Pointwise Convergence in Games with Conflicting Interest

TL;DR

This paper introduces non-negative weighted regret to analyze games with conflicting interests and demonstrates that optimistic no-regret algorithms converge to approximate Nash equilibria at a rate of O(1/ε²).

Contribution

It extends the concept of regret to weighted regret in games and proves convergence of optimistic algorithms to Nash equilibria in such settings.

Findings

Optimistic mirror descent converges to ε-approximate Nash equilibrium.

Convergence rate of O(1/ε²) for optimistic algorithms.

Algorithms are robust to player deviations.

Abstract

In this work, we introduce the concept of non-negative weighted regret, an extension of non-negative regret \cite{anagnostides2022last} in games. Investigating games with non-negative weighted regret helps us to understand games with conflicting interests, including harmonic games and important classes of zero-sum games.We show that optimistic variants of classical no-regret learning algorithms, namely optimistic mirror descent (OMD) and optimistic follow the regularized leader (OFTRL), converge to an $\epsilon$-approximate Nash equilibrium at a rate of $O(1/\epsilon^2)$.Consequently, they guarantee pointwise convergence to a Nash equilibrium if there are only finitely many Nash equilibria in the game. These algorithms are robust in the sense the convergence holds even if the players deviate Our theoretical findings are supported by empirical evaluations of OMD and OFTRL on the game of matching pennies and harmonic game instances.