We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.
The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, O(1)-competitive policies are achievable using only a single sample from the distribution of each agent. A notable portion of their results relies on reducing the design of single-sample prophet inequalities (SSPIs) to that of order-oblivious secretary (OOS) policies. The above reduction comes at the cost of not fully utilizing the available samples. However, to date, this is essentially the only method for proving SSPIs for many combinatorial sets (e.g., bipartite matching and various matroids). Very recently, Rubinstein et al. [ITCS'20] give a surprisingly simple algorithm which achieves the optimal competitive ratio for the single-choice SSPI problem -a result which is unobtainable going through the reduction to secretary problems.Motivated by this discrepancy, we study the competitiveness of simple SSPI policies directly, without appealing to results from OOS literature. In this direction, we first develop a framework for analyzing policies against a greedy-like prophet solution. Using this framework, we obtain the first SSPI for general (non-bipartite) matching environments, as well as improved competitive ratios for transversal and truncated partition matroids. Second, motivated by the observation that many OOS policies for matroids decompose the problem into independent rank-1 instances, we provide a meta-theorem which applies to any matroid satisfying this partition property. Leveraging the recent results by Rubinstein et al., we obtain improved competitive guarantees (most by a factor of 2) for a number of matroids captured by the reduction of Azar et al. (e.g., graphic, co-graphic, and low density matroids). Finally, we discuss applications of our SSPIs to the design of mechanisms for multi-dimensional limited information settings with improved revenue and welfare guarantees.
We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.
Considerable work has focused on optimal stopping problems where random IID offers arrive sequentially for a single available resource which is controlled by the decision-maker. After viewing the realization of the offer, the decision-maker irrevocably rejects it, or accepts it, collecting the reward and ending the game. We consider an important extension of this model to a dynamic setting where the resource is "renewable'' (a rental, a work assignment, or a temporary position) and can be allocated again after a delay period d. In the case where the reward distribution is known a priori, we design an (asymptotically optimal) 1/2-competitive Prophet Inequality, namely, a policy that collects in expectation at least half of the expected reward collected by a prophet who a priori knows all the realizations. This policy has a particularly simple characterization as a thresholding rule which depends on the reward distribution and the blocking period d, and arises naturally from an LP-relaxation of the prophet's optimal solution. Moreover, it gives the key for extending to the case of unknown distributions; here, we construct a dynamic threshold rule using the reward samples collected when the resource is not blocked. We provide a regret guarantee for our algorithm against the best policy in hindsight, and prove a complementing minimax lower bound on the best achievable regret, establishing that our policy achieves, up to poly-logarithmic factors, the best possible regret in this setting.
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