(In)Existence of Equilibria for 2-Player, 2-Value Games with Semistrictly Quasiconcave Cost Functions

Abstract

We consider 2-player, 2-value cost minimization games where the players’ costs take on two values, a,b, with a < b. The players play mixed strategies and their costs are evaluated by semistrictly quasiconcave cost functions representable by strictly quasiconcave, one-parameter functions $\mathsf {F}: [0, 1] \rightarrow \mathbb {R}$. Our main result is an impossibility result stating that:

• If the maximum of F is obtained in (0,1) and $\mathsf {F} \left (\frac {1}{2}\right )\ne b$, then there exists a 2-player, 2-value game without F-equilibrium.

The counterexample to the existence of equilibria game used for the impossibility result belongs to a new class of very sparse 2-player, 2-value bimatrix games which we call simple games. In an attempt to investigate the remaining case $\mathsf {F}\left (\frac {1}{2}\right ) = b$, we show that:

• Every simple, n-strategy game has an F-equilibrium when $\mathsf {F} \left (\frac {1}{2}\right ) = b$. We present a linear time algorithm for computing such an equilibrium.

• For 2-player, 2-value, 3-strategy games, we have that if $\mathsf {F} \left (\frac {1}{2}\right ) \le b$, then every 2-player, 2-value, 3-strategy game has an F-equilibrium; if $\mathsf {F} \left (\frac {1}{2}\right ) > b$, then there exists a simple 2-player, 2-value, 3-strategy game without F-equilibrium.

To the best of our knowledge, this work is the first to provide an (almost complete) answer on whether there is, for a given function F, a counterexample game without F-equilibrium.