EXPRESS: Learning to Price under Competition with Reference Price Effects

Abstract

This paper studies the algorithmic design of price competition in oligopolistic markets, with a focus on long-run market dynamics under reference price effects. We consider a sequential price competition framework with multiple sellers operating over a finite horizon, each lacking prior knowledge of the demand functions. Consumer demand depends not only on current prices but also on a reference price, defined as a weighted average of past prices that shapes consumer expectations. We characterize market stability through the concept of a stationary Nash equilibrium (SNE), where no seller has an incentive to deviate unilaterally and the reference price remains stable. To operate under incomplete demand information and limited observability of competitors' prices, we propose a Simultaneous Perturbation Stochastic Approximation with Callbacks (SPSAC) policy. Despite the lack of a monotonicity condition, which is typically required for convergence in online games, we show that this policy achieves a last-iterate convergence rate of O ( 1 / T ) for both prices and reference prices toward the SNE, along with a dynamic regret of O ( T ) over T periods. We further consider three extensions. In the first extension, when sellers have access to first-order feedback, the convergence rate improves to O ( 1 / T ) , highlighting the value of demand information in simplifying sequential price competition. In the second extension, where prices have long-term effects on the reference price and demand evolves as a nonstationary stochastic process, and in the third extension, where each seller maintains its own reference price, our policy continues to guarantee O ( 1 / T ) convergence and O ( T ) dynamic regret.