# THE TURN TOWARDS NONCOOPERATIVE GAME THEORY

In 1975, Selten (CGT, pp. 317-54) re-examined the concept of perfect equilibrium that he had introduced in German in 1965. This paper focuses on a refinement of Nash equilibrium called the trembling hand equilibrium. Selten had introduced subgame perfect equilibrium in his paper in German ten years before, but the 1975 paper brought it to the English language audience. This paper is regarded as the beginning of the literature on “refinements” of Nash equilibrium. Selten’s model will be discussed in Chapter 6 at Section 1.

In the late 1970s, work by Myerson and Maskin established mechanism design theory, following a program proposed by Hurwicz (1973). This work was honored by a Nobel Memorial Prize in 2007. See especially Maskin (1999) and Myerson (1979, 1986). The objective of mechanism design is to design a game so that its *noncooperative* equilibria correspond to the *cooperative* or other normative outcome that is desired. This seems of particular interest for public policy. Lloyd Shapley and Alvin Roth were honored by the 2012 Nobel for another sort of mechanism design, an algorithm that gives rise to pairs that are in the core of a cooperative game, and that has been implemented in a number of real clearing houses that place medical residents in positions and students in schools. Mechanism design theory will be discussed in Chapter 7.

The development of Selten’s conception of perfect equilibrium made possible some important progress on what has become known as the “folk theorem” in game theory. The “folk theorem” is the idea that, for games such as the Prisoner’s Dilemma (with very bad noncooperative results in one-off play) repeated play might lead to a cooperative outcome in some circumstances. As early as 1981, however, in a working paper of the UCLA department of economics, Fudenberg and Levine (1981, p. 19) sketched an analysis of repeated play of the Prisoner’s Dilemma in terms of perfect equilibria. A few years later Fudenberg and Maskin (1986) gave the general analysis that has now become standard. A quite different but related approach to repeated play in noncooperative games emerged with Axelrod’s (1981, 1984) computational studies. Coding simple rules for the selection of behavior strategies in repeated Prisoner’s Dilemmas, Axelrod played the rules one against another in a tournament, and found that tit-for-tat^{13} (a trigger strategy in which one plays cooperatively until the first defection by the other player, but responds with a single round of noncooperative play) did relatively well against a wide array of challengers. As much as the folk theorem work, this study contributed to the emergence of tit-for-tat and other trigger strategies as standard tools for understanding repeated play of noncooperative games.

In 1984 Bernheim introduced the concept of rationalizable strategies; a simultaneous paper of Pearce (1984) shared the innovation. One important departure of this paper is that Bernheim allows players to condition their decision rules on conjectures about the conjectures that others may make about them. This leads, in some cases, to a much larger set of stable strategies. In his Nobel Address, Aumann (2005) was to admit conjectures as to the *rules other players might use in selecting behavior strategies* among the conditions of a choice of strategies, with a further extension of the range of possible noncooperative equilibria in repeated games.

When we combine rationalizable strategies and correlated equilibrium, the case for Nash equilibria as predictors of behavior is very much reduced. If the game is played one-off, then players are not likely to have enough information to exclude non-Nash rationalizable equilibria, and the same will be true in the first plays of a repeated game. For later plays of a repeated game, though, correlated equilibria may emerge, and these, too, may be non-Nash. In an evolutionary model, where players are randomly matched to play one-off but can learn from the experience of one another, evolutionarily stable (Nash) equilibria seem a reasonable prediction. Even here, though, boundedly rational learning might result in correlated equilibria. More generally, where the Nash equilibrium in pure strategies is unique (including, but not limited to, the family of social dilemmas) correlated strategy equilibria can be excluded; and if in addition the Nash equilibrium is subject to some stringent stability conditions (Bernheim, 1984, p. 1020) then Nash equilibria are the only rationalizable strategies. All in all, Nash equilibria can no longer be treated as “solutions” to noncooperative games, but only as candidate solutions and as tools that may be useful in finding other (for example, correlated equilibrium) solutions.

In 1988, Harsanyi and Selten offered a framework to resolve the growing family of refinements of Nash equilibrium, suggesting a hierarchy of criteria for choosing among Nash equilibria. They rank the equilibria in terms of relative stability, so that, for example, Pareto-dominant equilibria are considered more stable than those that are not Pareto-dominant but are risk-dominant.

In 1989, the journal *Games and Economic Behavior* was founded.