Guiding Agents of Quantum Games to Equilibrium using Matrix Exponential Fixed-Point Iteration

2026-09-18

AI Summary

本文提出了一种基于矩阵指数固定点迭代的算法MEFPIA，用于解决多代理量子博弈中均衡策略计算的问题。

Abstract

In recent years, quantum game theory has gained significant attention as a framework for studying decision-making in multi-agent systems using quantum principles. However, computing equilibrium strategies is challenging because the dimension of the joint Hilbert space grows as the product of the players' local dimensions. In this paper, we consider an extended Gutoski-Watrous (EGW) game in which each player's quantum strategy is represented by a local density matrix. We derive tensor-contraction expressions for the payoff functions and their gradients, thereby avoiding the explicit construction of the full joint density matrix and its computationally expensive multiplication by the payoff operators. Building on the resulting effective Hamiltonians, we propose the Matrix Exponential Fixed-Point Iteration with Annealing (MEFPIA) algorithm to search for equilibrium points in EGW games. We compare MEFPIA with the Matrix Multiplicative Weights Update (MMWU) algorithm in terms of convergence. For the tested instances and parameter settings, both algorithms approach the same strategy profiles and payoffs, while MEFPIA achieves lower relative error in fewer iterations. These results indicate that MEFPIA is a promising numerical method for equilibrium search in multi-agent quantum games. Our findings provide important insights into the quantum game theory's potential for addressing complex decision-making processes, as well as opening up new paths for future research and exploration in multi-agent quantum systems.

Problem

• Quantum Game Theory
• Equilibrium Strategies
• Hilbert Space
• Multi-Agent Systems
• Tensor Contraction

Innovation

• Matrix Exponential Fixed-Point Iteration with Annealing (MEFPIA)
• tensor-contraction expressions
• equilibrium search