Semidefinite games

TL;DR

This work extends classical bimatrix and finite $N$-player games by replacing strategy simplices with spectrahedra $\mathcal{X}=\{X\succeq 0: \mathrm{tr}(X)=1\}$ and $\mathcal{Y}=\{Y\succeq 0: \mathrm{tr}(Y)=1\}$, and defining payoffs via a bisymmetric tensor $A$, $p_A(X,Y)=\sum X_{ij}A_{ijkl}Y_{kl}$. It develops a comprehensive SDP-based framework: semidefinite zero-sum games admit optimal strategies that are spectrahedra and can be computed via SDPs; their equilibria are almost equivalent to semidefinite programs via a semidefinite Dantzig construction; for general semidefinite games, Nash equilibria admit a spectrahedral characterization using projections of spectrahedra. The paper also provides a block-construction method to generate semidefinite games with many Nash equilibria, achieving more connected components than classical bimatrix constructions, thereby linking game-theoretic equilibrium geometry with semidefinite geometry. Overall, it bridges convex optimization, semidefinite geometry, and game theory, offering both computational techniques and structural insights, while highlighting rich open questions about spectrahedra-based equilibria and generalizations to broader game classes.

Abstract

We introduce and study the class of semidefinite games, which generalizes bimatrix games and finite $N$-person games, by replacing the simplex of the mixed strategies for each player by a slice of the positive semidefinite cone in the space of real symmetric matrices. For semidefinite two-player zero-sum games, we show that the optimal strategies can be computed by semidefinite programming. Furthermore, we show that two-player semidefinite zero-sum games are almost equivalent to semidefinite programming, generalizing Dantzig's result on the almost equivalence of bimatrix games and linear programming. For general two-player semidefinite games, we prove a spectrahedral characterization of the Nash equilibria. Moreover, we give constructions of semidefinite games with many Nash equilibria. In particular, we give a construction of semidefinite games whose number of connected components of Nash equilibria exceeds the long standing best known construction for many Nash equilibria in bimatrix games, which was presented by von Stengel in 1999.

Theorems & Definitions (39)

• Theorem 2.1
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof
• Lemma 4.5
• proof