Game, Set, Quantum: Parameterized Quantum Circuit for Correlated Equilibrium in Bayesian Games

Abstract

Strategic decision-making among many agents under incomplete information is central to economics, security, and multi-agent artificial intelligence (AI). Computing equilibria in such settings is challenging because the joint type-action space grows exponentially with the number of players. In binary-type, binary-action Bayesian games with $n$ players, an explicit representation over type-action profiles requires $O(2^{2n})$ entries, making direct linear-programming (LP) formulations increasingly costly as $n$ grows. We propose a hybrid quantum-classical framework for approximating Bayes correlated equilibrium (BCE) using a parameterized quantum circuit (PQC). The PQC represents the conditional distribution over joint actions using $O(nL)$ trainable parameters, where $L$ denotes the circuit depth; for the largest trained setting, $n=8$ and $L=2$, this corresponds to $48$ trainable angles. Each player count is trained independently by maximizing expected social welfare with a penalty on positive aggregated BCE obedience violations. On a strategically coupled Bayesian congestion game with $n=2,4,6,8$ players, feasible PQC solutions attain higher welfare than MCCFR and DCFR product-strategy baselines while satisfying $ε_{\max}\leq10^{-3}$, where $ε_{\max}$ denotes the maximum positive aggregated BCE obedience violation. Across five independent runs per setting, all runs are feasible for $n=2,4,6$, while four of five are feasible for $n=8$. PQC welfare remains below the exact LP optimum, with the absolute gap increasing with $n$, while classical state-vector simulation prevents PQC training beyond eight players. These results demonstrate the use of a compact PQC parameterization for approximate equilibrium computation and quantify its welfare, feasibility, and classical simulation scaling on the studied benchmark.