Guiding Agents of Quantum Games to Equilibrium using Matrix Exponential Fixed-Point Iteration
MEFPIA searches quantum-game equilibria faster, with lower error than matrix multiplicative-weights updates on tested instances.
The paper studies equilibrium search in extended Gutoski-Watrous quantum games where each player's strategy is a local density matrix. Tensor-contraction expressions for payoffs and gradients avoid constructing the full joint density matrix. The proposed Matrix Exponential Fixed-Point Iteration with Annealing (MEFPIA) is compared with Matrix Multiplicative Weights Update. On the tested instances both methods approach similar strategy profiles and payoffs, while MEFPIA reaches lower relative error in fewer iterations.
- Player strategies are local density matrices in extended Gutoski-Watrous games
- Tensor contractions avoid building the full joint density matrix
- MEFPIA reports lower relative error in fewer iterations than MMWU
Full article213 words · extracted from arxiv.org · click to collapse
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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.21944