[Paper Review] No-regret Learning in Price Competitions under Consumer Reference Effects

Negin Golrezaei, Patrick Jaillet+1 more|arXiv (Cornell University)|Nov 7, 2020

Advanced Bandit Algorithms Research20 references4 citations

TL;DR

This paper studies repeated price competition between two firms with consumer reference effects, where demand depends on current prices and a memory-based reference price. Using online mirror descent (OMD) with decreasing step sizes, the authors prove that firms' prices and reference prices converge to a Stable Nash Equilibrium (SNE), ensuring long-run market stability despite limited information and adversarial feedback.

ABSTRACT

We study long-run market stability for repeated price competitions between two firms, where consumer demand depends on firms' posted prices and consumers' price expectations called reference prices. Consumers' reference prices vary over time according to a memory-based dynamic, which is a weighted average of all historical prices. We focus on the setting where firms are not aware of demand functions and how reference prices are formed but have access to an oracle that provides a measure of consumers' responsiveness to the current posted prices. We show that if the firms run no-regret algorithms, in particular, online mirror descent(OMD), with decreasing step sizes, the market stabilizes in the sense that firms' prices and reference prices converge to a stable Nash Equilibrium (SNE). Interestingly, we also show that there exist constant step sizesunder which the market stabilizes. We further characterize the rate of convergence to the SNE for both decreasing and constant OMD step sizes.

Motivation & Objective

• To investigate whether firms using no-regret learning algorithms can achieve long-term market stability in repeated price competition with consumer reference effects.
• To model consumer reference prices as a weighted average of past prices, reflecting memory-based demand dynamics.
• To analyze market stability under incomplete information, where firms lack knowledge of demand functions or reference price formation.
• To establish conditions under which firms' strategies and reference prices converge to a Stable Nash Equilibrium (SNE).
• To characterize the convergence rate of prices and reference prices under both decreasing and constant OMD step sizes.

Proposed method

• Formalize the market as a dynamic game with two real firms and a virtual 'nature' firm representing the reference price dynamics.
• Model the reference price as a time-varying state formed by a weighted average of all past prices, evolving according to firms' pricing decisions.
• Apply online mirror descent (OMD) to each firm’s pricing strategy, using an oracle that provides real-time responsiveness feedback.
• Transform the two-firm game with a dynamic state into a three-firm game by introducing a virtual firm (nature) that learns with a constant step size.
• Use Bregman divergence and strong convexity to analyze regret and convergence, leveraging properties of OMD under varying step sizes.
• Prove convergence to SNE by showing that the distance to equilibrium decreases over time, even when the virtual firm learns faster than the real firms.

Experimental results

Research questions

• RQ1 Under what conditions do firms' prices and consumer reference prices converge to a Stable Nash Equilibrium (SNE) in repeated price competition with reference effects?
• RQ2 Can market stability be achieved when firms use no-regret learning algorithms like OMD without full knowledge of demand or reference price dynamics?
• RQ3 Does the convergence to SNE occur with decreasing step sizes in OMD, and what is the rate of convergence?
• RQ4 Can market stabilization still occur when the virtual firm (modeling reference price dynamics) uses a constant step size, despite its faster learning rate?
• RQ5 What structural properties does the Stable Nash Equilibrium (SNE) possess under the given model assumptions?

Key findings

• The paper establishes the existence of a unique Stable Nash Equilibrium (SNE) under the given model, characterized by first-order conditions where marginal profits are zero.
• With decreasing OMD step sizes, firms’ prices and reference prices converge to the SNE at a linear rate, as shown in Theorem 5.1.
• Even with a constant step size for the virtual firm (nature), the real firms can still stabilize the market by using decreasing step sizes in OMD.
• The convergence proof is non-trivial due to the presence of a fast-learning virtual firm, which invalidates standard multi-agent online learning convergence results.
• The analysis shows that the Bregman divergence and strong convexity of the regularizer ensure that the distance to equilibrium decreases over time, enabling convergence.
• Corollary 10.1.1 confirms that at the SNE, the gradient of the profit function with respect to each firm’s price is zero, confirming equilibrium conditions.