Quantum games use geometry to improve multiplayer strategy learning
Riemannian Optimization for Multi-Player Quantum Games on Product Unitary Manifolds
Artificial IntelligenceComputer Science and Game TheoryMachine LearningMultiagent Systems
Summary
Classical game theory sometimes leads to situations where players make choices that are bad for everyone, like in the Prisoner's Dilemma. Quantum game theory changes the rules by letting players use quantum strategies, which can lead to better outcomes. The authors extend this idea to games with many players who use a mix of quantum moves. They create a new method called USMEA that helps players learn the best set of quantum moves and how often to pick them. Their work shows how using math from geometry can help improve learning in complex multiplayer quantum settings.
quantum game theoryPrisoner's Dilemmaunitary matrixmixed strategyHermitian operatorquantum stateRiemannian optimizationmatrix exponentialmulti-agent learningquantum strategy
Authors
Alireza Habibi, Setareh Maghsudi
Abstract
Quantum game theory is an extension of classical game theory that uses quantum principles in game theory. The Eisert-Wilkens-Lewenstein (EWL) quantum game is an early example of the two-player classical Prisoner's Dilemma transformed into a quantum Prisoner's Dilemma. In the EWL game, the players choose pure quantum strategies represented by unitary matrices. This extension can resolve the classical dilemma by enabling cooperative equilibrium with higher payoff. In this paper, we first discuss the Extended EWL (EEWL) for multiplayer quantum games with mixed strategies. In EEWL, each player controls a set of unitary operators as quantum actions and uses a classical mixed strategy over these actions. The payoffs are defined as expectation values of Hermitian reward operators acting on a shared quantum state, which is generated and measured according to the EEWL protocol. We then propose the Unitary Strategy Matrix Exponential Algorithm (USMEA), a geometry-aware sequential algorithm for the EEWL mixed-strategy setting, in which each player jointly learns a trainable set of local unitary actions and the associated classical mixing probabilities. Thereby it acts as a learning-and-control layer for multi-agent quantum decision systems. We analyze the convergence properties of USMEA under standard smoothness and step-size conditions and validate the theory with numerical experiments. These results show how classical optimization methods can be systematically integrated into the design and analysis of engineered quantum strategic interactions.