On quasi-stable sets

Abstract

In this paper it is shown that for a bimatrix game each quasi-stable set is finite.

A generalization of the Shapley–Ichiishi result

“…The reason for this is that many Q-perturbed versions of (A, B) in fact impose the same conditions on S for it to be a Q-set. That way we can construct an equivalence relation on Q-perturbed versions of (A, B), by saying that two Q-perturbations are equivalent when they impose the same condition on S. It is shown in Vermeulen et al (1996) that there are only finitely many equivalence classes under this equivalence relation. Now, the strategy spaces of two equivalent Q-perturbations, viewed as cores of two TU-games, can be shown to have the same core structure.…”

“…These subsets take the form of the core of a TU-game, which gives the connection to the current paper. A closed set S of strategy pairs is a Q-set if every Q-perturbed version of (A, B) has an equilibrium close to S. Vermeulen et al (1996) showed that for bimatrix games, minimal Q-sets are finite.…”

“…Hillas (1990) defined quasi-stable sets. A closely related notion, in fact quasi-stability minus the invariance and minimality requirements, is the notion of a Q-set, defined in Vermeulen et al (1996). For a bimatrix game (A, B), a Q-perturbed version of (A, B) is a strategic form game in which the payoffs for the players are the same as in the original game, but each player is only allowed to select strategies from a-strictsubset of his strategy space.…”

“…1 A second application is in the context of stability of Nash equilibrium for bimatrix games. The main theorem of our paper can be used to construct a finite algorithm to determine whether a set of strategy pairs in a given bimatrix game contains a Q-set as defined in Vermeulen et al (1996). Details of this construction are discussed in the conclusion of this paper.…”