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Über dieses Buch

The last decade has seen a steady increase in the application of concepts from noncooperative game theory to such diverse fields as economics, political science, law, operations research, biology and social psychology. As a byproduct of this increased activity, there has been a growing awareness of the fact that the basic noncooperative solution concept, that of Nash equilibrium, suffers from severe drawbacks. The two main shortcomings of this concept are the following: (i) In extensive form games, a Nash strategy may prescribe off the equilibrium path behavior that is manifestly irrational. (Specifically, Nash equilibria may involve incredible threats), (ii) Nash equilibria need not be robust with respect to small perturbations in the data of the game. Confronted with the growing evidence to the detriment of the Nash concept, game theorists were prompted to search for more refined equilibrium notions with better properties and they have come up with a wide array of alternative solution concepts. This book surveys the most important refinements that have been introduced. Its objectives are fourfold (i) to illustrate desirable properties as well as drawbacks of the various equilibrium notions by means of simple specific examples, (ii) to study the relationships between the various refinements, (iii) to derive simplifying characterizations, and (iv) to discuss the plausibility of the assumptions underlying the concepts.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
In this chapter, we illustrate ( by means of a series of examples ) why the Nash equilibrium concept has to be refined and several refinements which have been proposed in the literature are introduced in an informal way. (No formal definitions are given in this chapter). First, in Sect. 1.1, it is motivated why the solution of a noncooperative game has to be a Nash equilibrium. In Sects. 1.2 – 1.4, we consider games in extensive form and discuss the following refinements of the Nash equilibrium concept: subgame perfect equilibria, sequential equilibria, perfect equilibria and stable equilibria. In Sects. 1.5 and 1.6, we consider refinements of the Nash equilibrium concept for normal form games, such as perfect equilibria, proper equilibria, persistent equilibria, essential equilibria and regular equilibria.
Eric van Damme

2. Games in Normal Form

Abstract
For normal form games the Nash equilibrium concept has to be refined since a Nash equilibrium of such a game need not be robust, i.e. it may be unstable against small perturbations in the data of the game. In this chapter, we will consider various refinements of the Nash concept for this class of games, all of which require an equilibrium to satisfy some particular robustness condition.
Eric van Damme

3. Matrix and Bimatrix Games

Abstract
In this chapter, we study 2-person normal form games, zero-sum games (matrix games) as well as nonzero-sum games (bimatrix games). It is our objective to investigate whether for this special class of games the results of the previous chapter can be refined and specialized.
Eric van Damme

4. Control Costs

Abstract
In this chapter, games with control costs1 are studied. These are normal form games in which each player, in addition to receiving his payoff from the game, incurs costs depending on how well he chooses to control his actions. Such a game models the idea that a player can reduce the probability of making mistakes, but that he can only do so by being extra prudent, hence, by spending an extra effort, which involves some costs. The goal of the chapter is to investigate what the consequences are of viewing an ordinary normal form game as a limiting case of a game with control costs, i.e. it is examined which equilibria are still viable when infinitesimal control costs are incorporated into the analysis of normal form games.
Eric van Damme

5. Incomplete Information

Abstract
Games with incomplete information are games in which some of the data are unknown to some of the players. In this chapter, a particular class of games with incomplete information, the class of disturbed games, is studied. A disturbed game is a normal form game in which each player, although knowing his own payoff function exactly, has only imprecise information about the payoff functions of his opponents. We study such games since we feel that it is more realistic to assume that each player always has some slight uncertainty about the payoffs of his opponents rather than to assume that he knows these payoffs exactly. Our objective in this chapter is to study what the consequences are of this more realistic point of view.
Eric van Damme

6. Extensive Form Games

Abstract
The comprehensive study of normal form games in Chaps. 2–5 has yielded a deeper insight into the relationships between various refinements of the Nash concept. The analysis has also shown that, for (generic) normal form games, there is actually little need to refine the Nash concept since, for almost all such games, all Nash equilibria possess all properties one might hope for.
Eric van Damme

7. Bargaining and Fair Division

Abstract
In this chapter surplus sharing problems are considered, i.e. it is assumed that synergetic gains can be obtained by cooperating and the question is how these gains should be divided. Although traditionally such problems belong to the realm of cooperative game theory, we will study them by non-cooperative methods. The objectives are twofold: (1) to show how concepts from cooperative game theory can be implemented by means of noncooperative methods and (2) to illustrate the strength of the subgame perfectness concept ( and the weakness of the Nash equilibrium concept ) in dynamic games with perfect information.
Eric van Damme

8. Repeated Games

Abstract
In this chapter we study whether repetition can lead to cooperation. Specifically, it is investigated which outcomes can be sustained by means of subgame perfect ( or Nash) equilibria when a game is repeated finitely or infinitely many times. The main result is the Perfect Folk Theorem, which states that, for almost all games, every outcome that is feasible and individually rational in the one-shot game can be approximated by subgame perfect equilibrium outcomes of the discounted supergame as the discount rate tends to zero, and that, for almost all games with more than one Nash equilibrium, any such outcome can be even approximated by a subgame perfect equilibrium payoff of the finitely repeated game as the number of repetitions tends to infinity.
Eric van Damme

9. Evolutionary Game Theory

Abstract
Game Theory has been developed as a theory of rational behavior in interpersonal conflict situations, with economics and the other social sciences being the intended fields of application. Since the theory is based on an idealized picture of human rationality, it is by no means obvious that it can be applied to situations in which the players cannot be attributed any intellectual capabilities. However, in their seminal paper ‘The logic of animal conflict’, Maynard Smith and Price showed that animal contests can be modeled as games and that game theory can be applied successfully in biology. The objective of this chapter is to review some of the developments in this biological branch of Game Theory and to point out the distinctions and similarities with the classical branch (also see Parker and Hammerstein [1985] ). The main emphasis will be on the mathematics involved, lack of space prevents an extensive discussion of the underlying biological assumptions as well as an analysis of specific examples. For these, the reader is referred to the very stimulating book ‘Evolution and the Theory of Games’ by John Maynard Smith.
Eric van Damme

10. Strategic Stability and Applications

Abstract
In this chapter, we study the concept of strategic stability that has been introduced in Kohlberg and Mertens [1986]. The first part of the chapter motivates this concept and studies its general properties. In the second half, the concept is applied to specific games.
Eric van Damme

Backmatter

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