Dynamic Service Fee Pricing under Strategic Behavior: Actions as Instruments and Phase Transition

We study a dynamic pricing problem for third-party platform service fees under strategic, far-sighted customers. In each time period, the platform sets a service fee based on historical data, observes the resulting transaction quantities, and collects revenue. The platform also monitors equilibrium prices influenced by both demand and supply. The objective is to maximize total revenue over a time horizon $T$. Our problem incorporates three practical challenges: (a) initially, the platform lacks knowledge of the demand side beforehand, necessitating a balance between exploring (learning the demand curve) and exploiting (maximizing revenue) simultaneously; (b) since only equilibrium prices and quantities are observable, traditional Ordinary Least Squares (OLS) estimators would be biased and inconsistent; (c) buyers are rational and strategic, seeking to maximize their consumer surplus and potentially misrepresenting their preferences. To address these challenges, we propose novel algorithmic solutions. Our approach involves: (i) a carefully designed active randomness injection to balance exploration and exploitation effectively; (ii) using non-i.i.d. actions as instrumental variables (IV) to consistently estimate demand; (iii) a low-switching cost design that promotes nearly truthful buyer behavior. We show an expected regret bound of $\tilde{\mathcal{O}} (\sqrt{T}\wedge\sigma_S^{-2})$ and demonstrate its optimality, up to logarithmic factors, with respect to both the time horizon $T$ and the randomness in supply $\sigma_S$. Despite its simplicity, our model offers valuable insights into the use of actions as estimation instruments, the benefits of low-switching pricing policies in mitigating strategic buyer behavior, and the role of supply randomness in facilitating exploration which leads to a phase transition of policy performance.