Single-Sample Prophet Inequalities Revisited.

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. 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. Finally, we discuss applications of our SSPIs to the design of mechanisms for multi-dimensional limited information settings with improved revenue and welfare guarantees.