Multi-dimensional mechanism design via random order contention resolution schemes.

Already in 1981 Myerson gave a characterization of the optimal mechanism for a single parameter Bayesian mechanism design. However, till today we have no idea for how such a characterization for the multi-dimensional setting could even look like. Moreover, it wasn't until no that long time ago that we could not develop mechanisms for such setting with any reasonable and provable performance guarantees. The seminal work of [Chawla et al. 2009] on sequential posted pricing mechanisms gave us an approach for approximately solving the Bayesian multi-parameter unit-demand mechanism design problem (BMUMD). The paper left open the question on how to obtain a constant approximation for the matroid setting. Two mathematically beautiful results from combinatorial optimization under uncertainty where devised in order to answer this question. First, Kleinberg and Weinberg in 2011 extended the classical Prophet Inequality result into the matroid setting to give a 2-approximation for BMUMD for a single matroid setting. Second, Feldman, Svensson and Zenklusen in 2016 adapted the Contention Resolution Scheme framework for online settings. We add to this line of work by considering the Contention Resolution Scheme framework in the random order setting. The most impressive implication of this research are the new algorithms for BMUMD which improve the previous results in the multimatroid setting. Although the range of implications of the CR Scheme framework in the random order is reasonably wide, we shall focus in this letter on presenting only the single matroid setting and how it is connected to BMUMD.