# zbMATH — the first resource for mathematics

A general theory of equilibrium selection in games. With a foreword by Robert Aumann. (English) Zbl 0693.90098
Cambridge, MA etc.: MIT Press. xviii, 378 p. \$ 32.50 (1988).
The classical theory of noncooperative games is constructed on one basic solution concept, that of equilibrium points. There are, however, two main problems posed by the concepts of equilibrium points. (1) The equilibrium selection problem. Almost every non-trivial noncooperative game possesses many (sometimes infinitely many) different equilibrium points. (2) The imperfectness problem. Many equilibrium points require some or all of the players to use highly irrational strategies. Such equilibrium points are said to be imperfect, in contradistinction to the perfect equilibrium points which involve no irrational strategies.
The authors have tried to solve the equilibrium selection problem. For that purpose, they propose a systematic general solution theory which always selects one equilibrium point as the solution of any non- cooperative game. Their one-point solution theory is based on the uniformly perfectly equilibrium points of any given game G - on those equilibria that can be obtained as the limits of the Nash equilibria of the uniformly perturbed standard form $$G_{\epsilon}$$ of game G when the perturbance parameter $$\epsilon$$ goes to zero. By requiring uniform perfectness, the new theory also overcomes the imperfectness problem posed by the concept of equilibrium points. In choosing between two equilibrium points as solution candidates, the new theory uses two independent criteria of rationality: risk dominance and payoff dominance. Risk dominance is based on individual rationality because it is an extension of Bayesian rationality from one-person decision problems involving uncertainty to n-person decision problems posed by n-person noncooperative games. In contrast, payoff dominance is based on collective rationality since it requires the players to cooperate in promoting their common interests. In cases where risk dominance and payoff dominance go in opposite directions, the new theory in general gives precedence to payoff dominance. To conclude, the new solution theory is primarily constructed for noncooperative games but it is also applicable to cooperative games, because each cooperative game can be remodeled as a bargaining game having the structure of a noncooperative game. The book consists of ten chapters of which Chapters 6 through 9 are devoted to applications of the new solution theory.

##### MSC:
 91A10 Noncooperative games 91A12 Cooperative games 91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance