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We consider a dynamic pricing problem where customer response to the current price is impacted by the customer price expectation, aka reference price. We study a simple and novel reference price mechanism where reference price is the average of the past prices offered by the seller. As opposed to the more commonly studied exponential smoothing mechanism, in our reference price mechanism the prices offered by seller have a longer-term effect on the future customer expectations. We show that under this mechanism, a markdown policy is near-optimal irrespective of the parameters of the model. This matches the common intuition that a seller may be better off by starting with a higher price and then decreasing it, as the customers feel like they are getting bargains on items that are ordinarily more expensive. For linear demand models, we also provide a detailed characterization of the near-optimal markdown policy along with an efficient way of computing it. We then consider a more challenging dynamic pricing and learning problem, where the demand model parameters are apriori unknown, and the seller needs to learn them online from the customers’ responses to the offered prices while simultaneously optimizing revenue. The objective is to minimize regret, i.e., the 𝑇-round revenue loss compared to a clairvoyant optimal policy. This task essentially amounts to learning a non-stationary optimal policy in a time-variant Markov Decision Process (MDP). For linear demand models, we provide an efficient learning algorithm with an optimal 𝑂(√𝑇 ) regret upper bound.more » « lessFree, publicly-accessible full text available July 11, 2025
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Free, publicly-accessible full text available July 8, 2025
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We present an algorithm based on posterior sampling (aka Thompson sampling) that achieves near-optimal worst-case regret bounds when the underlying Markov decision process (MDP) is communicating with a finite, although unknown, diameter. Our main result is a high probability regret upper bound of [Formula: see text] for any communicating MDP with S states, A actions, and diameter D. Here, regret compares the total reward achieved by the algorithm to the total expected reward of an optimal infinite-horizon undiscounted average reward policy in time horizon T. This result closely matches the known lower bound of [Formula: see text]. Our techniques involve proving some novel results about the anti-concentration of Dirichlet distribution, which may be of independent interest.more » « less
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We consider a stochastic inventory control problem under censored demand, lost sales, and positive lead times. This is a fundamental problem in inventory management, with significant literature establishing near optimality of a simple class of policies called “base-stock policies” as well as the convexity of long-run average cost under those policies. We consider a relatively less studied problem of designing a learning algorithm for this problem when the underlying demand distribution is unknown. The goal is to bound the regret of the algorithm when compared with the best base-stock policy. Our main contribution is a learning algorithm with a regret bound of [Formula: see text] for the inventory control problem. Here, [Formula: see text] is the fixed and known lead time, and D is an unknown parameter of the demand distribution described roughly as the expected number of time steps needed to generate enough demand to deplete one unit of inventory. Notably, our regret bounds depend linearly on L, which significantly improves the previously best-known regret bounds for this problem where the dependence on L was exponential. Our techniques utilize the convexity of the long-run average cost and a newly derived bound on the “bias” of base-stock policies to establish an almost black box connection between the problem of learning in Markov decision processes (MDPs) with these properties and the stochastic convex bandit problem. The techniques presented here may be of independent interest for other settings that involve large structured MDPs but with convex asymptotic average cost functions.more » « less
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We study revenue optimization in a repeated auction between a single seller and a single buyer. Traditionally, the design of repeated auctions requires strong modeling assumptions about the bidder behavior, such as it being myopic, infinite lookahead, or some specific form of learning behavior. Is it possible to design mechanisms which are simultaneously optimal against a multitude of possible buyer behaviors? We answer this question by designing a simple state-based mechanism that is simultaneously approximately optimal against a k-lookahead buyer for all k, a buyer who is a no-regret learner, and a buyer who is a policy-regret learner. Against each type of buyer our mechanism attains a constant fraction of the optimal revenue attainable against that type of buyer. We complement our positive results with almost tight impossibility results, showing that the revenue approximation tradeoffs achieved by our mechanism for different lookahead attitudes are near-optimal.more » « less