引导量子博弈智能体通过矩阵指数不动点迭代达到均衡

中文摘要 AI 辅助

近年来，量子博弈论作为利用量子原理研究多智能体系统中决策制定的框架，已获得显著关注。然而，计算均衡策略具有挑战性，因为联合希尔伯特空间的维度随玩家局部维度的乘积而增长。在本文中，我们考虑一种扩展的Gutoski-Watrous（EGW）博弈，其中每个玩家的量子策略由局部密度矩阵表示。我们推导了收益函数及其梯度的张量收缩表达式，从而避免了显式构造完整联合密度矩阵及其与收益算子的计算昂贵的乘法。基于由此产生的有效哈密顿量，我们提出了矩阵指数不动点迭代退火（MEFPIA）算法，用于搜索EGW博弈中的均衡点。我们将MEFPIA与矩阵乘法权重更新（MMWU）算法在收敛性方面进行比较。对于测试的实例和参数设置，两种算法接近相同的策略配置和收益，而MEFPIA在更少的迭代中实现了更低的相对误差。这些结果表明，MEFPIA是一种有前途的多智能体量子博弈均衡搜索数值方法。我们的发现为量子博弈论在解决复杂决策过程中的潜力提供了重要见解，并为多智能体量子系统的未来研究和探索开辟了新途径。

英文摘要

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.

发表机构

• Ruhr University Bochum（波鸿鲁尔大学）

机构由 AI 辅助整理，请以论文原文为准。