Pure strategy equilibria in symmetric two-player zero-sum games

Abstract

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric twoplayer zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.

Essays in games and decisions

“…I have showed that two-person symmetric zerosum games that possess this property have at least one equilibrium in pure strategies. Furthermore, this concept generalizes both generalized ordinal potentials (Monderer and Shapley, 1996) and quasiconcavity (Duersch et al, 2012a). Optimin equilibrium: An extension of maximin strategies…”

“…In this chapter, I will introduce three sufficient conditions for the existence of a pure Nash equilibrium in two-person symmetric zero-sum games: (1) a quasiconcavity notion based on the preferences of a decision maker in the equivalent decision problem of a game; (2) a functional representation of an axiom in Fishburn and Rosenthal (1986); (3) the so-called sign-quasiconcavity notion, which is a generalization of quasiconcavity in Duersch et al (2012a).…”

“…By exploiting the symmetry feature, Duersch et al (2012a) showed that if a two-person symmetric zero-sum game is quasiconcave, then it has a pure equilibrium. Moreover, if these games are quasiconcave, then the simple decision rule "imitateif-better" cannot be exploited indefinitely (Duersch et al, 2012b).…”

“…So, by Observation 2, the two-person symmetric zero-sum game (X, sgn(v |X )) 2 has no pure equilibrium. Based on Theorem 1 in Duersch et al (2012a), this implies that the game (X, sgn(v |X )) is not quasiconcave. If it is not possible to find a total order on X such that the game (X, sgn(v |X )) is quasiconcave, then it is not possible to find a total order on X so that the larger game (X, sgn(v)) is quasiconcave.…”

“…The most general condition that I will explore is the socalled sign-quasiconcavity, which generalizes both generalized ordinal potentials (Monderer and Shapley, 1996) and quasiconcavity (Duersch et al, 2012a).…”