Revenue Maximization Under Sequential Price Competition Via The Estimation Of s-Concave Demand Functions
This addresses revenue maximization for sellers in sequential price competition, offering incremental improvements in regret bounds and equilibrium analysis under shape constraints.
The paper tackles dynamic pricing competition among sellers with unknown nonlinear demand functions, proposing a policy that achieves O(T^{-1/7}) convergence to Nash equilibrium prices and O(T^{5/7}) regret relative to a dynamic benchmark.
We consider price competition among multiple sellers over a selling horizon of $T$ periods. In each period, sellers simultaneously offer their prices (which are made public) and subsequently observe their respective demand (not made public). The demand function of each seller depends on all sellers' prices through a private, unknown, and nonlinear relationship. We propose a dynamic pricing policy that uses semi-parametric least-squares estimation and show that when the sellers employ our policy, their prices converge at a rate of $O(T^{-1/7})$ to the Nash equilibrium prices that sellers would reach if they were fully informed. Each seller incurs a regret of $O(T^{5/7})$ relative to a dynamic benchmark policy. A theoretical contribution of our work is proving the existence of equilibrium under shape-constrained demand functions via the concept of $s$-concavity and establishing regret bounds of our proposed policy. Technically, we also establish new concentration results for the least squares estimator under shape constraints. Our findings offer significant insights into dynamic competition-aware pricing and contribute to the broader study of non-parametric learning in strategic decision-making.