Pure Saddle Points and Symmetric Relative Payoff Games

TL;DR

This paper characterizes when symmetric two-player zero-sum games have pure saddle points, showing they exist if the game isn't a generalized rock-paper-scissors game, and explores implications for evolutionary stability.

Contribution

It provides a complete characterization of pure saddle point existence in symmetric zero-sum games and links this to relative payoff games and evolutionary stability.

Findings

Pure saddle points exist iff the game is not a generalized rock-paper-scissors game.

Finite symmetric quasiconcave zero-sum games always have a pure saddle point.

Results apply to evolutionary stable strategies in finite populations.

Abstract

It is well known that the rock-paper-scissors game has no pure saddle point. We show that this holds more generally: A symmetric two-player zero-sum game has a pure saddle point if and only if it is not a generalized rock-paper-scissors game. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure saddle point. Further sufficient conditions for existence are provided. We apply our theory to a rich collection of examples by noting that the class of symmetric two-player 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 a finite population evolutionary stable strategies.