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In online sales, sellers usually offer each potential buyer a posted price in a take-it-or-leave fashion. Buyers can sometimes see posted prices faced by other buyers, and changing the price frequently could be considered unfair. The literature on posted-price mechanisms and prophet inequality problems has studied the two extremes of pricing policies, the fixed-price policy and fully dynamic pricing. The former is suboptimal in revenue but is perceived as fairer than the latter. This work examines the middle situation, where there are at most k distinct prices over the selling horizon. Using the framework of prophet inequalities with independent and identically distributed random variables, we propose a new prophet inequality for strategies that use at most k thresholds. We present asymptotic results in k and results for small values of k. For k = 2 prices, we show an improvement of at least 11% over the best fixed-price solution. Moreover, k = 5 prices suffice to guarantee almost 99% of the approximation factor obtained by a fully dynamic policy that uses an arbitrary number of prices. From a technical standpoint, we use an infinite-dimensional linear program in our analysis; this formulation could be of independent interest to other online selection problems. 

Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform’s algorithm may miss the users most disposed to do so. This mismatch decreases the platform’s revenue and the advertiser’s chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability p. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers k of the probability of choosing one of the top k candidates, given that one of these candidates will accept an offer. Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models. We further relax this linear program into an infinite counterpart, which we use to provide bounds for the objective and closed-form policies. For [Formula: see text], an optimal policy is a simple threshold rule that observes the first [Formula: see text] fraction of candidates and subsequently makes offers to the best candidate observed so far. Funding: Financial support from the U.S. National Science Foundation [Grants CCF-2106444, CCF-1910423, and CMMI 1552479] is gratefully acknowledged. 

Cloud computing has motivated renewed interest in resource allocation problems with new consumption models. A common goal is to share a resource, such as CPU or I/O bandwidth, among distinct users with different demand patterns as well as different quality of service requirements. To ensure these service requirements, cloud offerings often come with a service level agreement (SLA) between the provider and the users. A SLA specifies the amount of a resource a user is entitled to utilize. In many cloud settings, providers would like to operate resources at high utilization while simultaneously respecting individual SLAs. There is typically a trade-off between these two objectives; for example, utilization can be increased by shifting away resources from idle users to “scavenger” workload, but with the risk of the former then becoming active again. We study this fundamental tradeoff by formulating a resource allocation model that captures basic properties of cloud computing systems, including SLAs, highly limited feedback about the state of the system, and variable and unpredictable input sequences. Our main result is a simple and practical algorithm that achieves near-optimal performance on the above two objectives. First, we guarantee nearly optimal utilization of the resource even if compared with the omniscient offline dynamic optimum. Second, we simultaneously satisfy all individual SLAs up to a small error. The main algorithmic tool is a multiplicative weight update algorithm and a primal-dual argument to obtain its guarantees. We also provide numerical validation on real data to demonstrate the performance of our algorithm in practical applications.