On the Power of Deterministic Mechanisms for Facility Location Games

Abstract

Abstract. We consider K-Facility Location games, where n strategic agents report their locations in a metric space, and a mechanism maps them to K facilities. The agents seek to minimize their connection cost, namely the distance of their true location to the nearest facility, and may misreport their location. We are interested in deterministic mechanisms that are strategyproof, i.e., ensure that no agent can benefit from misreporting her location, do not resort to monetary transfers, and achieve a bounded approximation ratio to the total connection cost of the agents (or to the Lp norm of the connection costs, for some p ∈ [1, ∞) or for p = ∞). Our main result is an elegant characterization of deterministic strategyproof mechanisms with a bounded approximation ratio for 2-Facility Location on the line. In particular, we show that for instances with n ≥ 5 agents, any such mechanism either admits a unique dictator, or always places the facilities at the leftmost and the rightmost location of the instance. As a corollary, we obtain that the best approximation ratio achievable by deterministic strategyproof mechanisms for the problem of locating 2 facilities on the line to minimize the total connection cost is precisely n − 2. Another rather surprising consequence is that the TWO-EXTREMES mechanism of (Procaccia and Tennenholtz, EC 2009) is the only deterministic anonymous strategyproof mechanism with a bounded approximation ratio for 2-Facility Location on the line. The proof of the characterization employs several new ideas and technical tools, which provide new insights into the behavior of deterministic strategyproof mechanisms for K-Facility Location games, and may be of independent interest. Employing one of these tools, we show that for every K ≥ 3, there do not exist any deterministic anonymous strategyproof mechanisms with a bounded approximation ratio for K-Facility Location on the line, even for simple instances with K + 1 agents. Moreover, building on the characterization for the line, we show that there do not exist any deterministic strategyproof mechanisms with a bounded approximation ratio for 2-Facility Location on more general metric spaces, which is true even for simple instances with 3 agents located in a star.

On Voting and Facility Location

“…This model was extended by these authors and by others in many different ways. The metric space researched spanned from a line ( [14], [23]) to a circle ( [1], [2]), a tree ([1], [11]) or a general graph ( [1]). There are many papers regarding building several facilities (or electing a committee of candidates), where the cost of an agent is her distance to the closest facility ( [13], [14], [18], [23]).…”

“…The metric space researched spanned from a line ( [14], [23]) to a circle ( [1], [2]), a tree ([1], [11]) or a general graph ( [1]). There are many papers regarding building several facilities (or electing a committee of candidates), where the cost of an agent is her distance to the closest facility ( [13], [14], [18], [23]). As opposed to the voting scenario, the goal of the vast majority of these papers was to optimize over some global target function, and the most popular target functions were the utilitarian (social cost) and egalitarian (the maximal cost of an agent) (see, e.g, [1], [14], [23]), but there were also works regarding additional target functions like the L 2 norm (the sum of the squared distances of the agents, see [11]).…”

“…There are many papers regarding building several facilities (or electing a committee of candidates), where the cost of an agent is her distance to the closest facility ( [13], [14], [18], [23]). As opposed to the voting scenario, the goal of the vast majority of these papers was to optimize over some global target function, and the most popular target functions were the utilitarian (social cost) and egalitarian (the maximal cost of an agent) (see, e.g, [1], [14], [23]), but there were also works regarding additional target functions like the L 2 norm (the sum of the squared distances of the agents, see [11]). Some papers consider "obnoxious facility location" -a setting in which agents want to be as far away as possible from the facility, e.g., when selecting a location for a central garbage dump ( [7]).…”

Obvious Strategyproofness, Bounded Rationality and Approximation

“…On one hand, our bounds for OSP mechanisms without money are in stark contrast with the case of strategyproof mechanisms where the optimum is known to be truthful [Moulin, 1980]. On the other hand, our results for OSP mechanisms can be likened to truthful mechanisms for K-facility location, K ≥ 2, where there is a linear gap between truthful approximations with and without money Tennenholtz, 2013, Fotakis andTzamos, 2014] (incidentally, there are hints that the gap remains linear also for relaxed notions of truthfulness without money [Ferraioli et al, 2016]) -in the case of OSP mechanisms the price to pay to close the gap is not only money, but also monitoring.…”

Verifiably Truthful Mechanisms

“…The reason for choosing this specific domain is twofold. First, a slew of recent papers has brought about a good understanding of what quality guarantees are achievable via truthful facility location mechanisms [1,21,20,25,13,14,15,29,30,8,32]. Second, facility location has also served as a proof of concept for the approximate mechanism design without money agenda [27], whose principles were subsequently applied to a variety of other domains, including allocation problems [17,16,12,9], approval voting [2], kidney exchange [3,7], and scheduling [19].…”