(In)Existence of Equilibria for 2-Players, 2-Values Games with Concave Valuations

TL;DR

This paper investigates the existence of equilibria in 2-player, 2-values minimization games with concave valuation functions, providing impossibility results, algorithms, and conditions for equilibrium existence.

Contribution

It offers the first (almost complete) characterization of when such games lack equilibria based on the properties of the valuation function.

Findings

Existence of games without F-equilibrium under certain conditions.

Linear time algorithm for computing F-equilibrium in normal games.

Conditions for equilibrium existence in 3-strategy games.

Abstract

We consider 2-players, 2-values minimization games where the players' costs take on two values, $a,b$, $a<b$. The players play mixed strategies and their costs are evaluated by unimodal valuations. This broad class of valuations includes all concave, one-parameter functions $\mathsf{F}: [0,1]\rightarrow \mathbb{R}$ with a unique maximum point. Our main result is an impossibility result stating that: If the maximum is obtained in $(0,1)$ and $\mathsf{F}\left(\frac{1}{2}\right)\ne b$, then there exists a 2-players, 2-values game without $\mathsf{F}$-equilibrium. The counterexample game used for the impossibility result belongs to a new class of very sparse 2-players, 2-values bimatrix games which we call normal games. In an attempt to investigate the remaining case $\mathsf{F}\left(\frac{1}{2}\right) = b$, we show that: - Every normal, $n$-strategies game has an ${\mathsf{F}}$-equilibrium when ${\mathsf{F}}\left( \frac{1}{2} \right) = b$. We present a linear time algorithm for computing such an equilibrium. - For 2-players, 2-values games with 3 strategies we have that if $\mathsf{F}\left(\frac{1}{2}\right) \le b$, then every 2-players, 2-values, 3-strategies game has an $\mathsf{F}$-equilibrium; if $\mathsf{F}\left(\frac{1}{2}\right) > b$, then there exists a normal 2-players, 2-values, 3-strategies game without $\mathsf{F}$-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 concave function $\mathsf{F}$, a counterexample game without $\mathsf{F}$-equilibrium.