Decentralized Learning of Belief-Dependent Generalized Nash Equilibria under Partial Observability

We study distributed decision making under partial observability in constrained multi-agent systems. Agents operate with local sensing, limited communication, and noisy exogenous information while jointly satisfying coupled proximity constraints and learning an unknown latent parameter governing payoffs. We model the interaction as a belief-dependent generalized Nash equilibrium (GNE), defined as the equilibrium of an augmented belief-state game induced by agents' posterior beliefs. We develop a fully decentralized discrete-time learning scheme combining quadratic-penalty constraint enforcement with distributed Thompson sampling, enabling agents to simultaneously learn the payoff parameter and coordinate on constraint-feasible equilibria. Under mild regularity, constraint qualification, and posterior concentration assumptions, we prove that the induced stochastic dynamics converges almost surely to a belief-dependent GNE of the true constrained game. Finitetime performance is characterized through information-theoretic Bayesian regret bounds. Numerical experiments validate the theory and demonstrate that information quality, rather than information quantity, is the dominant driver of coordination.