Price Competition Under A Consider-Then-Choose Model With Lexicographic Choice

TL;DR

We address price competition under the Consider-then-Choose with Lexicographic Choice (CLC) model, capturing how platform search and filtering shape customer decisions. The authors introduce a pseudo-competitive property and a Local Nash Equilibrium (LNE) framework, along with a Valid Order-Price Pair (VOP) representation that yields tractable equilibrium analysis for small seller populations. Under a gradient-dominance condition, they prove the existence of a distinct-price LNE and show that distributed gradient ascent converges to an LNE; numerical experiments further demonstrate convergence even when gradient dominance weakens. The study provides practical insights for platform design and decentralized pricing strategies, linking sorting/filtering mechanisms to predictable, learnable pricing equilibria on competitive marketplaces.

Abstract

The sorting and filtering capabilities offered by modern e-commerce platforms significantly impact customers' purchase decisions, as well as the resulting prices set by competing sellers on these platforms. Motivated by this practical reality, we study price competition under a flexible choice model: Consider-then-Choose with Lexicographic Choice (CLC). In this model, a customer first forms a consideration set of sellers based on (i) her willingness-to-pay and (ii) an arbitrary set of criteria on items' non-price attributes; she then chooses the highest-ranked item according to a lexicographic ranking in which items with better performance on more important attributes are ranked higher. We provide a structural characterization of equilibria in the resulting game of price competition, and derive an economically interpretable condition, which we call gradient dominance, under which equilibria can be computed efficiently. For this subclass of CLC models, we prove that distributed gradient-based pricing dynamics converge to the set of equilibria. Extensive numerical experiments show robustness of our theoretical findings when gradient dominance does not hold.

Figures (2)

• Figure 1: Plot of $R_2(p_1, p_2)$, for different values of $p_1$, when customers satisfice in a duopoly. Here we assume an equal proportion of customers prefer seller 1 to seller 2 (and vice versa), and $w_c\sim\textup{Unif}[0,1]$. This induces a revenue function $R_2(p_1,p_2) = p_2\left(\frac{1}{2}(1-p_2) + \frac{1}{2}(p_1-p_2)^+\right)$ for seller 2. Observe that $R_2(p_1,p_2)$ is nonsmooth. Moreover, for $p_1 = 0.4$, it has two local maxima: one at 0.5 and one at 0.35.
• Figure 2: Sample non-convergent trajectories in Settings A-C

Theorems & Definitions (50)

• Example 1.1
• Definition 3.1: Lexicographic preferences
• Remark 3.2
• Definition 3.3: Consider-then-Choose with Lexicographic Choice
• Remark 3.4
• Proposition 4.1
• Definition 4.2: Local Nash Equilibrium
• Remark 4.3
• Definition 4.4: Pseudo-Competitive Property
• Theorem 4.5