Collecting probabilistic goods: a framework for sequential decision-making based on the Coupon Collector’s Problem

Abstract In this work, we investigate consumers’ sequential decision-making behavior under the setting of collection-based probabilistic selling. We formulate the sequential purchasing process as a draw-level Markov Decision Process (MDP) built on the classical Coupon Collector’s Problem. The Bellman recursion can be reduced exactly to a stage-level stopping problem, yielding an explicit value function and optimal stopping rule for forward-looking collectors. We further extend the analysis by characterizing collection goals (complete vs. partial) and quantifying the impact of consumer heterogeneity, demonstrating how risk preferences shape optimal stopping decisions. In our extensions, we analyze duplicate salvage, pity systems, and finite budget constraints and complement these results with discussions of batch purchasing and a demand-side completion premium. Finally, we provide numerical illustrations of the analytical results under selected parameter settings. This work provides a tractable analytical benchmark for understanding sequential stopping decisions in collection-based probabilistic selling.