Nash equilibrium, refinements of

Abstract

Game theory studies decisions by several persons in situations with significant interactions. Two features distinguish it from other theories of multi-person decisions. One is explicit consideration of each person’s available strategies and the outcomes resulting from combinations of their choices; that is, a complete and detailed specification of the ‘game’. Here a person’s strategy is a complete plan specifying his action in each contingency that might arise. In non-cooperative contexts, the other is a focus on optimal choices by each person separately. John Nash (1950; 1951) proposed that a combination of mutually optimal strategies can be characterized mathematically as an equilibrium. According to Nash’s definition, a combination is an equilibrium if each person’s choice is an optimal response to others’ choices. His definition assumes that a choice is optimal if it maximizes the person’s expected utility of outcomes, conditional on knowing or correctly anticipating the choices of others. In some applications, knowledge of others’ choices might stem from prior agreement or communication, or accurate prediction of others’ choices might derive from ‘common knowledge’ of strategies and outcomes and of optimizing behaviour. Because many games have multiple equilibria, the predictions obtained are incomplete. However, equilibrium is a weak criterion in some respects, and therefore one can refine the criterion to obtain sharper predictions (Harsanyi and Selten, 1988; Hillas and Kohlberg, 2002; Kohlberg, 1990; Kreps, 1990).