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Semidefinite network games: multiplayer minimax and complementarity problems

Constantin Ickstadt, Thorsten Theobald, Elias Tsigaridas, Antonios Varvitsiotis

TL;DR

This work extends network game theory to semidefinite strategies, where each player selects a density matrix to play edge games with neighbors. It shows that zero-sum Nash equilibria can be computed efficiently through semidefinite programming and that zero-sum recognition reduces to SDP feasibility checks, while general equilibria correspond to semidefinite linear complementarity problems. The approach relies on an operator representation of edge payoffs, Choi/Jamiolkowski techniques, and exploitability concepts, establishing a spectrahedral (SDP) description of equilibria and connecting these models to quantum games. Together, these results provide a cohesive framework for analyzing equilibria in quantum-inspired network settings and offer practical SDP/LCP tools for computation and recognition.

Abstract

Network games provide a powerful framework for modeling agent interactions in networked systems, where players are represented by nodes in a graph and their payoffs depend on the actions taken by their neighbors. Extending the framework of network games, we introduce and study semidefinite network games. In this model, each player selects a positive semidefinite matrix with trace equal to one, known as a density matrix, to engage in a two-player game with every neighboring node. The player's payoff is the cumulative payoff acquired from these edge games. Initially, we focus on the zero-sum setting, where the sum of all players' payoffs is equal to zero. We establish that, in this class of games, Nash equilibria can be characterized as the projection of a spectrahedron. Furthermore, we show that determining whether a semidefinite network game is a zero-sum game is equivalent to deciding if the value of a semidefinite program is zero. Beyond the zero-sum case, we characterize Nash equilibria as the solutions of a semidefinite linear complementarity problem.

Semidefinite network games: multiplayer minimax and complementarity problems

TL;DR

This work extends network game theory to semidefinite strategies, where each player selects a density matrix to play edge games with neighbors. It shows that zero-sum Nash equilibria can be computed efficiently through semidefinite programming and that zero-sum recognition reduces to SDP feasibility checks, while general equilibria correspond to semidefinite linear complementarity problems. The approach relies on an operator representation of edge payoffs, Choi/Jamiolkowski techniques, and exploitability concepts, establishing a spectrahedral (SDP) description of equilibria and connecting these models to quantum games. Together, these results provide a cohesive framework for analyzing equilibria in quantum-inspired network settings and offer practical SDP/LCP tools for computation and recognition.

Abstract

Network games provide a powerful framework for modeling agent interactions in networked systems, where players are represented by nodes in a graph and their payoffs depend on the actions taken by their neighbors. Extending the framework of network games, we introduce and study semidefinite network games. In this model, each player selects a positive semidefinite matrix with trace equal to one, known as a density matrix, to engage in a two-player game with every neighboring node. The player's payoff is the cumulative payoff acquired from these edge games. Initially, we focus on the zero-sum setting, where the sum of all players' payoffs is equal to zero. We establish that, in this class of games, Nash equilibria can be characterized as the projection of a spectrahedron. Furthermore, we show that determining whether a semidefinite network game is a zero-sum game is equivalent to deciding if the value of a semidefinite program is zero. Beyond the zero-sum case, we characterize Nash equilibria as the solutions of a semidefinite linear complementarity problem.
Paper Structure (13 sections, 6 theorems, 64 equations, 1 table)

This paper contains 13 sections, 6 theorems, 64 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear algebra.
  4. Semidefinite programming
  5. The Model
  6. Semidefinite games
  7. Semidefinite network games
  8. Nash equilibria in semidefinite network games
  9. Operator representation of semidefinite network games
  10. Semidefinite representability of Nash equilibria in semidefinite network games
  11. Recognizing zero-sum semidefinite network games
  12. Nash equilibria via semidefinite linear complementarity problems and the connection to quantum games
  13. Conclusion

Key Result

Proposition 4.1

Consider a zero-sum polymatrix game $G$. If $(y,w)$ is an optimal solution to the LP then $y$ is a Nash equilibrium of $G$. Conversely, if $y$ is Nash equilibrium of $G$, then there is a $w$ such that $(y,w)$ is an optimal solution of the LP eq:polymatrix-lp1.

Theorems & Definitions (17)

  • Remark 3.1
  • Example 3.2
  • Example 3.3
  • Proposition 4.1
  • Lemma 4.2
  • Remark 4.3
  • proof
  • Lemma 4.4
  • proof
  • Theorem 4.5
  • ...and 7 more