Monotone Near-Zero-Sum Games: A Generalization of Convex-Concave Minimax
Ruichen Luo, Sebastian U. Stich, Krishnendu Chatterjee
TL;DR
The paper introduces monotone near-zero-sum games as an intermediate class between monotone zero-sum and general-sum settings and proposes Iterative Coupling Linearization (ICL), a black-box reduction that transforms near-zero-sum games into sequences of zero-sum subproblems. It provides convergence guarantees and a refined gradient-query complexity that improves upon classical results when the near-zero-sum parameter δ is small, approaching zero-sum rates as δ→0. The authors demonstrate practical relevance through applications to regularized matrix games and competitive games with small incentives, supported by numerical experiments showing faster convergence under near-zero-sum conditions. This work offers a principled framework to leverage zero-sum solvers for broader classes of monotone games, with potential impact on algorithmic game theory and optimization in economics and AI.
Abstract
Zero-sum and non-zero-sum (aka general-sum) games are relevant in a wide range of applications. While general non-zero-sum games are computationally hard, researchers focus on the special class of monotone games for gradient-based algorithms. However, there is a substantial gap between the gradient complexity of monotone zero-sum and monotone general-sum games. Moreover, in many practical scenarios of games the zero-sum assumption needs to be relaxed. To address these issues, we define a new intermediate class of monotone near-zero-sum games that contains monotone zero-sum games as a special case. Then, we present a novel algorithm that transforms the near-zero-sum games into a sequence of zero-sum subproblems, improving the gradient-based complexity for the class. Finally, we demonstrate the applicability of this new class to model practical scenarios of games motivated from the literature.