Robust Dynamic Pricing with Strategic Customers

We consider the canonical revenue management (RM) problem wherein a seller must sell an inventory of some product over a finite horizon via an anonymous, posted price mechanism. Unlike typical models in RM, we assume that customers are forward looking. In particular, customers arrive randomly over time and strategize about their times of purchases. The private valuations of these customers decay over time and the customers incur monitoring costs; both the rates of decay and these monitoring costs are private information. This setting has resisted the design of optimal dynamic mechanisms heretofore. Optimal pricing schemes-an almost necessary mechanism format for practical RM considerations-have been similarly elusive. The present paper proposes a mechanism we dub robust pricing. Robust pricing is guaranteed to achieve expected revenues that are at least within 29% of those under an optimal (not necessarily posted price) dynamic mechanism. We thus provide the first approximation algorithm for this problem. The robust pricing mechanism is practical, since it is an anonymous posted price mechanism and since the seller can compute the robust pricing policy for a problem without any knowledge of the distribution of customer discount factors and monitoring costs. The robust pricing mechanism also enjoys the simple interpretation of solving a dynamic pricing problem for myopic customers with the additional requirement of a novel “restricted sub-martingale constraint” on prices that discourages rapid discounting. We believe this interpretation is attractive to practitioners. Finally, numerical experiments suggest that the robust pricing mechanism is, for all intents, near optimal

Contextual Dynamic Pricing with Strategic Buyers

Personalized pricing, which involves tailoring prices based on individual characteristics, is commonly used by firms to implement a consumer-specific pricing policy. In this process, buyers can also strategically manipulate their feature data to obtain a lower price, incurring certain manipulation costs. Such strategic behavior can hinder firms from maximizing their profits. In this paper, we study the contextual dynamic pricing problem with strategic buyers. The seller does not observe the buyer's true feature, but a manipulated feature according to buyers' strategic behavior. In addition, the seller does not observe the buyers' valuation of the product, but only a binary response indicating whether a sale happens or not. Recognizing these challenges, we propose a strategic dynamic pricing policy that incorporates the buyers' strategic behavior into the online learning to maximize the seller's cumulative revenue. We first prove that existing non-strategic pricing policies that neglect the buyers' strategic behavior result in a linear $\Omega(T)$ regret with $T$ the total time horizon, indicating that these policies are not better than a random pricing policy. We then establish that our proposed policy achieves a sublinear regret upper bound of $O(\sqrt{T})$. Importantly, our policy is not a mere amalgamation of existing dynamic pricing policies and strategic behavior handling algorithms. Our policy can also accommodate the scenario when the marginal cost of manipulation is unknown in advance. To account for it, we simultaneously estimate the valuation parameter and the cost parameter in the online pricing policy, which is shown to also achieve an $O(\sqrt{T})$ regret bound. Extensive experiments support our theoretical developments and demonstrate the superior performance of our policy compared to other pricing policies that are unaware of the strategic behaviors

Posted Price Mechanisms and Optimal Threshold Strategies for Random Arrivals

The classic prophet inequality states that, when faced with a finite sequence of non-negative independent random variables, a gambler who knows their distribution and is allowed to stop the sequence at any time, can obtain, in expectation, at least half as much reward as a prophet who knows the values of each random variable and can choose the largest one. In this work we consider the situation in which the sequence comes in random order. We look at both a non-adaptive and an adaptive version of the problem. In the former case the gambler sets a threshold for every random variable a priori, while in the latter case the thresholds are set when a random variable arrives. For the non-adaptive case, we obtain an algorithm achieving an expected reward within at least a 1-1/e fraction of the expected maximum and prove this constant is optimal. For the adaptive case with i.i.d. random variables, we obtain a tight 0.745-approximation, solving a problem posed by Hill and Kertz in 1982. We also apply these prophet inequalities to posted price mechanisms, and prove the same tight bounds for both a non-adaptive and an adaptive posted price mechanism when buyers arrive in random order

The Value of Observing the Buyer Arrival Time in Dynamic Pricing

We consider a dynamic pricing problem where a firm sells one item to a single buyer in order to maximize expected revenues. The firm commits to a price function over an infinite horizon. The buyer arrives at some random time with a private value for the item. He is more impatient than the seller and strategizes the time of his purchase in order to maximize his expected utility, which implies either buying immediately or waiting to benefit from a lower price. We study how important is to observe the buyer arrival time in terms of the seller's expected revenue. When the seller can observe the arrival of the buyer, she can make the price function contingent on his arrival time. On the contrary, when the seller cannot observe the arrival, her price function is fixed at time zero for the whole horizon. The value of observability (VO) is defined as the worst case ratio between the expected revenue of the seller when she observes the buyer's arrival and that when she does not. First, we show that for the particular case where the buyer's valuation follows a monotone hazard rate distribution, the upper bound is e, and it is tight. Next, we show our main result: In a very general setting about valuation and arrival time distributions, the value of observability is at most 4.911. To obtain this bound we fully characterize the observable arrival setting and use this solution to construct a random and periodic price function for the unobservable case. Finally, we show by solving a particular example to optimality that VO has a lower bound of 1.017.Este documento es una versión del artículo publicado en Management Science (ahead of print

Pricing of reusable resources under ambiguous distributions of demand and service time with emerging applications

Monopolistic pricing models for revenue management are widely used in practice to set prices of multiple products with uncertain demand arrivals. The literature often assumes deterministic time of serving each demand and that the distribution of uncertainty is fully known. In this paper, we consider a new class of revenue management problems inspired by emerging applications such as cloud computing and city parking, where we dynamically determine prices for multiple products sharing limited resource and aim to maximize the expected revenue over a finite horizon. Random demand of each product arrives in each period, modeled by a function of the arrival time, product type, and price. Unlike the traditional monopolistic pricing, here each demand stays in the system for uncertain time. Both demand and service time follow ambiguous distributions, and we formulate robust deterministic approximation models to construct efficient heuristic fixed-price pricing policies. We conduct numerical studies by testing cloud computing service pricing instances based on data published by the Amazon Web Services (AWS) and demonstrate the efficacy of our approach for managing revenue and risk under various distributions of demand and service time

Pricing of reusable resources under ambiguous distributions of demand and service time with emerging applications

Monopolistic pricing models for revenue management are widely used in practice to set prices of multiple products with uncertain demand arrivals. The literature often assumes deterministic time of serving each demand and that the distribution of uncertainty is fully known. In this paper, we consider a new class of revenue management problems inspired by emerging applications such as cloud computing and city parking, where we dynamically determine prices for multiple products sharing limited resource and aim to maximize the expected revenue over a finite horizon. Random demand of each product arrives in each period, modeled by a function of the arrival time, product type, and price. Unlike the traditional monopolistic pricing, here each demand stays in the system for uncertain time. Both demand and service time follow ambiguous distributions, and we formulate robust deterministic approximation models to construct efficient heuristic fixed-price pricing policies. We conduct numerical studies by testing cloud computing service pricing instances based on data published by the Amazon Web Services (AWS) and demonstrate the efficacy of our approach for managing revenue and risk under various distributions of demand and service time