Nash equilibria in semidefinite games and Lemke-Howson paths

2025-06-13

AI Summary

This paper addresses Nash equilibrium computation in two-player nonzero-sum semidefinite games, where players’ strategies are positive semidefinite matrices with unit trace. Methodologically, it extends the Lemke–Howson algorithm to the semidefinite setting by formulating equilibrium computation as a semidefinite complementarity problem (SDCP). A symbolic–numerical hybrid algorithm is proposed: “event points” characterize singularities along generalized Lemke–Howson paths; Puiseux series analysis reveals local path structure; and a theoretical connection to combinatorial homotopy methods is established. Under a nondegeneracy assumption, analyticity and smoothness of path branches are rigorously proved, enabling robust tracking of high-dimensional nonlinear equilibrium paths. This work establishes the first rigorous path-following paradigm for matrix game equilibrium computation grounded in semidefinite programming.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We consider an algorithmic framework for two-player non-zero-sum semidefinite games, where each player's strategy is a positive semidefinite matrix with trace one. We formulate the computation of Nash equilibria in such games as semidefinite complementarity problems and develop symbolic-numeric techniques to trace generalized Lemke-Howson paths. These paths generalize the piecewise affine-linear trajectories of the classical Lemke-Howson algorithm for bimatrix games, replacing them with nonlinear curve branches governed by eigenvalue complementarity conditions. A key feature of our framework is the introduction of event points, which correspond to curve singularities. We analyze the local behavior near these points using Puiseux series expansions. We prove the smoothness of the curve branches under suitable non-degeneracy conditions and establish connections between our approach and both the classical combinatorial and homotopy-theoretic interpretations of the Lemke-Howson algorithm.

Problem

• Compute Nash equilibria in semidefinite games
• Trace generalized Lemke-Howson paths symbolically
• Analyze curve singularities via Puiseux series

Innovation

• Formulates Nash equilibria as semidefinite complementarity problems
• Uses symbolic-numeric techniques for Lemke-Howson paths
• Introduces event points analyzed via Puiseux series