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.
翻译:Azar et al. [SODA'14] 对有限信息制度中的先知不平等问题的研究是由Azar et al. 开始的,这是在追求先前独立的上市价格机制时开始的。正如它们所显示的那样,美元(1)美元的竞争性政策仅使用每个代理商分布的单一样本是可以实现的。其显著的成果部分依赖于将单类先知不平等(SSPI)的设计减少到秩序模糊的秘书(OOS)政策。以上减少的代价是没有充分利用现有样本的直接成本。然而,迄今为止,这基本上是证明许多组合公司采用SSPI的唯一方法。最近,Rubinstein et al. [ITS'20] 给出了一个令人惊讶的简单算法,它能实现单类先知SSPI问题的最佳竞争比率(SSPI) 的配置。 美元- smaril不平等(SSPI ) 的简化后,我们通过SAL 文献机制直接研究简单的SPI 政策的竞争力,我们首先开发一个框架来分析与类似贪婪的市级的货币汇率政策, 将SAL- scaraldeal deal deal des des des des des ladeal deal des des des pal des des laut the laut the laut laut des abild des the laut laut the laut laut laut the ex laut the ex fir ex laus laut des des abil abil des abil des laut laut laut laut laut laut des des laut lautus.