game theory the theory of the structure of, and the rational strategies for performing in, games or gamelike human interactions. Although there were forerunners, game theory was virtually invented by the mathematician John von Neumann and the economist Oskar Morgenstern in the early 1940s. Its most striking feature is its compact representation of interactions of two or more choosers, or players. For example, two players may face two choices each, and in combination these choices produce four possible outcomes. Actual choices are of strategies, not of outcomes, although it is assessments of outcomes that recommend strategies. To do well in a game, even for all choosers to do well, as is often possible, generally requires taking all other players’ positions and interests into account. Hence, to evaluate strategies directly, without reference to the outcomes they might produce in interaction with others, is conspicuously perverse. It is not surprising, therefore, that in ethics, game theory has been preeminently applied to utilitarian moral theory. As the numbers of players and strategies rise, the complexity of games increases geometrically. If two players have two strategies each and each ranks the four possible outcomes without ties, there are already seventy-eight strategically distinct games. Even minor real-life interactions may have astronomically greater complexity. One might complain that this makes game theory useless. Alternatively, one can note that this makes it realistic and helps us understand why real-life choices are at least as complex as they sometimes seem. To complicate matters further, players can choose over probabilistic combinations of their ‘pure’ strategies. Hence, the original four outcomes in a simple 2 $ 2 game define a continuum of potential outcomes.
After noting the structure of games, one might then be struck by an immediate implication of this mere description. A rational individual may be supposed to attempt to maximize her potential or expected outcome in a game. But if there are two or more choosers in a game, in general they cannot all maximize simultaneously over their expected outcomes while assuming that all others are doing likewise. This is a mathematical principle: in general, we cannot maximize over two functions simultaneously. For example, the general notion of the greatest good of the greatest number is incoherent. Hence, in interactive choice contexts, the simple notion of economic rationality is incoherent. Virtually all of early game theory was dedicated to finding an alternative principle for resolving game interactions. There are now many so-called solution theories, most of which are about outcomes rather than strategies (they stipulate which outcomes or range of outcomes is game-theoretically rational). There is little consensus on how to generalize from the ordinary rationality of merely choosing more rather than less (and of displaying consistent preferences) to the general choice of strategies in games.
Payoffs in early game theory were almost always represented in cardinal, transferable utilities. Transferable utility is an odd notion that was evidently introduced to avoid the disdain with which economists then treated interpersonal comparisons of utility. It seems to be analogous to money. In the language of contemporary law and economics, one could say the theory is one of wealth maximization. In the early theory, the rationality conditions were as follows. (1) In general, if the sums of the payoffs to all players in various outcomes differ, it is assumed that rational players will manage to divide the largest possible payoff among themselves. (2) No individual will accept a payoff below the ‘security level’ obtainable even if all the other players form a coalition against the individual. (3) Finally, sometimes it is also assumed that no group of players will rationally accept less than it could get as its group security level – but in some games, no outcome can meet this condition. This is an odd combination of individual and collective elements. The collective elements are plausibly thought of as merely predictive: if we individually wish to do well, we should combine efforts to help us do best as a group. But what we want is a theory that converts individual preferences into collective results. Unfortunately, to put a move doing just this in the foundations of the theory is questionbegging. Our fundamental burden is to determine whether a theory of individual rationality can produce collectively good results, not to stipulate that it must. In the theory with cardinal, additive payoffs, we can divide games into constant sum games, in which the sum of all players’ payoffs in each outcome is a constant, and variable sum games. Zerosum games are a special case of constant sum games. Two-person constant sum games are games of pure conflict, because each player’s gain is the other’s loss. In constant sum games with more than two players and in all variable sum games, there is generally reason for coalition formation to improve payoffs to members of the coalition (hence, the appeal of assumptions 1 and 3 above). Games without transferable utility, such as games in which players have only ordinal preferences, may be characterized as games of pure conflict or of pure coordination when players’ preference orderings over outcomes are opposite or identical, respectively, or as games of mixed motive when their orderings are partly the same and partly reversed. Mathematical analysis of such games is evidently less tractable than that of games with cardinal, additive utility, and their theory is only beginning to be extensively developed. Despite the apparent circularity of the rationality assumptions of early game theory, it is the game theorists’ prisoner’s dilemma that makes clear that compelling individual principles of choice can produce collectively deficient outcomes. This game was discovered about 1950 and later given its catchy but inapt name. If they play it in isolation from any other interaction between them, two players in this game can each do what seems individually best and reach an outcome that both consider inferior to the outcome that results from making opposite strategy choices. Even with the knowledge that this is the problem they face, the players still have incentive to choose the strategies that jointly produce the inferior outcome. Prisoner’s dilemma involves both coordination and conflict. It has played a central role in contemporary discussions of moral and political philosophy. Games that predominantly involve coordination, such as when we coordinate in all driving on the right or all on the left, have a similarly central role. The understanding of both classes of games has been read into the political philosophies of Hobbes and Hume and into mutual advantage theories of justice.
See also DECISION THEORY, PRISONER ‘S DILEMMA , UTILITARIANIS. R.Har.