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.
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Notes
An algorithm to find perfect equilibria in n-person normal form games has been given in Van den Elzen [1991, Chap. 8].
Corollary 2.2.6 has been generalized in Okada [1988]. Okada introduces a concept of lexicographic dominance and he shows that a perfect equilibrium is lexicographically undominated and that a lexicographically undominated strategy combination is undominated. Okada also gives examples to illustrate that in n-player games with n ≥ 3 the reverse implications are not correct.
The concepts of perfect and proper equilibria are defined only for finite games. For generalizations of these concepts to infinite games, see Simon [1987] and Simon and Stinchcombe [1990]. For applications see Levin and Harstad [1984] and Chatterjee and Samuelson [1990].
A path following algorithm to find a proper equilibrium of an n-person normal form game has been given in Yamamoto [1990].
In Blume, Brandenburger and Dekel [1991 a] refinements of Nash equilibrium are rationalized by the assumption that players have lexicographic preferences. Nash, perfect and proper equilibria can be characterized by means of different versions of lexicographic expected utility maximization.
Garcia Jurado and Prada SÃ nchez [1990] introduce a refinement of properness (called strongly proper equilibrium) that requires, in addition to (2.3.1), that strategies that are equally good are chosen with the same probability.
It would have been more convenient to define regularity by means of the system that is obtained by replacing (2.5.4) with the equation x ki [Ri(x\x)-Ri(x)]=0. This definition is proposed in Ritzberger and Vogelsberger [1989]. The modified definition of regularity has the same properties as the one used in the book, but the smoothness of the modified system allows some new insights into the structure of the Nash equilibrium correspondence. In Jansen [1987] an alternative definition of regularity is given for bimatrix games. Jansen exploits the fact that finding the equilibria of a bimatrix game is equivalent to solving a linear complimentarity problem.
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© 1991 Springer-Verlag Berlin Heidelberg
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van Damme, E. (1991). 2 Games in Normal Form. In: Stability and Perfection of Nash Equilibria. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58242-4_2
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DOI: https://doi.org/10.1007/978-3-642-58242-4_2
Publisher Name: Springer, Berlin, Heidelberg
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