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

Abstract: We consider price competition among multiple sellers over a selling horizon of $T$ periods. In each period, sellers simultaneously offer their prices and subsequently observe their respective demand that is unobservable to competitors. The demand function for each seller depends on all sellers' prices through a private, unknown, and nonlinear relationship. To address this challenge, we propose a semi-parametric least-squares estimation of the nonlinear mean function, which does not require sellers to communicate demand information. We show that when all 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.

Optimal Nonlinear Online Learning under Sequential Price Competition via (s)-Concavity

The paper "Optimal Nonlinear Online Learning under Sequential Price Competition via (s)-Concavity" by Daniele Bracale and colleagues addresses the complex problem of price competition among multiple sellers in a dynamic market setting. The focus is on developing a robust semi-parametric estimation framework to determine optimal pricing policies without requiring communication of demand information between competitors. The challenge is compounded by the nonlinear nature of demand functions, which depend on both private and unknown parameters.

Key Contributions and Methodology:

1. Nonlinear Demand Modeling: The authors model the demand function as a monotone single-index model, (\lambda_i(\mathbf{p}) = \psi_i(\theta_i^{\top}\mathbf{p})), where (\psi_i) is a non-decreasing and nonlinear function. This framework extends beyond traditional linear models, thereby accommodating more realistic demand scenarios where the relationship between price and demand is inherently nonlinear.

2. Shape-Constrained Estimation: A noteworthy theoretical contribution is the introduction of (s)-concavity as a shape constraint on the demand function (\psi_i). This condition not only guarantees the existence of a Nash Equilibrium (NE) but also ensures the convergence of the sellers’ pricing policies towards this equilibrium. The (s)-concavity condition is a generalization of log-concavity and is tied to the economic concept of virtual valuation function, (\varphi_i(u) = u + \frac{\psi_i(u)}{\psi_i'(u)}), being increasing.

3. Algorithm Development: The authors propose a semi-parametric estimation then best-response (SPE-BR) algorithm. This algorithm is structured into an initial exploration phase where sellers gather data to estimate the parameters of their demand functions, followed by an exploitation phase where these estimates are used to inform pricing decisions. The algorithm guarantees that each firm in a competitive market setting achieves sublinear regret relative to a dynamic benchmark policy.

4. Convergence and Regret Analysis: The paper proves that the regret incurred by each seller is (O(T^{5/7})), which is minimal given the complexity of the problem. Additionally, it demonstrates that the joint pricing strategies of sellers converge to the Nash Equilibrium at a rate of (O(T^{-1/7})).

5. Numerical Simulations: Experiments demonstrate the practical applicability of the SPE-BR algorithm, showing that it consistently converges to the Nash Equilibrium even in high-dimensional settings with multiple sellers.

Implications and Future Directions:

The study has significant implications for dynamic pricing in competitive markets. It provides a framework that balances the trade-offs between exploration (learning the market demand curve) and exploitation (optimizing pricing decisions based on learned information). Notably, the application of (s)-concavity enables the formulation of a fully data-driven, tuning-parameter-free algorithm, which is particularly beneficial in real-world applications where setting of parameters can be challenging.

Future research may expand on this work by addressing adaptive exploration-exploitation strategies where the exploration phase is not fixed but dynamically optimized through learning. Furthermore, the real-time application of such algorithms in digital marketplaces and the incorporation of additional factors such as inventory constraints and dynamic externalities would enrich this line of research, enhancing its practical viability. Additionally, exploring how (s)-concavity and related shape constraints can be harnessed in other domains beyond pricing, such as in optimal auction design and resource allocation, could provide further theoretical and practical insights.

The paper successfully melds theoretical advancements with practical algorithmic design, offering a comprehensive approach to managing competition in nonlinear pricing environments.