Dynamic Pricing of an Expiring Item under Strategic Buyers with Stochastic Arrival

2025-09-29

AI Summary

This paper studies optimal dynamic pricing for time-sensitive goods (e.g., expiring vouchers) in a setting with strategic buyers possessing private valuations and stochastic arrival times. The central challenge lies in the strategic tension between the seller’s urgency to liquidate inventory before expiration and buyers’ incentive to delay purchase in anticipation of price reductions. To address this, we propose a Value-Based Threshold (VBT) policy that decouples the buyer’s two-dimensional private type—valuation and arrival time—and rigorously establish existence and constructive characterization of the Bayesian Nash equilibrium. Leveraging stochastic processes, dynamic game theory, and ordinary differential equations, we develop an analytically tractable equilibrium framework. Numerical analysis reveals: (i) linear discounting is near-optimal in thick markets, whereas fixed pricing dominates in thin markets; (ii) high buyer time-sensitivity favors linear markdowns, while patient sellers benefit from a quasi-auction mechanism—charging premium prices early and steeply slashing prices near expiration. Our results fundamentally characterize how market thickness and seller patience jointly determine optimal pricing structure.

Technology Category

• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• Reasoning under Uncertainty: Sequential Decision Making
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We study the optimal dynamic pricing of an expiring ticket or voucher, sold by a time-sensitive seller to strategic buyers who arrive stochastically with private values. The expiring nature creates a conflict: the seller's urgency to sell before expiration drives price reductions, which in turn incentivize buyers to wait. We seek the seller's optimal pricing policy that resolves this tension. The main analytical challenge is that buyer type is two-dimensional (valuation and arrival time), which makes equilibrium intractable under general strategies. To address this, we introduce the Value-Based Threshold (VBT) strategy, a tractable framework that decouples these two dimensions. Using this framework, we prove equilibrium existence via an ordinary differential equation and provide a constructive procedure for its characterization. We then derive near-optimal pricing policies for two stylized regimes: a constant price in thin markets and a linear discount in thick markets. Numerical frontier analysis confirms these benchmarks and shows how optimal policy adapts as the seller's time sensitivity changes. Our findings clarify the conflict between quick sales and strategic waiting. Sellers facing thick markets or high time sensitivity benefit from linear discounts, while in thin markets a constant price neutralizes buyers' incentive to wait. We also show this simple policy remains robust across broad conditions. For patient sellers, a quasi-auction schedule that maintains a high price until a sharp final drop is most effective in aggregating demand.

Problem

• Optimizing dynamic pricing for expiring items
• Resolving seller urgency versus buyer waiting incentives
• Developing tractable strategies for two-dimensional buyer types

Innovation

• Value-Based Threshold strategy decouples buyer dimensions
• Constructive procedure characterizes equilibrium via differential equation
• Linear discounts for thick markets, constant price for thin