Encyclopedia of Nonlinear Science
Humans play games. From the formalized warfare of chess to the Machiavellian machinations of politics to the subtleties of sexual pursuit, interactions between individuals with desires and priorities that are often conflicting and contradictory lies at the heart of human society. The formal study of games, however, is a relatively recent phenomenon and has moved rapidly from its beginnings as a mathematical tool to aid gamblers to its current status as an essential paradigm in fields as diverse as economics and evolutionary biology.
Today, game theory can be defined as “the analysis of rational behavior under circumstances of strategic interdependence, when an individual’s best strategy depends upon what his opponents are likely to do” (Varoufakis, 2001). Individuals may be people, corporations, nations, animals, species, or any other entity that can be said to exhibit strategic behavior.
Formal game theory began in 1713, when Pierre-Rémond de Montmort first proposed the concept of a minimax solution to a card game called Le Her. A minimax solution to a two-player game is one in which an individual chooses his strategy so as to minimize the maximum loss or risk that he/she will incur. It was James Waldegrave, the originator of Le Her, who actually produced a minimax solution to the game.
Other concepts of fundamental importance to modern game theory also emerged in the 18th century, a result of work by Daniel Bernoulli in his 1738 analysis of the St. Petersburg paradox (Dimand & Dimand, 1992). These concepts included utility (a measure of the desirability to the player of each possible outcome of the game), the maximization of expected utility, diminishing marginal utility (the decrease in the amount of benefit derived from consuming each additional unit of a product or service), and risk aversion as a parameter of a utility function.
The work on minimax game solutions remained an isolated curiosity until the 1920s, when Émile Borel published a series of short papers on strategic games in the Proceedings of the French Academy of Sciences between 1921 and 1927. In these papers, he defined the normal form of a game: a matrix representation of the game in which each player tries to work out the best strategy independent of the sequence of moves. Borel later claimed to have proven the minimax theorem, but this does not appear to be the case. In fact, he may not even have stated the theorem.
The first formal proof of the minimax theorem for two-person games with any finite number of pure strategies was given by John von Neumann in a paper presented to the Gottingen Mathematical Society on December 7, 1926 (Dimand & Dimand, 1992). He proved that in a zero-sum game (one in which one player’s gain is the other’s loss), there exists a unique set of mixed strategies, one per player, which equalizes the payoffs that each player can gain regardless of the strategy adopted by the other player.
At about the same time, the Princeton economist Oskar Morgenstern was pondering mixed strategy game theoretic issues, as exemplified by that master of bluff, Sherlock Holmes, and the strategies he should adopt to avoid his arch enemy, Professor Moriarty. In 1944, von Neumann and Morgenstern collaborated to produce The Theory of Games and Economic Behavior, the seminal publication in this area. In 1947, they revised the book to include expected utility theory, under which games are expressed in terms of the players’ perceptions of the inherent desirability and likelihood of their outcomes, and players never expect other players to hold mistaken beliefs—the assumption of complete rationality, which has been fundamental to much ensuing work. Economists were initially reluctant to accept game theory, but since the publication of von Neumann and Morgenstern’s book, the theory has undergone extensive development, and has been applied to an enormous variety of problems in economics, to the point where Leonard (1992) could assert that “game theory plays a central role in economic theory.”
Minimax theory assumes that each player has perfect knowledge about the game and that the game is zero-sum. In such a case, the best strategy for each player is independent of the strategy adopted by the other player. In most real-world situations, this is not the case; more often, the best strategy for one player depends on what the other players choose to do. The extension of minimax theory to n-player, noncooperative games was achieved by the Princeton mathematician John Nash in a paper published in 1950. In this paper, he defined the Nash equilibrium. A Nash equilibrium is a set of strategies such that no player could improve his/her payoff, given the strategies of all other players in the game, by changing his/her strategy. Nash proved that all noncooperative games have a Nash equilibrium and, thereby, established an analytical structure within which all situations of conflict and cooperation could be studied. For this work, he received the 1994 Nobel Prize for Economics, together with John Harsanyi and Reinhard Selten.
Game theory was also applied with considerable success to other fields of research, perhaps most notably to evolutionary biology. The concepts of game theory transfer readily to evolutionary biology—the values of different outcomes, which in economic theory are measured as utility, are readily interpreted in terms of Darwinian fitness. Moreover, the somewhat sweeping assumption of complete rationality in the behavior of the agents is replaced by the concept of evolutionary stability.
Game-theoretic concepts were first explicitly applied to the study of evolution by Lewontin (1961), who saw the agents in the game as a species on the one hand, against nature on the other. The utility of this approach was quickly recognized, and the focus shifted to modeling interactions between individuals. In this context, the concept of the Nash equilibrium was rediscovered independently by John Maynard Smith and G.R.Price in 1973 as the Evolutionarily Stable Strategy (ESS—“a strategy such that, if all the members of a population adopt it, no other strategy can invade” (Maynard Smith, 1982, p. 204). An ESS represents the solution to a game.
In the last three decades, game theory has been used to provide a framework for the analysis of a wide range of biological phenomena, including the evolution of sex ratios, parental investment in offspring, patterns of animal dispersal, competition for resources (see, e.g., Maynard Smith, 1982) and the evolution of cooperation (Axelrod, 1984). In biology, as in economics, game theory has become an essential tool for the theorist, providing a structured framework for the analysis of a wide range of phenomena.
JENNIFER HALLINAN
See also Artificial life; Biological evolution; Economic system dynamics
Further Reading
Axelrod, R. 1984. The Evolution of Cooperation, New York: Basic Books
Dimand, R.W. & Dimand, M. 1992. The early history of the theory of strategic games from Waldegrave to Borel. In Toward a History of Game Theory, edited by E.R.Weintraub, Durham: Duke University Press
Leonard, R.J. 1992. Creating a context for game theory. In Toward a History of Game Theory, edited by E.R.Weintraub, Durham: Duke University Press
Lewontin, R.C. 1961. Evolution and the theory of games. Journal of Theoretical Biology, 1:382–403
Maynard Smith, J. 1982. Evolution and the Theory of Games, Cambridge: Cambridge University Press
Nash, J. 1950. Equilibrium points in N-person games. Proceedings of the National Academy of Sciences, 36: 48–49
Varoufakis, Y. 2001. General introduction: game theory’s quest for a single, unifying framework for the social sciences. In Game Theory: Critical Concepts in the Social Sciences, edited by Y.Varoufakis, vol. 1. London and New York: Routledge
von Neumann, J. & Morganstern, O. 1953. Theory of Games and Economic Behavior, 3rd edition, New York: Wiley (1st edition, 1944)
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