Nash equilibria in semidefinite games and Lemke-Howson paths

TL;DR

This paper introduces a novel algorithmic framework for computing Nash equilibria in two-player semidefinite games, extending classical methods by tracing nonlinear paths governed by eigenvalue conditions.

Contribution

It formulates Nash equilibrium computation as semidefinite complementarity problems and develops symbolic-numeric techniques to trace generalized Lemke-Howson paths with analysis of singularities.

Findings

Paths are governed by eigenvalue complementarity conditions.

Curve smoothness is established under non-degeneracy.

Connections to classical and homotopy interpretations are demonstrated.

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.