Strategy-Proof Facility Location for Concave Cost Functions

We consider k-Facility Location games, where n strategic agents report their locations on the real line, and a mechanism maps them to k facilities. Each agent seeks to minimize his connection cost, given by a nonnegative increasing function of his distance to the nearest facility. Departing from previous work, that mostly considers the identity cost function, we are interested in mechanisms without payments that are (group) strategyproof for any given cost function, and achieve a good approximation ratio for the social cost and/or the maximum cost of the agents. We present a randomized mechanism, called Equal Cost, which is group strategyproof and achieves a bounded approximation ratio for all k and n, for any given concave cost function. The approximation ratio is at most 2 for Max Cost and at most n for Social Cost. To the best of our knowledge, this is the first mechanism with a bounded approximation ratio for instances with k > 2 facilities and any number of agents. Our result implies an interesting separation between deterministic mechanisms, whose approximation ratio for Max Cost jumps from 2 to unbounded when k increases from 2 to 3, and randomized mechanisms, whose approximation ratio remains at most 2 for all k. On the negative side, we exclude the possibility of a mechanism with the properties of Equal Cost for strictly convex cost functions. We also present a randomized mechanism, called Pick the Loser, which applies to instances with k facilities and n = k+1 agents, and for any given concave cost function, is strongly group strategyproof and achieves an approximation ratio of 2 for Social Cost

Facility location with double-peaked preference

We study the problem of locating a single facility on a real line based on the reports of self-interested agents, when agents have double-peaked preferences, with the peaks being on opposite sides of their locations. We observe that double-peaked preferences capture real-life scenarios and thus complement the well-studied notion of single-peaked preferences. We mainly focus on the case where peaks are equidistant from the agents' locations and discuss how our results extend to more general settings. We show that most of the results for single-peaked preferences do not directly apply to this setting; this makes the problem essentially more challenging. As our main contribution, we present a simple truthful-in-expectation mechanism that achieves an approximation ratio of 1+b/c for both the social and the maximum cost, where b is the distance of the agent from the peak and c is the minimum cost of an agent. For the latter case, we provide a 3/2 lower bound on the approximation ratio of any truthful-in-expectation mechanism. We also study deterministic mechanisms under some natural conditions, proving lower bounds and approximation guarantees. We prove that among a large class of reasonable mechanisms, there is no deterministic mechanism that outperforms our truthful-in-expectation mechanism

Facility Location Games with Ordinal Preferences

We consider a new setting of facility location games with ordinal preferences. In such a setting, we have a set of agents and a set of facilities. Each agent is located on a line and has an ordinal preference over the facilities. Our goal is to design strategyproof mechanisms that elicit truthful information (preferences and/or locations) from the agents and locate the facilities to minimize both maximum and total cost objectives as well as to maximize both minimum and total utility objectives. For the four possible objectives, we consider the 2-facility settings in which only preferences are private, or locations are private. For each possible combination of the objectives and settings, we provide lower and upper bounds on the approximation ratios of strategyproof mechanisms, which are asymptotically tight up to a constant. Finally, we discuss the generalization of our results beyond two facilities and when the agents can misreport both locations and preferences

Verifiably Truthful Mechanisms

It is typically expected that if a mechanism is truthful, then the agents would, indeed, truthfully report their private information. But why would an agent believe that the mechanism is truthful? We wish to design truthful mechanisms, whose truthfulness can be verified efficiently (in the computational sense). Our approach involves three steps: (i) specifying the structure of mechanisms, (ii) constructing a verification algorithm, and (iii) measuring the quality of verifiably truthful mechanisms. We demonstrate this approach using a case study: approximate mechanism design without money for facility location