Near-Optimal Quantum Algorithms for Computing (Coarse) Correlated Equilibria of General-Sum Games

Abstract

Computing Nash equilibria of zero-sum games in classical and quantum settings is extensively studied. For general-sum games, computing Nash equilibria is PPAD-hard and the computing of a more general concept called correlated equilibria has been widely explored in game theory. In this paper, we initiate the study of quantum algorithms for computing $\epsilon$-approximate correlated equilibria (CE) and coarse correlated equilibria (CCE) in multi-player normal-form games. Our approach utilizes quantum improvements to the multi-scale Multiplicative Weight Update (MWU) method for CE calculations, achieving a query complexity of $\tilde{O}(m\sqrt{n})$ for fixed $\epsilon$. For CCE, we extend techniques from quantum algorithms for zero-sum games to multi-player settings, achieving query complexity $\tilde{O}(m\sqrt{n}/{\epsilon }^{2.5})$. Both algorithms demonstrate a near-optimal scaling in the number of players $m$ and actions $n$, as confirmed by our quantum query lower bounds.