Revenue Maximization Under Sequential Price Competition Via The Estimation Of s-Concave Demand Functions

TL;DR

The paper tackles revenue maximization in dynamic, multi-seller price competition under unknown nonlinear demand. It introduces a tuning-free, shape-constrained semi-parametric policy leveraging monotone single-index models with $s$-concave link functions, estimated in two exploration phases before exploiting a learned best-response mapping. The authors establish existence and uniqueness of Nash equilibrium under these constraints, derive regret bounds with an optimal exploration parameter, and prove NE convergence rates, complemented by concentration results for the $s$-concave regression estimator. Numerical experiments validate the theoretical rates and demonstrate robustness to misspecification, while providing practical insights into dynamic, learning-enabled pricing with nonlinear demand. The framework advances nonparametric learning in strategic settings by combining shape constraints with online learning in a coherent equilibrium-focused analysis.

Abstract

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.

Figures (10)

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• Figure 4: Illustration of our policy (\ref{['algo:semiparametric']}) with $N=4$ sellers in sequential price competition under nonlinear demands. For each seller $i \in\{1,2,3,4\}$, in their exploration phase (dotted line) of length $\tau_i$, they offer randomized prices following their distribution $\mathscr{D}_i$. Within the exploration phase, each seller has a private phase for estimating $\boldsymbol{\theta}_i$ (blue box with dotted border), with length $\tau_i\kappa_i$ and a private phase for estimating $\psi_i$ (yellow box with continued border line), with length $\tau_i(1-\kappa_i)$. The $i$-th seller's price experiment ends at period $t= \tau_i$ (black circles). Subsequently, in their exploitation phase (represented by the continued black line), seller $i$ offers prices based on the estimators generated in the exploration phase.
• Figure 5: Performance of Algorithm \ref{['algo:semiparametric']} in sequential price competition with $N \in \{2,4,6\}$ sellers for different values of the contraction constant $L_{\boldsymbol{\Gamma}}$.

Theorems & Definitions (64)

• Lemma 3.3: Existence of NE
• Proposition 3.5
• Lemma 3.7
• Remark 3.8
• Remark 4.1: The common exploration phase
• Remark 4.2: Need for two different phases for model estimation
• Remark 4.3: Selection of $\tau$ and $\kappa_i$
• Proposition 5.2: Informal version of \ref{['prop:concentration_ineq_theta']}
• Theorem 5.3: Informal version of \ref{['thm:Dumbgen']}
• Theorem 5.4