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

Obviously Strategyproof Single-Minded Combinatorial Auctions

We consider the setting of combinatorial auctions when the agents are single-minded and have no contingent reasoning skills. We are interested in mechanisms that provide the right incentives to these imperfectly rational agents, and therefore focus our attention to obviously strategyproof (OSP) mechanisms. These mechanisms require that at each point during the execution where an agent is queried to communicate information, it should be "obvious" for the agent what strategy to adopt in order to maximise her utility. In this paper we study the potential of OSP mechanisms with respect to the approximability of the optimal social welfare. We consider two cases depending on whether the desired bundles of the agents are known or unknown to the mechanism. For the case of known-bundle single-minded agents we show that OSP can actually be as powerful as (plain) strategyproofness (SP). In particular, we show that we can implement the very same algorithm used for SP to achieve a √m-approximation of the optimal social welfare with an OSP mechanism, m being the total number of items. Restricting our attention to declaration domains with two values, we provide a 2-approximate OSP mechanism, and prove that this approximation bound is tight. We also present a randomised mechanism that is universally OSP and achieves a finite approximation of the optimal social welfare for the case of arbitrary size finite domains. This mechanism also provides a bounded approximation ratio when the valuations lie in a bounded interval (even if the declaration domain is infinitely large). For the case of unknown-bundle single-minded agents, we show how we can achieve an approximation ratio equal to the size of the largest desired set, in an OSP way. We remark this is the first known application of OSP to multi-dimensional settings, i.e., settings where agents have to declare more than one parameter. Our results paint a rather positive picture regarding the power of OSP mechanisms in this context, particularly for known-bundle single-minded agents. All our results are constructive, and even though some known strategyproof algorithms are used, implementing them in an OSP way is a non-trivial task

Sum of Us: Strategyproof Selection from the Selectors

We consider directed graphs over a set of n agents, where an edge (i,j) is taken to mean that agent i supports or trusts agent j. Given such a graph and an integer k\leq n, we wish to select a subset of k agents that maximizes the sum of indegrees, i.e., a subset of k most popular or most trusted agents. At the same time we assume that each individual agent is only interested in being selected, and may misreport its outgoing edges to this end. This problem formulation captures realistic scenarios where agents choose among themselves, which can be found in the context of Internet search, social networks like Twitter, or reputation systems like Epinions. Our goal is to design mechanisms without payments that map each graph to a k-subset of agents to be selected and satisfy the following two constraints: strategyproofness, i.e., agents cannot benefit from misreporting their outgoing edges, and approximate optimality, i.e., the sum of indegrees of the selected subset of agents is always close to optimal. Our first main result is a surprising impossibility: for k \in {1,...,n-1}, no deterministic strategyproof mechanism can provide a finite approximation ratio. Our second main result is a randomized strategyproof mechanism with an approximation ratio that is bounded from above by four for any value of k, and approaches one as k grows

The Pareto Frontier for Random Mechanisms

We study the trade-offs between strategyproofness and other desiderata, such as efficiency or fairness, that often arise in the design of random ordinal mechanisms. We use approximate strategyproofness to define manipulability, a measure to quantify the incentive properties of non-strategyproof mechanisms, and we introduce the deficit, a measure to quantify the performance of mechanisms with respect to another desideratum. When this desideratum is incompatible with strategyproofness, mechanisms that trade off manipulability and deficit optimally form the Pareto frontier. Our main contribution is a structural characterization of this Pareto frontier, and we present algorithms that exploit this structure to compute it. To illustrate its shape, we apply our results for two different desiderata, namely Plurality and Veto scoring, in settings with 3 alternatives and up to 18 agents.Comment: Working Pape