Safe Equilibrium

TL;DR

The paper introduces a new game-theoretic solution concept called safe equilibrium, which balances rational and irrational opponent behaviors, providing a more robust strategy than Nash or maximin, with proven existence and computational methods.

Contribution

It proposes the safe equilibrium concept, proves its existence and computational hardness, and offers algorithms for exact and approximate solutions in multi-player games.

Findings

Safe equilibrium exists in all strategic-form games.

Computing a safe equilibrium is PPAD-hard.

Provides algorithms for exact and scalable approximate solutions.

Abstract

The standard game-theoretic solution concept, Nash equilibrium, assumes that all players behave rationally. If we follow a Nash equilibrium and opponents are irrational (or follow strategies from a different Nash equilibrium), then we may obtain an extremely low payoff. On the other hand, a maximin strategy assumes that all opposing agents are playing to minimize our payoff (even if it is not in their best interest), and ensures the maximal possible worst-case payoff, but results in exceedingly conservative play. We propose a new solution concept called safe equilibrium that models opponents as behaving rationally with a specified probability and behaving potentially arbitrarily with the remaining probability. We prove that a safe equilibrium exists in all strategic-form games (for all possible values of the rationality parameters), and prove that its computation is PPAD-hard. We present exact algorithms for computing a safe equilibrium in both 2 and $n$-player games, as well as scalable approximation algorithms.