A Direct Second-Order Method for Solving Two-Player Zero-Sum Games
David Yang, Yuan Gao, Tianyi Lin, Christian Kroer
TL;DR
This paper presents the first direct second-order method for efficiently computing Nash equilibria in two-player zero-sum games, combining local superlinear convergence with global efficiency.
Contribution
The authors develop a novel second-order algorithm using Douglas-Rachford splitting and semi-smooth Newton methods, enhancing convergence speed over existing first-order methods.
Findings
Order-of-magnitude speedups over PRM$^+$ for high-precision solutions
Local superlinear convergence of the proposed method
Effective hybrid approach combining global and local convergence benefits
Abstract
We introduce, to our knowledge, the first direct second-order method for computing Nash equilibria in two-player zero-sum games. To do so, we construct a Douglas-Rachford-style splitting formulation, which we then solve with a semi-smooth Newton (SSN) method. We show that our algorithm enjoys local superlinear convergence. In order to augment the fast local behavior of our SSN method with global efficiency guarantees, we develop a hybrid method that combines our SSN method with the state-of-the-art first-order method for game solving, Predictive Regret Matching (PRM). Our hybrid algorithm leverages the global progress provided by PRM, while achieving a local superlinear convergence rate once it switches to SSN near a Nash equilibrium. Numerical experiments on matrix games demonstrate order-of-magnitude speedups over PRM for high-precision solutions.
Peer Reviews
Decision·Submitted to ICLR 2026
The authors achieve their goal of designing a provably superlinear converging solver by what appears to be expert use of advanced techniques of non-smooth and convex analysis.
The first thing that comes to mind when reading the authors' claim of designing "the first direct second-order method for computing Nash equilibria in two-player zero-sum games" is that two-player matrix games are well-known to reduce to linear programming, for which a wide range of methods are already in existence. The authors claim that "While this approach works in principle, it is impractical for large-scale games; even with state-of-the-art commercial solvers, the LP reformulation inflate
On the positive side, solving bilinear saddle-point problems is a central problem in game theory and optimization. Designing practical, scalable algorithms for solving such problems is an important and active research topic. The use of second-order Newton-type methods is relatively unexplored in this area. The paper contributes to filling this gap. While second-order methods have a significantly higher per-iteration complexity than first-order methods, such as PRM+, the paper proposes a natural
On the negative side, there are very basic flaws in the experimental evaluation of the paper, and the results are quite underwhelming. Concerning the results shown in Tables 1 and 2, the first basic issue is that there is no comparison with LP solvers. A commercial solver such as Gurobi would instantly solve exactly such small games. So on the whole, while the method is supposed to be superior in the high-precision regime, the paper fails to benchmark against the best approach in that regime, wh
The (hybrid) second-order approach for computing NE in the two-player zero-sum game setting seems novel. The experimental results on random matrix games demonstrate the effectiveness of this method (compared to first-order methods), especially in the high-precision regime.
Overall, certain parts of the presentation are somewhat sloppy, which makes interpreting the paper's main theoretical contributions more challenging. In particular, the proposed second-order method relies on using a first-order method as a warm-start, but the main (local) convergence guarantee of the paper (Theorem 4) does not concretely connect the performance of the first-order method with the time to reach the local convergence regime. I believe the presentation would be strengthened if the a
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Taxonomy
TopicsStochastic Gradient Optimization Techniques · Advanced Optimization Algorithms Research · Matrix Theory and Algorithms