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Competitive Demand Learning: A Non-cooperative Pricing Algorithm with Coordinated Price Experimentation

Yongge Yang, Yu-Ching Lee, Po-An Chen

TL;DR

This paper addresses dynamic, competitive pricing when firms face unknown demand. It introduces the Coordinated Demand Learning (CDL) algorithm, which orchestrates price experiments across firms to reveal demand responses while steering prices toward the clairvoyant Nash equilibrium. The authors establish convergence via contraction properties, and derive sublinear regret bounds of $O(F\sqrt{T})$ and revenue-difference bounds of $O(F^2 T^{3/4})$, with extensions to partially clairvoyant settings. Numerical experiments on linear and multinomial logit demands corroborate the theoretical guarantees and illustrate how learning costs scale with the number of firms. The work offers a practical mechanism for platform-mediated, information-rich, coordinated pricing in competitive markets with unknown demand functions.

Abstract

We consider a periodical equilibrium pricing problem for multiple firms over a planning horizon of T periods. At each period, firms set their selling prices and receive stochastic demand from consumers. Firms do not know their underlying demand curve, but they wish to determine the selling prices to maximize total revenue under competition. Hence, they have to do some price experiments such that the observed demand data are informative to make price decisions. However, uncoordinated price updating can render the demand information gathered by price experimentation less informative or inaccurate. We design a nonparametric learning algorithm to facilitate coordinated dynamic pricing, in which competitive firms estimate their demand functions based on observations and adjust their pricing strategies in a prescribed manner. We show that the pricing decisions, determined by estimated demand functions, converge to underlying equilibrium as time progresses. We obtain a bound of the revenue difference that has an order of O(F^2 T^3/4) and a regret bound that has an order of O(F T^1/2) with respect to the number of the competitive firms F and T . We also develop a modified algorithm to handle the situation where some firms may have the knowledge of the demand curve.

Competitive Demand Learning: A Non-cooperative Pricing Algorithm with Coordinated Price Experimentation

TL;DR

This paper addresses dynamic, competitive pricing when firms face unknown demand. It introduces the Coordinated Demand Learning (CDL) algorithm, which orchestrates price experiments across firms to reveal demand responses while steering prices toward the clairvoyant Nash equilibrium. The authors establish convergence via contraction properties, and derive sublinear regret bounds of and revenue-difference bounds of , with extensions to partially clairvoyant settings. Numerical experiments on linear and multinomial logit demands corroborate the theoretical guarantees and illustrate how learning costs scale with the number of firms. The work offers a practical mechanism for platform-mediated, information-rich, coordinated pricing in competitive markets with unknown demand functions.

Abstract

We consider a periodical equilibrium pricing problem for multiple firms over a planning horizon of T periods. At each period, firms set their selling prices and receive stochastic demand from consumers. Firms do not know their underlying demand curve, but they wish to determine the selling prices to maximize total revenue under competition. Hence, they have to do some price experiments such that the observed demand data are informative to make price decisions. However, uncoordinated price updating can render the demand information gathered by price experimentation less informative or inaccurate. We design a nonparametric learning algorithm to facilitate coordinated dynamic pricing, in which competitive firms estimate their demand functions based on observations and adjust their pricing strategies in a prescribed manner. We show that the pricing decisions, determined by estimated demand functions, converge to underlying equilibrium as time progresses. We obtain a bound of the revenue difference that has an order of O(F^2 T^3/4) and a regret bound that has an order of O(F T^1/2) with respect to the number of the competitive firms F and T . We also develop a modified algorithm to handle the situation where some firms may have the knowledge of the demand curve.

Paper Structure

This paper contains 25 sections, 21 theorems, 133 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Our Results and Main Contributions
  3. Related Work
  4. Model and Preliminaries
  5. Design Ideas of the Algorithm
  6. Algorithm
  7. Performance Measures
  8. Discussions: Our and Related Models and Algorithms
  9. Analyses: Convergence, Revenue Difference, and Regret
  10. Limit Point Analysis
  11. Convergence Analysis
  12. Revenue Difference and Regret
  13. Extension: Partially-Clairvoyant Model and Results
  14. Numerical Experiments
  15. Conclusions
  16. ...and 10 more sections

Key Result

Lemma 1

Suppose that ${\varepsilon^i_t=0}$, ${\forall i}$ and $t$, and that the sequence $\lbrace \hat{\mathbf{p}}_n \rbrace$, assuming nonzero demand and that the price is away from the limits, generated by CDL converges to a limit point $\tilde{\mathbf{p}}$, which satisfies ${\tilde{p}^i=-\frac{\lambda^i(

Figures (5)

  • Figure 1: The prices sequences of firm 1, firm 2 and firm 3.
  • Figure 2: The averaged fraction of revenue loss and averaged fraction of revenue difference of a firm who has no knowledge of the underlying demand for different numbers $F$. (The upper panels are the results of linear underlying model and the lower panels are the results of multinomial logit underlying model.)
  • Figure 3: The log-log plots of averaged fractions of revenue loss and revenue difference of a firm who has no knowledge of the underlying demand as a function of $\log (T)$ when $F=2$ and the underlying demand curves are linear models (upper panels) or multinomial logit models (lower panels).
  • Figure 4: The averaged fractions of revenue loss and revenue difference of a firm who has no knowledge of the underlying demand curve (upper panels) and a firm who knows the underlying demand curve (lower panels). The total number of firms $F=5$ and the underlying demand curves are multinomial logit models.
  • Figure 5: The log-log plots of averaged fractions of revenue loss and revenue difference of the firms who have no knowledge of the underlying demand as a function of $\log (T)$ (upper panels) and those of the clairvoyant firms as a function of $\log (T)$ (lower panels). $F^\prime=3, F=5$, and the underlying demand curves are multinomial logit models.

Theorems & Definitions (27)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Lemma 2
  • Proposition 3
  • Theorem 1
  • ...and 17 more