Monotone Near-Zero-Sum Games: A Generalization of Convex-Concave Minimax
Ruichen Luo, Sebastian U. Stich, Krishnendu Chatterjee
TL;DR
This paper introduces monotone near-zero-sum games, a new class that generalizes zero-sum games, and proposes an algorithm to solve them more efficiently, with applications in practical game scenarios.
Contribution
The paper defines a new class of monotone near-zero-sum games and develops an algorithm that reduces their complexity by transforming them into zero-sum subproblems.
Findings
The new class includes zero-sum games as a special case.
The proposed algorithm improves gradient complexity for near-zero-sum games.
Demonstrates practical applicability in real-world game scenarios.
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…
Peer Reviews
Decision·ICLR 2026 Poster
The paper is well written and examines a concrete problem. To my knowledge, the gap in the number of gradients needed between monotone zero-sum and monotone general-sum has been unexplored. This gap can be substantial, which motivates the main contribution of the paper. The paper also does a good job at explaining the technical steps and highlighting the technical contribution compared to existing techniques. The proposed algorithm is quite natural and clean; I didn't find any notable issues in
Overall, I believe the paper is lacking in motivating the significance of the new class of problems and providing good applications of it. Some of the examples discussed in Section 4 look pretty artificial. For example, this setting of matrix games with transaction fees hasn't been studied before, to my knowledge, and its significance is unclear. Concerning Example 2, combing competition with cooperation is, of course, very interesting. But the precise way it is formulated in that example appear
1. The problem is theoretically well-motivated. The authors clearly identify a gap between the gradient complexities of monotone zero-sum and monotone general-sum games, and take a principled step toward bridging this gap through the introduction of a new intermediate class of monotone near-zero-sum games. 2. The presentation is clear and well-structured. The decomposition of the potential function into a jointly convex coupling part and a convex–concave zero-sum part provides strong intuition t
My main concern is generalizability of the method presented in the paper and the overall contribution of the paper. 1. The main theoretical results hinge on the near-zero-sum smoothness parameter $\delta$ that measures how strongly coupled the game is. However, it remains unclear in which practical settings the $\delta$-smoothness assumption naturally holds or how to estimate it empirically. Although it is formally bounded by the overall smoothness $L$, it is uncertain how often $\delta$ would
On a conceptual level, the paper does take a step in filling the gap between zero sum and general sum games, introducing an interesting class. Moreover, from the technical side, the proofs require technical work. On the application side, the transaction fee games are interesting as a class so I appreciate a method that solves them efficiently.
I think that the weaker aspect of the paper lies in its applications. Aside from the example mentioned above, I am not sure about the practical relevance of the classes that can be captured within the presented framework.
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Taxonomy
TopicsStochastic Gradient Optimization Techniques · Advanced Optimization Algorithms Research · Sparse and Compressive Sensing Techniques