Strategic Buying Agents

Abstract

The emergence of agentic AI is shifting online shopping from passive search and recommendation toward autonomous buying agents that monitor markets, reason about uncertainty, and make purchase decisions on behalf of consumers. We study the design of strategic buying agents that decide when to purchase an item within a finite shopping window. The central challenge is to translate price observations, the shopping window, and information about future price changes into a purchase-timing rule. We formulate this problem across a hierarchy of informational environments. In a stationary environment, prices are assumed to adjusted following an exogenously given Poisson price, and refreshed prices are drawn from a stable distribution; the optimal policy is a dynamic purchase threshold characterized by an ordinary differential equation. In a Bayesian learning environment, the price update rate is assumed to be known, but the price adjustment distribution is uncertain; the optimal rule remains threshold-based, with the threshold depending on posterior beliefs. We also bound the value of knowing the true price adjustment distribution. In a model-free robust environment, the agent relies only on price bounds and seeks worst-case protection; randomized static thresholds yield sharp guarantees for both competitive ratios and minimax regret. Finally, using Keepatracked Amazon product price histories of 367 items and 48,933 time-stamped posted-price observations, we evaluate our three policies and examine how they can be incorporated in language-model buying agents. The results show that these policies are competitive on the test instances and suggest that language models are more effective when used to select a policy and calibrate inputs than when used to make buy-or-wait decisions directly.