AN OUTLINE OF THE HISTORY OF GAME THEORY
1 April 1995
This outline was written by Paul Walker in the early 1990s and originally circulated
by e-mail, and I coded it for html as a self-teaching project in 1995. For some time
Dr. Walker had an improved version served from the University of Canterbury in New
Zealand, but it seems no longer to be served. Several sites link to this version,
and accordingly I will maintain it for now. RM
Timeline:
Ancient Times
1700
1800
1900
1950
1960
1970
1980
1990
The Present
- 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 socalled 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'sResearches 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.
- 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 chess is strictly determined, ie:
chess has only one individually rational payoff profile in pure strategies. This
theorem was published by E. Zermelo in his paper Uber eine Anwendung
der Mengenlehre auf die Theorie des Schachspiels and hence is referred to as
Zermelo's Theorem.
- 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</ a> 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.
- 1950-53
- 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-PersonGames 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 Surveyby
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 paperAcceptable 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 paperVon 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
inEvolution 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 paperMarkets 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
- Carlton E. Lemke and J.T. Howson, Jr., describe an algorithm for finding a Nash
equilibrium in a bimatrix game, thereby giving a constructive proof of the existence
of an equilibrium point, in their paper Equilibrium Points in Bimatrix
Games. The paper also shows that, except for degenerate situations, the number
of equilibria in a bimatrix game is odd.
- 1965
- Publication of Rufus Isaacs's Differential Games: A Mathematical
Theory with Applications to Warfare and Pursuit, Control and Optimization.
- 1965
- R. Selten, Spieltheoretische Behandlung eines Oligopolmodellsmit
Nachfragetraegheit. In this article Selten introduced the idea of refinements
of the Nash equilibrium with the concept of (subgame) perfect equilibria.
- 1965
- The concept of the Kernel is due to M. Davis and M. Maschler, The
Kernel of a Cooperative Game. The kernel is always included in the bargaining
set but is often much smaller.
- 1966
- Infinitely repeated games with incomplete information were born in a paper by
R. J.Aumann and M. Maschler, Game-Theoretic Aspects of Gradual Disarmament.
- 1966
- In his paper A General Theory of Rational Behavior in GameSituations
John Harsanyi gave the, now, most commonly used definition to distinguish between
cooperative and non-cooperative games. A game is cooperative if commitments--agreements,
promises, threats--are fully binding and enforceable. It is non-cooperative if commitments
are not enforceable.
- 1967
- Lloyd Shapley, independently of O.N. Bondareva, showed that the core of a TU
game is non-empty iff it is balanced in his paper On Balanced Sets
and Cores.
- 1967
- In the articleThe Core of a N-Person Game, H. E. Scarf extended
the notion of balancedness to NTU games, then showed that every balanced NTU game
has a non-empty core.
- 1967-68
- In a series of three papers,Games with Incomplete Information
Played by 'Bayesian' Players, Parts I, II and III, John Harsanyi constructed
the theory of games of incomplete information. This laid the theoretical groundwork
for information economics that has become one of the major themes of economics and
game theory.
- 1968
- The long-standing question as to whether stable sets always exist was answered
in the negative by William Lucas in his paper A Game with no Solution.
- 1969
- David Schmeidler introduced the Nucleolus in this paper The
Nucleolus of a Characteristic Game. The Nucleolus always exists, is unique, is
a member of the Kernel and for any non- empty core is always in it.
- 1969
- Shapley defined a value for NTU games in his article Utility
Comparison and the Theory of Games.
- 1969
- For a coalitional game to be a market game it is necessary that it and all its
subgames have non-empty cores, ie: that the game be totally balanced. In Market
Games L. S. Shapley and Martin Shubik prove that this necessary condition is
also sufficient.
- 1972
- International Journal of Game Theory was founded by Oskar Morgenstern.
- 1972
- The concept of an Evolutionarily Stable Strategy (ESS), was introduced to evolutionary
game theory by John Maynard Smith in an essay Game Theory and The
Evolution of Fighting. The ESS concept has since found increasing use within
the economics (and biology!) literature.
- 1973
- In the traditional view of strategy randomization, the players use a randomising
device to decide on their actions. John Harsanyi was the first to break away from
this view with his paper Games with Randomly Disturbed Payoffs:
A New Rationale for Mixed Strategy Equilibrium Points. For Harsanyi nobody really
randomises. The appearance of randomisation is due to the payoffs not being exactly
known to all; each player, who knows his own payoff exactly, has a unique optimal
action against his estimate of what the others will do.
- 1973
- The major impetus for the use of the ESS concept was the publication of J. Maynard
Smith and G. Price's paper The Logic of Animal Conflict.
- 1973
- The revelation principle can be traced back to Gibbard's paper Manipulation
of Voting Schemes: A General Result
- 1974
- Publication of R. J. Aumann and L. S. Shapley's book Values
of Non-Atomic Games. It deals with values for large games in which all the players
are individually insignificant (non-atomic games).
- 1974
- R. J. Aumann proposed the concept of a correlated equilibrium in his paper Subjectivity and Correlation in Randomized Strategies.
- 1975
- The introduction of trembling hand perfect equilibria occurred in the paper Reexamination of the Perfectness Concept for Equilibrium Points in
Extensive Games by Reinhard Selten. This paper was the true catalyst for the
'refinement industry' that has developed around the Nash equilibrium.
- 1975
- E. Kalai and M. Smorodinsky, in their articleOther Solutions
to Nash's Bargaining Problem, replace Nash's independence of irrelevant alternatives
axiom with a monotonicity axiom. The resulting solution is known as the Kalai-Smorodinsky
solution.
- 1975
- In his paper Cross-Subsidization: Pricing in Public Enterprises,
G. Faulhaber shows that the set of subsidy-free prices are those prices for which
the resulting revenue (ri = piqi for given demand levels qi) vector lies in the core
of the cost allocation game.
- 1976
- An event is common knowledge among a set of agents if all know it and all know
that they all know it and so on ad infinitum. Although the idea first appeared in
the work of the philosopher D. K. Lewis in the late 1960s it
was not until its formalisation in Robert Aumann's Agreeing to Disagree
that game theorists and economists came to fully appreciate its importance.
- 1977
- S. C. Littlechild and G. F. Thompson are among the first to apply the nucleolus
to the problem of cost allocation with their articleAircraft Landing
Fees:A Game Theory Approach. They use the nucleolus, along with the core and
Shapley value, to calculate fair and efficient landing and take-off fees for Birmingham
airport.
- 1981
- Elon Kohlberg introduced the idea of forward induction in a conference paper
Some Problems with the Concept of Perfect Equilibria.
- 1981
- R. J. Aumann published a Survey of Repeated Games.This survey
firstly proposed the idea of applying the notion of an automaton to describe a player
in a repeated game. A second idea from the survey is to study the interactive behaviour
of bounded players by studying a game with appropriately restricted set of strategies.
These ideas have given birth to a large and growing literature.
- 1982
- David M. Kreps and Robert Wilson extend the idea of a subgame perfect equilibrium
to subgames in the extensive form that begin at information sets with imperfect information.They
call this extended idea of equilibrium sequential. It is detailed in their paper
Sequential Equilibria.
- 1982
- A. Rubinstein considered a non-cooperative approach to bargaining in his paperPerfect Equilibrium in a Bargaining Model. He considered an alternating-offer
game were offers are made sequentially until one is accepted. There is no bound on
the number of offers that can be made but there is a cost to delay for each player.Rubinstein
showed that the subgame perfect equilibrium is unique when each player's cost of
time is given by some discount factor delta.
- 1982
- Publication of Evolution and the Theory of Games by John
Maynard Smith.
- 1984
- Following the work of Gale and Shapley, A. E. Roth applied the core to the problem
of the assignment of interns to hospitals. In his paper The Evolution
of the Labour Market for Medical Interns and Residents: A Case Study in Game Theoryhe
found that American hospitals developed in 1950 a method of assignment that is a
point in the core.
- 1984
- The idea of a rationalizability was introduced in two papers; B. D. Bernheim,
Rationalizable Strategic Behavior and D. G. Pearce,Rationalizable
Strategic Behavior and the Problem of Perfection.
- 1984
- Publication of The Evolution of Cooperation by Robert Axelrod.
- 1985
- For a Bayesian game the question arises as to whether or not it is possible to
construct a situation for which there is no sets of types large enough to contain
all the private information that players are supposed to have. In their paper,Formulation of Bayesian Analysis for Games with Incomplete Information,
J.-F. Mertens and S. Zamir show that it is not possible to do so.
- 1985-86
- Following Aumann, the theory of automata is now being used to formulate the idea
of bounded rationality in repeated games. Two of the first articles to take this
approach were A. Neyman's 1985 paper Bounded Complexity Justifies
Cooperation in the Finitely Repeated Prisoner's Dilemma and A. Rubinstein's 1986
article Finite Automata Play the Repeated Prisoner's Dilemma.
- 1986
- In their paper On the Strategic Stability of EquilibriaElon
Kohlberg and Jean-Francois Mertens deal with the problem of he refinement of Nash
equilibria in the normal form, rather than the extensive form of a game as with the
Selten and Kreps and Wilson papers. This paper is also one of the first, published,
discussions of the idea of forward induction.
- 1988
- John C. Harsanyi and Reinhard Selten produced the first general theory of selecting
between equilibria in their book A General Theory of Equilibrium
Selection in Games. They provide criteria for selecting one particular equilibrium
point for any non-cooperative or cooperative game.
- 1988
- With their paper The Bayesian Foundations of Solution Concepts
of Games Tan and Werlang are among the first to formally discuss the assumptions
about a player's knowledge that lie behind the concepts of Nash equilibria and rationalizability.
- 1988
- One interpretation of the Nash equilibrium is to think of it as an accepted (learned)
'standard of behaviour' which governs the interaction of various agents in repetitions
of similar situations. The problem then arises of how agents learn the equilibrium.
One of the earliest works to attack the learning problem was Drew Fudenberg and David
Kreps's A Theory of Learning, Experimentation and Equilibria, (MIT and Stanford Graduate
School of Business, unpublished), which uses an learning process similar to Brown's
fictitious play, except that players occasionally experiment by choosing strategies
at random, in the context of iterated extensive form games. Evolutionary game models
are also commonly utilised within the learning literature.
- 1989
- The journal Games and Economic Behavior founded.
- 1990
- The first graduate level microeconomics textbook to fully integrate game theory
into the standard microeconomic material was David M. Krep's A Course
in Microeconomic Theory.
- 1990
- In the article Equilibrium without Independence Vincent
Crawford discusses mixed strategy Nash equilibrium when the players preferences do
not satisfy the assumptions necessary to be represented by expected utility functions.
- 1991
- An early published discussion of the idea of a Perfect Bayesian Equilibrium is
the paper by D. Fudenberg and J. Tirole, Perfect Bayesian Equilibrium
and Sequential Equilibrium.
- 1992
- Publication of the Handbook of Game Theory with Economic Applications,
Volume 1 edited by Robert J. Aumann and Sergiu Hart.
- 1994
- Game Theory and the Law by Douglas G. Baird, Robert H.
Gertner and Randal C. Picker is one of the first books in law and economics to take
an explicitly game theoretic approach to the subject.
- 1994
- Publication of the Handbook of Game Theory with Economic Applications,
Volume 2 edited by Robert J. Aumann and Sergiu Hart.
- 1994
- The Central Bank of Sweden Prize in Economic Science in Memory of Alfred Nobel
was award to John Nash, John C. Harsanyi and Reinhard Selten for their contributions
to Game Theory.
Bibliography and Notes
0 - 500AD
- The Talmud results are from Aumann, R. J. and M. Maschler,( 1985), Game Theoretic
Analysis of a Bankruptcy Problem from the Talmud, Journal of Economic Theory 36,
195-213.
- 1713
- On Waldegrave see Kuhn, H. W.(1968), Preface to Waldegrave's Comments: Excerpt
from Montmort's Letter to Nicholas Bernoulli, pp. 3-6 in Precursors in Mathematical
Economics: An Anthology (Series of Reprints of Scarce Works on Political Economy,
19) (W. J. Baumol and S. M. Goldfeld,eds.), London: London School of Economics and
Political Science and Waldegrave's Comments: Excerpt from Montmort's Letter to Nicholas
Bernoulli, pp. 7-9 in Precursors in Mathematical Economics: An Anthology (Series
of Reprints of Scarce Works on Political Economy, 19) (W. J. Baumol and S. M. Goldfeld,
eds.), London: London School of Economics and Political Science, 1968.
- 1838
- Cournot, Augustin A. (1838),Recherches sur les Principes Mathematiquesde la Theorie
des Richesses. Paris: Hachette. (English translation: Researches into the Mathematical
Principles of the Theory of Wealth. New York: Macmillan, 1897. (Reprinted New York:
Augustus M. Kelley, 1971)).
- 1881
- Edgeworth, Francis Ysidro (1881), Mathematical Psychics: An Essay on the Application
of Mathematics to the Moral Sciences. London: Kegan Paul. (Reprinted New York: Augustus
M. Kelley, 1967).
- 1913
- Zermelo, E. (1913), Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels,
pp. 501-504 in Proceedings of the Fifth International Congress of Mathematicians,
Volume II (E. W. Hobson and A. E. H. Love, eds.), Cambridge: Cambridge University
Press.
- 1921-27
- This follows Dimand, Robert W. and Mary Ann Dimand (1992), The Early History
of the Theory of Games from Waldegrave to Borel, pp. 15-27 in Toward a History of
Game Theory (Annual Supplement to Volume 24 History of Political Economy) (E. Roy
Weintraub, ed.), Durham: Duke University Press. Frechet, Maurice (1953), Emile Borel,
Initiator of the Theory of Psychological games and its Application, Econometrica
21, 95-96, credits Borel with seven notes on game theory between 1921 and 1927. The
Frechet seven are: (1) La theorie du jeu et les equations integrales a noyan symetrique
gauche, Comptes Rendus Academie des Sciences, Vol. 173, 1921, pp. 1304-1308. (2)
Sur les jeux ou interviennent l'hasard et l'habilete des joueurs, Association Francaise
pour l'Advancement des Sciences,1923, pp. 79-85. (3) Sur les jeux ou interviennent
l'hasard et l'habilete des joueurs, Theorie des Probabilites. Paris: Librairie Scientifique,
J. Hermann, (1924), pp. 204-224. (4) Un theoreme sur les systemes de formes lineaires
a determinant symetrique gauche, Comptes Rendus Academie des Sciences, Vol. 183,
1926, pp. 925-927, avec erratum, p. 996 . (5)Algebre et calcul des probabilites,
Comptes Rendus Academie des Sciences, Vol. 184, 1927,pp. 52-53. (6) Traite du calcul
des probabilites et de ses applications, Applications des jeux de hasard. Paris:
Gauthier-Villars, Vol. IV, 1938, Fascicule 2, 122 pp. (7) Jeux ou la psychologie
joue un role fondamental, see (6) pp. 71-87. Dimand and Dimand note that (6) and
(7) are dated 1938 and so are outside the 1921-1927 time frame while article (2)
has the same title as the chapter from the book (3). Three of Borel's notes were
translated and published in Econometrica 21(1953). (1) was published as Theory of
Play and Integral Equations with Skew Symmetric Kernels, pp. 91-100. (3) was published
as On Games that involve Chance and the Skill of the Players, pp. 101-115. (5) was
published as On Systems of Linear Forms of Skew Symmetric Determinant and the General
Theory of Play, pp.116-117.
- 1928
- von Neumann, J. (1928), Zur Theorie der Gesellschaftsspiele, Mathematische Annalen100,
295-320. (Translated as "On the Theory of Games of Strategy", pp.13-42
in Contributions to the Theory of Games, Volume IV (Annals of Mathematics Studies,
40) (A.W. Tucker and R. D. Luce, eds.), Princeton University Press, Princeton, 1959).
- 1930
- Zeuthen, F. (1930), Problems of Monopoly and Economic Warfare. London: George
Routledge and Sons. The mathematical equivalence of Zeuthen's and Nash's solutions
was shown by Harsanyi, J. C. (1956), Approaches to the Bargaining Problem Before
and After the Theory of Games: A Critical Discussion of Zeuthen's, Hicks' and Nash's
Theories, Econometrica 24, 144-157.
- 1934
- Fisher, R. A. (1934), Randomisation, and an Old Enigma of Card Play, Mathematical
Gazette 18, 294-297.
- 1938
- Ville, Jean (1938), Note sur la theorie generale des jeux ou intervient l'habilite
desjouers, pp. 105-113 in Applications aux jeux de hasard, Tome IV, Fascicule II
of Traite du calcul des probabilities et de ses applications (Emile Borel), Paris:
Gauthier-Villars.
- 1944
- von Neumann, J., and O. Morgenstern (1944), Theory of Games and Economic Behavior.
Princeton: Princeton University Press.
- 1945
- Simon, H. A. (1945), Review of the Theory of Games and Economic Behavior by J.
von Neumann and O. Morgenstern, American Journal of Sociology 27, 558-560.
- 1946
- Loomis, L. H. (1946), On a Theorem of von Neumann, Proceedings of the National
Academy of Sciences of the United States of America 32, 213-215.
- 1950
- Kuhn, H. W. and A. W. Tucker, eds. (1950), Contributions to the Theory of Games,Volume
I (Annals of Mathematics Studies, 24). Princeton: Princeton University Press.
- 1950
- Publication of Tucker's (1950) memo occurred in 1980 under the title On Jargon:
The Prisoner's Dilemma, UMAP Journal 1, 101.
- 1950-1953
- Nash, J. F. (1950), Equilibrium Points in N-Person Games,
Proceedings of the National Academy of Sciences of the United States of America 36,
48-49.
- Nash, J. F.(1951),Non-Cooperative Games,
Annals of Mathematics 54, 286-295.
- Nash, J. F.(1950), The Bargaining Problem,
Econometrica 18, 155-162.
- Nash, J. F.(1953), Two Person Cooperative
Games, Econometrica 21, 128-140.
- 1951
- Brown, G. W. (1951), Iterative Solution of Games by Fictitious Play, pp. 374-376
in Activity Analysis of Production and Allocation (T. C. Koopmans, ed.), New York:
Wiley.
- 1952
- McKinsey, John Charles C. (1952), Introduction to the Theory of Games. New York:
McGraw-Hill Book Co.
- 1952
- Flood's 1952 Rand memorandum was published in Flood, M. A.(1958), Some Experimental
Games, Management Science 5, 5-26.
- 1952
- Some of the experimental papers from the conference appear in Thrall, R. M.,
C. H. Coombs and R. C. Davis, eds. (1954), Decision Processes. New York: Wiley.
- 1952-53
- Gillies published version of the core concept appears in his paper, Gillies,
D. B. (1959), Solutions to General Non-Zero-Sum Games, pp. 47-85 in Contributions
to the Theory of Games, Volume IV (Annals of Mathematics Studies, 40) (A. W. Tucker
and R. D. Luce, eds.), Princeton:Princeton University Press.
- 1953
- Shapley, L. S. (1953), A Value for n-Person Games, pp. 307-317 in Contributions
to the Theory of Games, Volume II (Annals of Mathematics Studies, 28) (H. W. Kuhn
and A. W.Tucker, eds.), Princeton: Princeton University Press.
- 1953
- Shapley, L. S. (1953), Stochastic Games, Proceedings of the National Academy
of Sciences of the United States of America 39, 1095-1100.
- 1953
- Kuhn, H. W. (1953), Extensive Games and the Problem of Information, pp. 193-216
in Contributions to the Theory of Games, Volume II (Annals of Mathematics Studies,
28) (H.W. Kuhn and A. W. Tucker, eds.), Princeton: Princeton University Press.
- 1953
- Kuhn, H. W. and A. W. Tucker, eds. (1953), Contributions to the Theory of Games,Volume
II (Annals of Mathematics Studies, 28). Princeton: Princeton University Press
- 1954
- Shapley, L. S. and M. Shubik (1954), A Method for Evaluating The Distribution
of Power in a Committee System, American Political Science Review 48, 787-792.
- 1955
- Braithwaite, R. B. (1955), Theory of Games as a Tool for the Moral Philosopher.
Cambridge: Cambridge University Press.
- 1957
- Luce, R. Duncan and Howard Raiffa (1957), Games and Decisions: Introduction and
Critical Survey. New York: Wiley. (Reprinted New York: Dover, 1989).
- 1957
- Dresher, Melvin, A. W. Tucker and P. Wolfe, eds. (1957), Contributions to the
Theory of Games, Volume III (Annals of Mathematics Studies, 39). Princeton: Princeton
University Press.
- 1959
- Aumann, R. J. (1959), Acceptable Points in General Cooperative N-Person Games,
pp.287-324 in Contributions to the Theory of Games, Volume IV (Annals of Mathematics
Studies, 40) (A. W. Tucker and R. D. Luce, eds.), Princeton: Princeton University
Press.
- 1959
- Shubik, M. (1959), Edgeworth Market Games, pp. 267-278 in Contributions to the
Theory of Games, Volume IV (Annals of Mathematics Studies, 40) (A. W. Tucker and
R. D.Luce, eds.), Princeton: Princeton University Press.
- 1959
- Tucker, A. W. and R. D. Luce, eds. (1959), Contributions to the Theory of Games,Volume
IV (Annals of Mathematics Studies, 40). Princeton: Princeton University Press.
- 1959
- Shubik, M. (1959), Strategy and Market Structure: Competition, Oligopoly, and
the Theory of Games. New York: Wiley.
- 1960
- Aumann, R. J. and B. Peleg (1960), Von Neumann-Morgenstern Solutions to Cooperative
Games without Side Payments, Bulletin of the American Mathematical Society 66, 173-179.
- 1960
- Schelling, T. C. (1960), The Strategy of Conflict. Cambridge, Mass.: Harvard
University Press.
- 1961
- Lewontin, R. C. (1961), Evolution and the Theory of Games, Journal of Theoretical
Biology 1, 382-403.
- 1961
- Aumann, R. J. (1961), The Core of a Cooperative Game Without Side Payments, Transactions
of the American Mathematical Society 98, 539-552.
- 1962
- Gale, D. and L. S. Shapley (1962), College Admissions and the Stability of Marriage,
American Mathematics Monthly 69, 9-15.
- 1962
- Shubik, M. (1962), Incentives, Decentralized Control, the Assignment of Joint
Costs and Internal Pricing, Management Science 8, 325-343.
- 1962
- Borch, Karl (1962), Application of Game Theory to Some Problems in Automobile
Insurance, The Astin Bulletin 2 (part 2), 208-221.
- 1963
- Debreu, G. and H. Scarf (1963), A Limit Theorem on the Core of an Economy, International
Economic Review 4, 235-246.
- 1964
- Aumann, R. J. (1964), Markets with a Continuum of Traders, Econometrica 32,39-50.
- 1964
- Aumann, R. J. and M. Maschler (1964), The Bargaining Set for Cooperative Games,
pp.443-476 in Advances in Game Theory (Annals of Mathematics Studies, 52) (M. Dresher,
L. S. Shapley and A. W. Tucker, eds.), Princeton: Princeton University Press.
- 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 and667-689.
- 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 Disturbed 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 California 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.
Paul Walker.................. p.walker@econ.canterbury.ac.nz
Department of Economics...... Phone +64 3 3642 033
University of Canterbury......Fax +64 3 3642 635
Private Bag 4800
Christchurch
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