum game with finitely many pure strategies for each player is determined, ie: when mixed strategies are admitted, this variety of game has precisely one individually rational payoff vector. The proof makes involved use of some topology and of functional calculus. This paper also introduced the extensive form of a game. 1930 Publication of F. Zeuthen''s book Problems of Monopoly and Economic Warfare. In chapter IV he proposed a solution to the bargaining problem which Harsanyi later showed is equivalent to Nash''s bargaining solution. 1934 R.A. Fisher independently discovers Waldegrave''s solution to the card game le Her. Fisher reported his work in the paper Randomisation and an Old Enigma of Card Play. 1938 Ville gives the first elementary, but still partially topological, proof of the minimax theorem. Von Neumann and Morgenstern''s (1944) proof of the theorem is a revised, and more elementary, version of Ville''s proof. 1944 Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern is published. As well as expounding two-person zero sum theory this book is the seminal work in areas of game theory such as the notion of a cooperative game, with transferable utility (TU), its coalitional form and its von Neumann-Morgenstern stable sets. It was also the account of axiomatic utility theory given here that led to its wide spread adoption within economics. 1945 Herbert Simon writes the first review of von Neumann-Morgenstern. 1946 The first entirely algebraic proof of the minimax theorem is due to L. H. Loomis''s, On a Theorem of von Neumann, paper. 1950 Contributions to the Theory of Games I, H. W. Kuhn and A. W. Tucker eds., published. 1950 In January 1950 Melvin Dresher and Merrill Flood carry out, at the Rand Corporation, the experiment which introduced the game now known as the Prisoner''s Dilemma. The famous story associated with this game is due to A. W. Tucker, A Two-Person Dilemma, (memo, Stanford University). Howard Raiffa independently conducted, unpublished, experiments with the Prisoner''s Dilemma. 1950 John McDonald''s Strategy in Poker, Business and War published. This was the first introduction to game theory for the general reader. 1950-53 In four papers between 1950 and 1953 John Nash made seminal contributions to both non-cooperative game theory and to bargaining theory. In two papers, Equilibrium Points in N- Person Games (1950) and Non-cooperative Games (1951), Nash proved the existence of a strategic equilibrium for non-cooperative games-the Nash equilibrium-and proposed the "Nash program", in which he suggested approaching the study of cooperative games via their reduction to non-cooperative form. In his two papers on bargaining theory, The Bargaining Problem (1950) and Two-Person Cooperative Games (1953), he founded axiomatic bargaining theory, proved the existence of the Nash bargaining solution and provided the first execution of the Nash program. 1951 George W. Brown described and discussed a simple iterative method for approximating solutions of discrete zero-sum games in his paper Iterative Solutions of Games by Fictitious Play. 1952 The first textbook on game theory was John Charles C. McKinsey, Introduction to the Theory of Games. 1952 Merrill Flood''s report, (Rand Corporation research memorandum, Some Experimental Games, RM-789, June), on the 1950 Dresher/Flood experiments appears. 1952 The Ford Foundation and the University of Michigan sponsor a seminar on the "Design of Experiments in Decision Processes" in Santa Monica. This was the first experimental economics/experimental game theory conference 1952-53 The notion of the Core as a general solution concept was developed by L. S. Shapley (Rand Corporation research memorandum, Notes on the N-Person Game III: Some Variants of the von-Neumann-Morgenstern Definition of Solution, RM- 817, 1952) and D. B. Gillies (Some Theorems on N-Person Games, Ph.D. thesis, Department of Mathematics, Princeton University, 1953). The core is the set of allocations that cannot be improved upon by any coalition. 1953 Lloyd Shapley in his paper A value for N-Person Games characterised, by a set of axioms, a solution concept that associates with each coalitional game,v, a unique out-come, v. This solution in now known as the Shapley value. 1953 Lloyd Shapley''s paper Stochastic Games showed that for the strictly competitive case, with future payoff discounted at a fixed rate, such games are determined and that they have optimal strategies that depend only on the game being played, not on the history or even on the date, ie: the strategies are stationary. 1953 Extensive form games allow the modeller to specify the exact order in which players have to make their decisions and to formulate the assumptions about the information possessed by the players in all stages of the game. H. W. Kuhn''s paper, Extensive Games and the Problem of Information includes the formulation of extensive form games which is currently used, and also some basic theorems pertaining to this class of games. 1953 Contributions to the Theory of Games II, H. W. Kuhn and A. W. Tucker eds., published. 1954 One of the earliest applications of game theory to political science is L. S. Shapley and M. Shubik with their paper A Method for Evaluating the Distribution of Power in a Committee System. They use the Shapley value to determine the power of the members of the UN Security Council. 1954-55 Differential Games were developed by Rufus Isaacs in the early 1950s. They grew out of the problem of forming and solving military pursuit games. The first publications in the area were Rand Corporation research memoranda, by Isaacs, RM-1391 (30 November 1954), RM-1399 (30 November 1954), RM-1411 (21 December 1954) and RM-1486 (25 March 1955) all entitled, in part, Differential Games. 1955 One of the first applications of game theory to philosophy is R. B. Braithwaite''s Theory of Games as a Tool for the Moral Philosopher. 1957 Games and Decisions: Introduction and Critical Survey by Robert Duncan Luce and Howard Raiffa published. 1957 Contributions to the Theory of Games III, M. A. Dresher, A. W. Tucker and P. Wolfe eds., published. 1959 The notion of a Strong Equilibrium was introduced by R. J. Aumann in the paper Acceptable Points in General Cooperative N-Person Games. 1959 The relationship between Edgeworth''s idea of the contract curve and the core was pointed out by Martin Shubik in his paper Edgeworth Market Games. One limitation with this paper is that Shubik worked within the confines of TU games whereas Edgeworth''s idea is more appropriately modelled as an NTU game. 1959 Contributions to the Theory of Games IV, A. W. Tucker and R. D. Luce eds., published. 1959 Publication of Martin Shubik''s Strategy and Market Structure: Competition, Oligopoly, and the Theory of Games. This was one of the first books to take an explicitly non-cooperative game theoretic approach to modelling oligopoly. It also contains an early statement of the Folk Theorem. Late 50''s Near the end of this decade came the first studies of repeated games. The main result to appear at this time was the Folk Theorem. This states that the equilibrium outcomes in an infinitely repeated game coincide with the feasible and strongly individually rational outcomes of the one-shot game on which it is based. Authorship of the theorem is obscure. 1960 The development of NTU (non-transferable utility) games made cooperative game theory more widely applicable. Von Neumann and Morgenstern stable sets were investigated in the NTU context in the Aumann and Peleg paper Von Neumann and Morgenstern Solutions to Cooperative Games Without Side Payments. 1960 Publication of Thomas C. Schelling''s The Strategy of Conflict. It is in this book that Schelling introduced the idea of a focal-point effect. 1961 The first explicit application to evolutionary biology was by R. C. Lewontin in Evolution and the Theory of Games. 1961 The Core was extended to NTU games by R. J. Aumann in his paper The Core of a Cooperative Game Without Side Payments. 1962 In their paper College Admissions and the Stability of Marriage, D. Gale and L. Shapley asked whether it is possible to match m women with m men so that there is no pair consisting of a woman and a man who prefer each other to the partners with whom they are currently matched. Game theoretically the question is, does the appropriately defined NTU coalitional game have a non-empty core? Gale and Shapley proved not only non-emptiness but also provided an algorithm for finding a point in it. 1962 One of the first applications of game theory to cost allocation is Martin Shubik''s paper Incentives, Decentralized Control, the Assignment of Joint Costs and Internal Pricing. In this paper Shubik argued that the Shapley value could be used to provide a means of devising incentive-compatible cost assignments and internal pricing in a firm with decentralised decision making. 1962 An early use of game theory in insurance is Karl Borch''s paper Application of Game Theory to Some Problems in Automobile Insurance. The article indicates how game theory can be applied to determine premiums for different classes of insurance, when required total premium for all classes is given. Borch suggests that the Shapley value will give reasonable premiums for all classes of risk. 1963 O. N. Bondareva established that for a TU game its core is non-empty iff it is balanced. The reference, which is in Russian, translates as Some Applications of Linear Programming Methods to the Theory of Cooperative Games. 1963 In their paper A Limit Theorem on the Core of an Economy G. Debreu and H. Scarf generalised Edgeworth, in the context of a NTU game, by allowing an arbitrary number of commodities and an arbitrary but finite number of types of traders. 1964 Robert J. Aumann further extended Edgeworth by assuming that the agents constitute a (non-atomic) continuum in his paper Markets with a Continuum of Traders. 1964 The idea of the Bargaining Set was introduced and discussed in the paper by R. J. Aumann and M. Maschler, The Bargaining Set for Cooperative Games. The bargaining set includes the core but unlike it, is never empty for TU games.

1964 Lemke, Carlton E. and J. T. Howson, Jr. (1964), Equilibrium Points of Bimatrix Games, Society for Industrial and Applied Mathematics Journal of Applied Mathematics 12, 413-423. 1965 Isaacs, Rufus (1965), Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. New York: Wiley. 1965 Selten, R. (1965), Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit, Zeitschrift fur die gesamte Staatswissenschaft 121, 301-324 and 667-689. 1965 Davis, M. and M. Maschler (1965), The Kernel of a Cooperative Game, Naval Research Logistics Quarterly 12, 223-259. 1966 Aumann, R. J. and M. Maschler (1966), Game-Theoretic Aspects of Gradual Disarmament, Chapter V in Report to the U.S. Arms Control and Disarmament Agency ST-80. Princeton: Mathematica. 1966 Harsanyi, J. C. (1966), A General Theory of Rational Behavior in Game Situations, Econometrica 34, 613-634. 1967 Shapley, L. S. (1967), On Balanced Sets and Cores, Naval Research Logistics Quarterly 14, 453-460. 1967 Scarf, H. E. (1967), The Core of a N-Person Game, Econometrica 35, 50-69. 1967-68 Harsanyi, J. C. (1967-8), Games with Incomplete Information Played by ''Bayesian'' Players, Parts I, II and III, Management Science 14, 159-182, 320-334 and 486-502. 1968 Lucas, W. F. (1968), A Game with No Solution, Bulletin of the American Mathematical Society 74, 237-239. 1969 Schmeidler, D. (1969), The Nucleolus of a Characteristic Function Game, Society for Industrial and Applied Mathematics Journal of Applied Mathematics 17, 1163-1170. 1969 Shapley, L. S. (1969), Utility Comparison and the Theory of Games, pp. 251-263 in La Decision, Paris: Editions du Centre National de la Recherche Scientifique. (Reprinted on pp. 307-319 of The Shapley value (Alvin E. Roth, ed.), Cambridge: Cambridge University Press, 1988). 1969 Shapley, L. S. and M. Shubik (1969), On Market Games, Journal of Economic Theory 1, 9-25. 1972 Maynard Smith, John (1972), Game Theory and the Evolution of Fighting, pp.8-28 in On Evolution (John Maynard Smith), Edinburgh: Edinburgh University Press. 1973 Harsanyi, J. C. (1973), Games with Randomly Distured Payoffs: A New Rationale for Mixed Strategy Equilibrium Points, International Journal of Game Theory 2, 1-23. 1973 Maynard Smith, John and G. A. Price (1973), The Logic of Animal Conflict, Nature 246, 15-18. 1973 Gibbard, A. (1973), Manipulation of Voting Schemes: A General Result, Econometrica 41, 587-601. 1974 Aumann, R. J. and L. S. Shapley (1974), values of Non-Atomic Games. Princeton: Princeton University Press. 1974 Aumann, R. J. (1974), Subjectivity and Correlation in Randomized Strategies, Journal of Mathematical Economics 1, 67-96. 1975 Selten, R. (1975), Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games, International Journal of Game Theory 4, 25-55. 1975 Kalai, E. and M. Smorodinsky (1975), Other Solutions to Nash''s Bargaining Problem, Econometrica 43, 513-518. 1975 Faulhaber, G. (1975), Cross-Subsidization: Pricing in Public Enterprises, American Economic Review 65, 966-977. 1976 Lewis, D. K. (1969), Convention: A Philosophical Study. Cambridge Mass.: Harvard University Press. 1976 Aumann, R. J. (1976), Agreeing to Disagree, Annals of Statistics 4, 1236-1239. 1977 Littlechild, S. C. and G. F. Thompson (1977), Aircraft Landing Fees: A Game Theory Approach, Bell Journal of Economics 8, 186-204. 1981 Kohlberg, Elon (1981), Some Problems with the Concept of Perfect Equilibria, Rapporteurs'' Report of the NBER Conference on the Theory of General Economic Equilibrium by Karl Dunz and Nirvikar Sing, University of Californa Berkeley. 1981 Aumann, R. J. (1981), Survey of Repeated Games, pp.11-42 in Essays in Game Theory and Mathematical Economics in Honor of Oskar Morgenstern (R. J. Aumann et al), Zurich: Bibliographisches Institut. (This paper is a slightly revised and updated version of a paper originally presented as background material for a one-day workshop on repeated games that took place at the Institute for Mathematical Studies in the Social Sciences (Stanford University) summer seminar on mathematical economics on 10 August 1978.) (A slightly revised and updated version of the 1981 version is reprinted as Repeated Games on pp. 209-242 of Issues in Contemporary Microeconomics and Welfare (George R Feiwel, ed.), London: Macmillan.) 1982 Kreps, D. M. and R. B. Wison (1982), Sequential Equilibria, Econometrica 50, 863-894. 1982 Rubinstein, A. (1982), Perfect Equilibrium in a Bargaining Model, Econometrica 50, 97-109. 1982 Maynard Smith, John (1982), Evolution and the Theory of Games. Cambridge: Cambridge University Press. 1984 Roth, A. E. (1984), The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory, Journal of Political Economy 92, 991-1016. 1984 Bernheim, B. D. (1984), Rationalizable Strategic Behavior, Econometrica 52, 1007-1028. 1984 Pearce, D. G. (1984), Rationalizable Strategic Behavior and the Problem of Perfection, Econometrica 52, 1029-1050. 1984 Axelrod, R. (1984), The Evolution of Cooperation. New York: Basic Books. 1985 Mertens, J.-F. and S. Zamir (1985), Formulation of Bayesian Analysis for Games with Incomplete Information, International Journal of Games Theory 14, 1-29. 1985-86 Neyman, A. (1985), Bounded Complexity Jusifies Cooperation in the Finitely Repeated Prisoner''s Dilemma, Economic Letters 19, 227-229. 1985-86 Rubinstein, A. (1986), Finite Automata Play the Repeated Prisoner''s Dilemma, Journal of Economic Theory 39, 83-96. 1986 Kohlberg, E. and J.-F. Mertens (1986), On the Strategic Stability of Equilibria, Econometrica 54, 1003-1037. 1988 Harsanyi, J. C. and R. Selten (1988), A General Theory of Equilibrium Selection in Games. Cambridge Mass.: MIT Press. 1988 Tan, T. and S. Werlang (1988), The Bayesian Foundations of Solution Concepts of Games, Journal of Economic Theory 45, 370-391. 1990 Kreps, D. M. (1990), A Course in Microeconomic Theory. Princeton: Princeton University Press. 1990 Crawford, V. P. (1990), Equilibrium without Independence, Journal of Economic Theory 50,127-154. 1991 Fudenberg, D. and J. Tirole (1991), Perfect Bayesian Equilibrium and Sequential Equilibrium, Journal of Economic Theory 53, 236-260. 1992 Aumann, R. J. and S. Hart, eds. (1992), Handbook of Game Theory with Economic Applications, Volume 1. Amsterdam: North-Holland. 1994 Baird, Douglas G., Robert H. Gertner and Randal C. Picker (1994), Game Theory and the Law. Cambridge Mass.: Harvard University Press. 1994 Aumann, R. J. and S. Hart, eds. (1994), Handbook of Game Theory with Economic Applications, Volume 2. Amsterdam: North-Holland.

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