Computing Equilibrium beyond Unilateral Deviation

2026-04-30

AI Summary

This work addresses the limitation of traditional equilibrium concepts, which only guard against unilateral deviations and fail to account for profitable coordinated deviations by coalitions. We propose a novel framework of multilateral stable equilibria that explicitly quantifies coalition deviation incentives as an optimizable objective. By minimizing the average (or weighted average, or maximum) gain achievable by any deviating coalition, our approach ensures both the existence and computability of equilibria. Integrating game-theoretic modeling, complexity lower-bound analysis, and matching algorithm design, we establish computational hardness results for both average- and maximum-gain objectives and present efficient algorithms that match these lower bounds. The framework is successfully applied to compute the Exploitability Welfare Frontier—the maximal social welfare attainable under a given level of exploitability.

Abstract

Most familiar equilibrium concepts, such as Nash and correlated equilibrium, guarantee only that no single player can improve their utility by deviating unilaterally. They offer no guarantees against profitable coordinated deviations by coalitions. Although the literature proposes solution concepts that provide stability against multilateral deviations (\emph{e.g.}, strong Nash and coalition-proof equilibrium), these generally fail to exist. In this paper, we study an alternative solution concept that minimizes coalitional deviation incentives, rather than requiring them to vanish, and is therefore guaranteed to exist. Specifically, we focus on minimizing the average gain of a deviating coalition, and extend the framework to weighted-average and maximum-within-coalition gains. In contrast, the minimum-gain analogue is shown to be computationally intractable. For the average-gain and maximum-gain objectives, we prove a lower bound on the complexity of computing such an equilibrium and present an algorithm that matches this bound. Finally, we use our framework to solve the \emph{Exploitability Welfare Frontier} (EWF), the maximum attainable social welfare subject to a given exploitability (the maximum gain over all unilateral deviations).

Problem

• equilibrium
• coalitional deviation
• exploitability
• social welfare
• game theory

Innovation

• coalition-proof equilibrium
• average-gain minimization
• exploitability welfare frontier
• computational complexity
• multi-agent equilibrium