# Game Theory

## The Origins Of Game Theory

Writings by several nineteenth-century economists, such as A. A. Cournot and Joseph Bertrand on duopoly and F. Y. Edgeworth on bilateral monopoly, and later work in the 1930s by F. Zeuthen on bargaining and H. von Stackelberg on oligopoly, were later reinterpreted in game-theoretic terms, sometimes in problematic ways (Leonard, 1994; Dimand and Dimand). Game theory emerged as a distinct subdiscipline of applied mathematics, economics, and social science with the publication in 1944 of *Theory of Games and Economic Behavior,* a work of more than six hundred pages written in Princeton by two Continental European emigrés, John von Neumann, a Hungarian mathematician and physicist who was a pioneer in fields from quantum mechanics to computers, and Oskar Morgenstern, a former director of the Austrian Institute for Economic Research. They built upon analyses of two-person, zero-sum games published in the 1920s.

In a series of notes from 1921 to 1927 (three of which were translated into English in *Econometrica* in 1953), the French mathematician and probability theorist Emile Borel developed the concept of a mixed strategy (assigning a probability to each feasible strategy rather than a pure strategy selecting with certainty a single action that the opponent could then predict) and showed that for some particular games with small numbers of possible pure strategies, rational choices by the two players would lead to a minimax solution. Each player would choose the mixed strategy that would minimize the maximum payoff that the other player could be sure of achieving. The young John von Neumann provided the first proof that this minimax solution held for all two-person, constant-sum games (strictly competitive games) in 1928, although the proof of the minimax theorem used by von Neumann and Morgenstern in 1944 was based on the first elementary (that is, nontopological) proof of the existence of a minimax solution, proved by Borel's student Jean Ville in 1938 (Weintraub; Leonard, 1995; Dimand and Dimand). For games with variable sums and more players, where coalitions among players are possible, von Neumann and Morgenstern proposed a more general solution concept, the stable set, but could not prove its existence. In the 1960s, William Lucas proved by counterexample that existence of the stable set solution could not be proved because it was not true in general.

Although von Neumann's and Morgenstern's work was the subject of long and extensive review articles in economics journals, some of which predicted widespread and rapid application, game theory was developed in the 1950s primarily by A. W. Tucker and his students in Princeton's mathematics department (see Shubik's recollections in Weintraub) and at the RAND Corporation, a nonprofit corporation based in Santa Monica, California, whose only client was the U.S. Air Force (Nasar). Expecting that the theory of strategic games would be as relevant to military and naval strategy as contemporary developments in operations research were, the U.S. Office of Naval Research supported much of the basic research, and Morgenstern was named as an editor of the *Naval Research Logistics Quarterly.*

Much has been written about the influence of game theory and related forms of rational-choice theory such as systems analysis on nuclear strategy (although General Curtis LeMay complained that RAND stood for Research And No Development) and of how the Cold War context and military funding helped shape game theory and economics (Heims; Poundstone; Mirowski), mirrored by the shaping of similar mathematical techniques into "planometrics" on the other side of the Cold War (Campbell). Researchers in peace studies, publishing largely in the *Journal of Conflict Resolution* in the late 1950s and the 1960s, drew on Prisoner's Dilemma games to analyze the Cold War (see Schelling), while from 1965 to 1968 (while ratification of the Nuclear Non-Proliferation Treaty was pending) the U.S. Arms Control and Disarmament Agency sponsored important research on bargaining games with incomplete information and their application to arms races and disarmament (later declassified and published as Mayberry with Harsanyi, Scarf, and Selten; and Aumann and Maschler with Stearns).

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- Game Theory - Nash Equilibrium, The Nash Bargaining Solution, And The Shapley Value
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