Lying to a Bayesian: Optimal Randomisation in Repeated Pricing

Reinforcement Learning has succeeded at a wide range of games ( Chess, Go, Atari ). Here we apply RL methods to a game-theoretic problem in economics : a repeated pricing game between a Bayesian-updating producer and a consumer with private willingness-to-pay. We established existence of the mixed strategy Nash equilibrium for the continuous-action extension through Reny (1999), which we then approximate computationally via three RL methods. We have used three different approaches (Monte Carlo, Q-learning, Deep Q-Network) to simulate the benefits of randomisation for consumer’s behaviour. All three approaches show a consistent result that $\epsilon^{*}<1$, supporting randomisation as an obfuscation strategy. Tabular Q-learning shows no clear monotonic trend in $\epsilon^{*}(\beta)$, while DQN recovers the monotone decrease predicted by Monte Carlo, reconciling the previously-observed discrepancy. We empirically validate the Karp-Kleinberg lower bound on posterior variance, confirming that mixed strategies prevent posterior collapse.