Game Theory

John Nash, the outstanding figure among the Princeton and RAND game theorists (Nasar; Giocoli), developed, in articles from his dissertation, both the Nash equilibrium for noncooperative games, where the players cannot make binding agreements enforced by an outside agency, and the Nash bargaining solution for cooperative games where such binding agreements are possible (Nash). Nash equilibrium, by far the most widely influential solution concept in game theory, applied to games with any number of players and with payoffs whose sum carried with the combination of strategies chosen by the players, while von Neumann's minimax solution was limited to two-person, constant-sum games. A Nash equilibrium is a strategy combination in which each player's chosen strategy is a best response to the strategies of the other players, so that no player can get a higher expected payoff by changing strategy as long as the strategies of the other players stay the same. No player has an incentive to be the first to deviate from a Nash equilibrium.

Nash proved the existence of equilibrium but not uniqueness: a game will have at least one strategy combination that is a Nash equilibrium, but it may have many or even an infinity of Nash equilibria (especially if the choice of action involves picking a value for a continuous variable). Cournot's 1838 analysis of duopoly has been interpreted in retrospect as a special case of Nash equilibrium, just as Harsanyi perceived the congruity of Zeuthen's 1930 discussion of bargaining and the Nash bargaining solution. Refinements of Nash equilibrium, which serve to rule out some of the possible equilibria, include the concept of a subgame perfect equilibrium (see Harsanyi and Selten), which is a Nash equilibrium both for an entire extended game (a game in which actions must be chosen at several decision nodes in a game tree) and for any game starting from any decision node in the game tree, including points that would never be reached in equilibrium, so that any threats to take certain actions if another player were to deviate from the equilibrium path would be credible (rational in terms of self-interest once that point in the game had been reached). A further refinement rules out some subgame perfect Nash equilibria by allowing for the possibility of a "trembling hand," that is, a small probability that an opposing player, although rational, may make mistakes (Harsanyi and Selten). Thomas Schelling has suggested that if there is some clue that would lead players to regard one Nash equilibrium as more likely than others, that equilibrium will be a focal point.

Nash equilibrium, with its refinements, remains at the heart of noncooperative game theory. Applied to the study of market structure by Martin Shubik (1959), this approach has come to dominate the field of industrial organization, as indicated by Jean Tirole (1988) in a book widely accepted as the standard economics textbook on industrial organization and as a model for subsequent texts. More recently, noncooperative game theory has found economic applications ranging from strategic trade policy in international trade to the credibility of anti-inflationary monetary policy and the design of auctions for broadcast frequencies. From economics, noncooperative game theory based on refinements of Nash equilibrium has spread to business school courses on business strategy (see Ghemawat, applying game theory in six Harvard Business School cases for MBA students). Some economists view business strategy as an application of game theory, with ideas flowing in one direction, rather than as a distinct field (Shapiro).

However, scholars of strategic management remain sharply divided over whether game theory provides useful insights or just a rationalization for any conceivable observed behavior (see the papers by Barney, Saloner, Camerer, and Postrel in Rumelt, Schendel, and Teece, especially Postrel's paper, which verifies Rumelt's Flaming Trousers Conjecture by constructing a game-theoretic model with a subgame perfect Bayesian Nash equilibrium in which bank presidents publicly set their pants on fire, a form of costly signaling that is profitable only for a bank that can get repeat business, that is, a high-quality bank).

Nash proposed the Nash bargaining solution for two-person cooperative games, that the players maximize the product of their gains over what each would receive at the threat point (the Nash equilibrium of the noncooperative game that they would play if they failed to reach agreement on how to divide the gains), and showed it to be the only solution possessing all of a particular set of intuitively appealing properties (efficiency, symmetry, independence of unit changes, independence of irrelevant alternatives). Feminist economists such as Marjorie McElroy and Notburga Ott have begun to apply bargaining models whose outcome depends critically on the threat point (the outcome of the noncooperative game that would be played if bargaining does not lead to agreement), as well as Prisoner's Dilemma games, to bargaining within the household (see Seiz for a survey).

Another influential solution concept for cooperative games, the Shapley value for n-person games (Shapley), allots to each player the average of that player's marginal contribution to the payoff each possible coalition would receive and, for a class of games with large numbers of players, coincides with the core of a market (the set of undominated imputations or allocations), yet another solution concept discovered by graduate students at Princeton in the early 1950s (in this case, Shapley and D. B. Gillies) and then rediscovered by Shubik in Edgeworth's 1881 analysis. There is a large literature in accounting applying the Shapley value to cost allocation (Roth and Verrecchia).