Coarse Q-learning: Indifference, Indeterminacy, and Instability

We introduce Coarse Q-learning (CQL), a reinforcement-learning model for bandit problems with stochastically varying menus. Alternatives are exogenously partitioned into similarity classes, and feedback from sampled alternatives is pooled within classes into class-level valuations. Choices follow multinomial logit over class valuations, and valuations update toward realized payoffs as in Q-learning. Using stochastic approximation, we derive the mean-field dynamics and characterize the steady states as smooth analogues of Valuation Equilibria. The model yields novel long-run phenomena in the high payoff-sensitivity limit: depending on the environment, CQL may exhibit multiple stable strict equilibria, a unique globally stable mixed equilibrium with indifference across classes, or no stable equilibrium at all, with valuations and choice probabilities converging instead to a stable limit cycle. These outcomes are driven by coarse aggregation and do not arise in the standard alternative-level benchmark.