We consider the problem of locating a single facility on a vertex in a given graph based on agents' preferences, where the domain of the preferences is either single-peaked or single-dipped. Our main interest is the existence of deterministic social choice functions (SCFs) that are Pareto efficient and false-name-proof, i.e., resistant to fake votes. We show that regardless of whether preferences are single-peaked or single-dipped, such an SCF exists (i) for any tree graph, and (ii) for a cycle graph if and only if its length is less than six. We also show that when the preferences are single-peaked, such an SCF exists for any ladder (i.e., 2-by-m grid) graph, and does not exist for any larger hypergrid.
Over the last decade, false-name-proofness has been scrutinized in various mechanism design problems @cite_27 @cite_18 @cite_25 @cite_8 , as a refinement of truthfulness for such open and anonymous environments, as the internet. Bu @cite_19 clarified a connection between false-name-proofness and population monotonicity in general social choice. @cite_2 also addressed FNP SCFs that are associated with monetary compensation. @cite_21 considered the case of locating two homogeneous facilities. @cite_4 studied some discrete structures, but focused on randomized SCFs. One of the most similar works to this paper is @cite_17 , who also clarified the network structures under which FNP and PE SCFs exist for single-peaked preferences. One clear difference from ours is that, in their paper proposed a new class of graphs, called a ZV-line, as a generalization of path graphs. In our paper we investigate well-known existing structures, namely tree, hypergrid, and cycle graphs. ZV-line graphs contain any tree and ladder (i.e., @math -grid for arbitrary @math ), but do not cover any other graphs considered in this paper, such as larger (hyper-)grid graphs and cycle graphs with lengths over three.
@cite_18 @cite_4 @cite_8 @cite_21 @cite_19 @cite_27 @cite_2 @cite_25 @cite_17
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In many real-life scenarios, a group of agents needs to agree on a common action, e.g., on a location for a public facility, while there is some consistency between their preferences, e.g., all preferences are derived from a common metric space. The facility location problem models such scenarios and it is a well-studied problem in social choice. We study mechanisms for facility location on unweighted undirected graphs, which are resistant to manipulations (strategyproof, abstention-proof, and false-name-proof ) by both individuals and coalitions and are efficient (Pareto optimal). We define a family of graphs, ZV -line graphs, and show a general facility location mechanism for these graphs which satisfies all these desired properties. Moreover, we show that this mechanism can be computed in polynomial time, the mechanism is anonymous, and it can equivalently be defined as the first Pareto optimal location according to some predefined order. Our main result, the ZV -line graphs family and the mechanism we present for it, unifies the few current works in the literature of false-name-proof facility location on discrete graphs, including the preliminary (unpublished) works we are aware of. Finally, we discuss some generalizations and limitations of our result for problems of facility location on other structures. We consider the problem of locating facilities on a discrete acyclic graph, where agents’ locations are publicly known and the agents are requested to report their demands, i.e., which facilities they want to access. In this paper, we study the effect of manipulations by agents that utilize vacant vertices. Such manipulations are called rename or false-name manipulations in game theory and mechanism design literature. For locating one facility on a path, we carefully compare our model with traditional ones and clarify their differences by pointing out that some existing results in the traditional model do not carry over to our model. For locating two facilities, we analyze the existing and new mechanisms from a perspective of approximation ratio and provide non-trivial lower bounds. Finally, we introduce a new mechanism design model where richer information is available to the mechanism designer and show that under the new model false-name-proofness does not always imply population monotonicity. The class of Groves mechanisms has been attracting much attention in called social welfare maximization) and dominant strategy incentive compatibility. However, when strategic agents can create multiple fake identities and reveal more than one preference under them, a refined characteristic called false-name-proofness is required. Utilitarian efficiency and false-name-proofness are incompatible in combinatorial auctions, if we also have individual rationality as a desired condition. However, although individual rationality is strongly desirable, if participation is mandatory due to social norms or reputations, a mechanism without individual rationality can be sustained. In this paper we investigate the relationship between utilitarian efficiency and false-name-proofness in a social choice environment with monetary transfers. We show that in our modelization no mechanism simultaneously satisfies utilitarian efficiency, false-name-proofness, and individual rationality. Considering this fact, we ignore individual rationality and design various mechanisms that simultaneously satisfy the other two properties. We also compare our different mechanisms in terms of the distance to individual rationality. Finally we illustrate our mechanisms on a facility location problem. We consider the problem of makespan minimization on m unrelated machines in the context of algorithmic mechanism design, where the machines are the strategic players. This is a multidimensional scheduling domain, and the only known positive results for makespan minimization in such a domain are O(m)-approximation truthful mechanisms [22, 20]. We study a well-motivated special case of this problem, where the processing time of a job on each machine may either be "low" or "high", and the low and high values are public and job-dependent. This preserves the multidimensionality of the domain, and generalizes the restricted-machines (i.e., pj,∞ ) setting in scheduling. We give a general technique to convert any c-approximation algorithm to a 3c-approximation truthful-in-expectation mechanism. This is one of the few known results that shows how to export approximation algorithms for a multidimensional problem into truthful mechanisms in a black-box fashion. When the low and high values are the same for all jobs, we devise a deterministic 2-approximation truthful mechanism. These are the first truthful mechanisms with non-trivial performance guarantees for a multidimensional scheduling domain. Our constructions are novel in two respects. First, we do not utilize or rely on explicit price definitions to prove truthfulness; instead we design algorithms that satisfy cycle monotonicity. Cycle monotonicity [23] is a necessary and sufficient condition for truthfulness, is a generalization of value monotonicity for multidimensional domains. However, whereas value monotonicity has been used extensively and successfully to design truthful mechanisms in single-dimensional domains, ours is the first work that leverages cycle monotonicity in the multidimensional setting. Second, our randomized mechanisms are obtained by first constructing a fractional truthful mechanism for a fractional relaxation of the problem, and then converting it into a truthful-in-expectation mechanism. This builds upon a technique of [16], and shows the usefulness of fractional mechanisms in truthful mechanism design.