Approximate mechanism design for distributed facility location

We consider the distributed facility location problem, in which there is a set of agents positioned on the real line, who are also partitioned into multiple symmetric districts. The goal is to choose a single location (where a public facility is to be built) so as to minimize the total distance of the agents from that location. Importantly, this process is distributed: the positions of the agents in each district are first aggregated into a representative location for the district, and then one of the representatives is chosen as the facility location. This indirect access to the positions of the agents inevitably leads to inefficiency, which is captured by the notion of distortion. We study both the discrete version of the problem, where the set of alternative locations if finite, as well as the continuous one, where every point of the line is an alternative. For both versions, we paint an almost complete picture of the distortion landscape of distributed mechanisms. We start from the discrete setting, for which we show a tight bound of 3 on the distortion of general mechanisms and a tight bound of 7 for strategyproof mechanisms. For the continuous setting, we show that the distortion of general mechanisms lies between 2 and 3, whereas the distortion of strategyproof mechanisms is exactly 3.