Mechanism design with strategic mediators

We consider the problem of designing mechanisms that interact with strategic agents through strategic intermediaries (or mediators), and investigate the cost to society due to the mediators' strategic behavior. Selfish agents with private information are each associated with exactly one strategic mediator, and can interact with the mechanism exclusively through that mediator. Each mediator aims to optimize the combined utility of his agents, while the mechanism aims to optimize the combined utility of all agents. We focus on the problem of facility location on a metric induced by a publicly known tree. With nonstrategic mediators, there is a dominant strategy mechanism that is optimal. We show that when both agents and mediators act strategically, there is no dominant strategy mechanism that achieves any approximation. We, thus, slightly relax the incentive constraints, and define the notion of a two-sided incentive compatible mechanism. We show that the 3-competitive deterministic mechanism suggested by Procaccia and Tennenholtz [2013] and Dekel et al. [2010] for lines extends naturally to trees, and is still 3-competitive as well as two-sided incentive compatible. This is essentially the best possible (follows from Dekel et al. [2010] and Procaccia and Tennenholtz [2013]). We then show that by allowing randomization one can construct a 2-competitive randomized mechanism that is two-sided incentive compatible, and this is also essentially tight. This result also reduces a gap left in the work of Procaccia and Tennenholtz [2013] and Lu et al. [2009] for the problem of designing strategy-proof mechanisms for weighted agents with no mediators on a line. We also investigate a generalization of the preceding setting where there are multiple levels of mediators.