Approximately Optimal Mechanisms for Strategyproof Facility Location: Minimizing L_p Norm of Costs

Abstract

We consider the problem of locating a single facility on the real line. This facility serves a set of agents, each of whom is located on the line, and incurs a cost equal to his distance from the facility. An agent's location is private information that is known only to him. Agents report their location to a central planner who decides where to locate the facility. The planner's objective is to minimize a "social" cost function that depends on the agent-costs. However, agents might not report truthfully; to address this issue, the planner must restrict himself to strategyproof mechanisms, in which truthful reporting is a dominant strategy for each agent. A mechanism that simply chooses the optimal solution is generally not strategyproof, and so the planner aspires to to use a mechanism that effectively approximates his objective function. This general class of problems was first studied by Procaccia and Tennenholtz and has been the subject of much research since then.In our paper, we study the problem described above with the social cost function being the L p norm of the vector of agent-costs. We show that the median mechanism (which is known to be strategyproof) provides a 2 1− 1 p approximation ratio, and that is the optimal approximation ratio among all deterministic strategyproof mechanisms. For randomized mechanisms, we present two results. First, we present a negative result: we show that for integer ∞ > p > 2, no mechanism-from a rather large class of randomized mechanisms-has an approximation ratio better than that of the median mechanism. This is in contrast to the case of p = 2 and p = ∞ where a randomized mechanism provably helps improve the worst case approximation ratio. Second, for the case of 2 agents, we show that a mechanism called LRM, first designed by Procaccia and Tennenholtz for the special case of L ∞ , provides the optimal approximation ratio among all randomized mechanisms.

Characterization of Group-strategyproof Mechanisms for Facility Location in Strictly Convex Space

“…For randomized mechanisms, we introduce a condition called translational invariance, which says that if we apply a translation to the inputs, the mechanism must output a location that is the result of the same translation to the original output. This condition is quite natural in the facility location domain (also known as shift invariance in [9] and position invariance in [12]), and we indeed find some strange mechanisms (e.g., Mechanism 2) that are not translation-invariant and related to some constant. Our main theorem states as follow.…”

Manipulation-resistant false-name-proof facility location mechanisms for complex graphs

“…For instance, bounds on the sum of costs (the Harsanyi social welfare) were derived for false-name-proof facility location mechanisms on the continuous line and continuous trees by Todo et al [67], strategy-proof facility location on the continuous cycle by Alon et al [3], and for strategy-proof facility location on the discrete cycle by Dokow et al [18]. Alon et al [2,3], Fotakis and Tzamos [28], and Schummer and Vohra [60] proved bounds on the maximum cost of an agent due to requiring strategyproofness, Feldman and Wilf [24] bounded the approximation of the optimal L 2 norm (sum of the squared distances of the agents) due to requiring strategy-proofness, and Feigenbaum et al [23] bounded the approximation of the optimal L p norm. The approximation bounds for the problem of positioning several facilities (where the cost of an agent is her distance to the closest facility) was studied by (to name a few) Fotakis and Tzamos [29], Fotakis and Tzamos [28], Lu et al [40], and Procaccia and Tennenholtz [53].…”

Facility location with double-peaked preferences

“…A series of papers have studied generalizations of the problem to more general metric spaces [1,11,32], multiple facilities [14,23,26,27] or even enhancing strategyproof mechanisms with additional capabilities [21,22]. Most of the related work actually considers the same objectives that we do here, namely the social cost or the maximum cost, with the notable exceptions of the least-squares objective [18], the L p norm of costs [17] or the minimax envy [5]. In a recent paper, Procaccia et al [29] use the facility location problem to explore the trade-offs between the approximation guarantees and the variance of truthful-in-expectation mechanisms.…”

“…This property was independently defined by [17] where it was referred to as shift invariance. One can view position invariance as an analogue to neutrality in problems like the one studied here, where there is a continuum of outcomes instead of a finite set.…”

“…Here, we studied the objective functions of the social cost and the maximum cost, following the original literature on the problem with single-peaked preferences [30]. Since then, several other objectives functions have been considered in the literature such as the least-squares objective, [18], the L p norm of costs [17] or the minimax envy [5]; it would make sense to consider the same or at least similar objectives for the case of double-peaked preferences as well. Finally, it would be meaningful to consider the problem under the verification framework [21,22], especially if it turns out that strong inapproximability bounds apply, at least for the case of deterministic strategyproof mechanisms.…”