The Complexity of Computing Robust Mediated Equilibria in Ordinal Games

TL;DR

This paper explores the computational complexity of finding robust mediated equilibria in ordinal games, demonstrating that mixed strategies can be effectively used under folk theorems to achieve stability despite utility uncertainties.

Contribution

It introduces a framework for computing robust equilibria in ordinal games using folk theorems, and analyzes the complexity of these computations.

Findings

Robust equilibria can be identified using mixed strategies in ordinal games.

Computational complexity varies across different game settings.

Folk theorems enable stability of equilibria despite utility uncertainties.

Abstract

Usually, to apply game-theoretic methods, we must specify utilities precisely, and we run the risk that the solutions we compute are not robust to errors in this specification. Ordinal games provide an attractive alternative: they require specifying only which outcomes are preferred to which other ones. Unfortunately, they provide little guidance for how to play unless there are pure Nash equilibria; evaluating mixed strategies appears to fundamentally require cardinal utilities. In this paper, we observe that we can in fact make good use of mixed strategies in ordinal games if we consider settings that allow for folk theorems. These allow us to find equilibria that are robust, in the sense that they remain equilibria no matter which cardinal utilities are the correct ones -- as long as they are consistent with the specified ordinal preferences. We analyze this concept and study the computational complexity of finding such equilibria in a range of settings.