Book Review of A Game of Thrones

A Game of Thrones is the first book in A Song of Ice and Fireseries by American Author George R. R. Martin. It waspublished in 1996, won the 1997 Locus Award(1997 Award Winners & Nominees, 1997),and was nominated for both the 1997 Nebula Award[1] and the 1997 World Fantasy Award(1997 Award Winners & Nominees, 1997).

It is a great masterpiece. Martin created a new fantasy world with a variety of magicalelements including dragons, monster races called the Others and evenabnormal seasons that winter or summer can last for tens of years.The story starts on the land called Westeros and tella cruel political conflictamong the greed Queen Cersei Lannister together with her son who tried to rule the Seven Kingdom, the Starks who were willing to protect the King’s Iron Thrones and the Targaryens who wanted to recapture their lands. The title comes from Queen Cersei’s quotes on page 471: "When you play the game of thrones, you win or you die. There is no middle ground."(A Game of Thrones, 2013)

In the book, what extremely amaze readers was how real the fictive world was, and I attribute it to the author’s excellent writing skills on plot building and character development(Contrarian, 2013).

Martin told the story by following three principal storylines simultaneously(A Game of Thrones, 2013), in the Seven Kingdoms, on the Wall and in the East. Skipping among storylines cut the story into several splendid plots. The story is filled with suspensebecause it is hard for readers to followall the information or details at once. However, at the end of the book, when the three storylines which seem separated before intersected, all the haze fade away. Additionally, the story was told from multiple POVs (Point of View)(A Game of Thrones Book Review, 2011). Every chapter was told by one of the main characters (8 in total). This method of telling stories add reality by putting readers into the plots and on the positions of characters in the book so that readers could imagine themselves as the roles and totally comprehendcharacters’ mind, choices and actions.The following part is taken as an example.

“The ground was so far below him he could barely make it out through the grey mists that whirled around him, but he could feel how fast he was falling, and he knew what was waiting for him down here. Even in dreams, you could not fall forever. He would wake up in the instant before he hit the ground, he knew. You always woke up in the instant before you hit the ground.The ground was closer now, still far far away, a thousand miles away, but closer than it had been. It was cold here in the darkness. There was no sun, no stars, only the ground below coming up to smash him, and the grey mists. He wanted to cry”(Martin, 1996)

This paragraph was whatBran, Eddard’s little 7-year-child, thought while he was falling from a high tower.Reading the sentences, I felt like I was falling with Bran. The fear, the loneliness, the hopelessness, all these negative emotion came to me when the ground got closer and closer. There are a number of similar paragraphs in the book, greatly increased the reality and attraction.

It is more brilliant that by fiercely contrast between different characters’ point of view, readers could thoroughly seize roles’ trails and personality. Eddard Stark, Warden of the North and Lord of Winterfell, impressed the readers as a faithful, straight and clemency gentleman. He is a considerate husband to his wife, a loving father to his children and a merciful lord to inhabitants on his land and men under his hands. As the Hand of the King, when the King died and the greedy Queen attempt to usurp the throne,Eddard gave up the chance that detaining the Queen and her brother before they could act(A Game of Thrones, 2013). He refused to use his household guardsmen to take control of the throne during the King’s Died Night because of his loyalty to his friend, King Robert. He could not bear to do anything beyond his authority or harmful to the Kingdom andas a result, he was arrested and executed. Eddard’s death extremely shocked readers and his faith and loyalty were extruded while contrasted with the Queen’s greed and evil.

Martin’s book is not only a fantasy fiction but also a reflection of the real world. The core of the book are humanity and reality. All the characters symbolize real people and all plot’s development follow real regulation. It is the reality of the fiction that attract amount of readers from all over the world to read the book.

Reference

1997 Award Winners & Nominees. (1997). from: Worlds Without End:

https:///books_year_index.asp?year=1997

A Game of Thrones. (2013-10-22). from: Wikipedia:

/wiki/A_Game_of_Thrones#cite_note-WWE-1997-1

A Game of Thrones Book Review. (2011). from: The Literary Snob:

/post/5787280732/a-game-of-thrones-book-review

Contrarian Glenn. (2013-5-13). Book Review: A Game of Thrones Series by George R. R. Martin.

from: Blogcritics: http://blogcritics.org/book-review-a-game-of-thrones1/

MartinR. R.George. (1996). A Game of Thrones.

 

第二篇：博弈编年史 A Chronology of Game Theory

A Chronology of Game Theory (zz)

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

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 Oligopolmodells mit 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 Game Situations 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 article Other 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 article Aircraft 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 paper Perfect 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 Theory he 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 Equilibria Elon 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 player 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 Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel was award to John Nash, John C. Harsanyi and Reinhard Selten "for their pioneering analysis of equilibria in the theory of non-cooperative games".

2005

The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel was award to Robert J. Aumann and Thomas C. Schelling "for having enhanced our understanding of conflict and cooperation through game-theory analysis".

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)).

1871

Darwin, C. (1871), The Descent of Man, and Selection in Relation to Sex. London: John Murray. This theory of the evolution of the sex ratio is normally attributed to R. A. Fisher (The Genetical Theory of Natural Selection. Oxford: Clarendon Press, 1930). Before presenting the theory Fisher quotes a paragraph from the second (1874) edition of Darwin's Descent of Man in which Darwin cannot see how a 1:1 sex ratio could be the result of natural section. Fisher appears not to have noticed that the paragraph he quotes comes from a section which replaces the section in the first edition which contains the essence of Fisher's own theory. The fact that Darwin had anticipated Fisher by some 60 years was first noted by Michael Bulmer in his 1994 book, Theoretical Evolutionary Ecology. Sunderland, MA: Sinauer Associates Publishers. See chapter 10, pages 207 - 208. This fact is also discussed in an unpublished paper by Martin Osborne; Darwin, Fisher, and a Theory of the Evolution of the Sex Ratio. See martin j. osborne's recent research

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. The reference for the Konig paper is Konig, Denes (1927), Uber eine Schlussweise aus dem Endlichen ins Unendliche, Acta Sci. Math. Szeged 3, 121-130 while the Kalmar reference is Kalmar, Laszlo (1928/29), Zur Theorie der abstrakten Spiele, Acta Sci. Math. Szeged 4, 65-85. The English translation of Zermelo's paper and discussion of all three papers is in Schwalbe, U. and P. Walker (2001), Zermelo and the Early History of Game Theory, Games and Economic Behavior v34 no1, 123-37.

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 Annalen 100, 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 des jouers, 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

McDonald, John (1950), Strategy in Poker, Business and War. New York: Norton. This book based on two articles McDonald wrote for Fortune magazine. The first, Poker, An American Game (March, 1948) and the second, A Theory of Strategy (June, 1949).

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 ans 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 theTheory 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 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.