False-Name-Proof Facility Location on Discrete Structures

07/21/2019 ∙ by Taiki Todo, et al. ∙ 0

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.

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1 Introduction

Social choice theory analyzes how collective decisions are made based on the preferences of individual agents. Its typical application field is voting, where each agent reports a preference ordering over a set of alternatives and a social choice function (SCF) selects one. One of the most well-studied criteria for SCFs is robustness to agents’ manipulations. An SCF is truthful if no agent can benefit by telling a lie, and false-name-proof if it is no agent cann benefit by casting multiple votes. The Gibbard-Satterthwaite theorem implies that any deterministic, truthful, and Pareto efficient (PE) SCF must be dictatorial. Also, any false-name-proof (FNP) SCF must be randomized [7].

Overcoming such negative results has been a promising research direction. One of the most popular approaches is to restrict agents’ preferences. For example, when their preferences are restricted to being single-peaked, the well-known median voter schemes are truthful, PE, and anonymous [17], and their strict subclass is also FNP [27]. The model where agents’ preferences are single-peaked has also been called the facility location problem, where each agent has an ideal point on an interval, e.g., on a street, and the SCF locates a public good, e.g., a train station, to which agents want to stay close.

Tree Cycle Hypergrid
Single-Peaked Any ✓ [18] ✓ (Thm. 3) Ladder ✓ [18]
✗ (Thm. 4) Other ✗ (Thm. 5 and 6)
Single-Dipped Any ✓ (Thm. 7) ✓ (Thm. 8) Open
✗ (Thm. 9)
Table 1: Summary of our contributions. ✓ indicates an FNP and PE SCF, and ✗ indicates that no such SCF exists. It remains open to clarify whether such an SCF exists for general hypergrid graphs when agents’ preferences are single-dipped.

Moulin [17], as well as many other works on facility location problems, considered an interval as the set of possible alternatives, where any point in the interval can be chosen by an SCF. In several practical situations, however, the set of possible alternatives is discrete and has slightly more complex underlying network structure, which the agents’ preferences also respect. For example, in multi-criteria voting with two criteria, each of which has only three options, the underlying network is a three-by-three grid. When we need to choose a time-slot to organize a joint meeting, the problem resembles choosing a point on a discrete cycle. Dokow et al. [8] studied truthful SCFs on discrete lines and cycles. Ono et al. [20] considered FNP SCFs on a discrete line. However, there has been very few works on FNP SCFs on more complex structures (see Related Works section).

In this paper we tackle the following question: for which graph structures does an FNP and PE SCF exist? When the mechanism designer can arbitrarily modify the network structure of the set of possible alternatives, the problem is simplified. The network structure, however, is a metaphor of a common feature among agents’ preferences, where modifying the network structure equals changing the domain of agents’ preferences. This is almost impossible in practice because agents’ preferences are their own private information. The mechanism designer, therefore, first faces the problem of verifying whether, under a given network structure (or equally, a given preference domain), a desirable SCF exists.

Locating a bad is another possible extension of the facility location problem, where an SCF is required to locate a public bad, e.g., a nuclear plant or a disposal station, which each agent wants to avoid. Agents’ preferenes are therefore assumed to be single-dipped, which is sometimes called obnoxious. Actually, some existing works have studied truthful facility location with single-dipped preferences [5]. Nevertheless, to the best of our knowledge, no work has dealt with both false-name-proofness and more complex structures than a path, such as cycles.

Table 1 summarizes our contribution. Regardless of whether the preferences are single-peaked or single-dipped, there is an FNP and PE SCF for any tree graph and any cycle graph of length less than six, and there is no such SCF for any larger cycle graph. For hypergrid graphs, when preferences are single-peaked, such an SCF exists if and only if the given hypergrid graph is a ladder, i.e., of dimension two and at least one of which has at most two vertices.

2 Related Works

Moulin [17] proposed generalized median voter schemes, which are the only deterministic, truthful, PE, and anonymous SCFs. Procaccia and Tennenholtz [21] proposed a general framework of approximate mechanism design, which evaluates the worst case performance of truthful SCFs from the perspective of competitive ratio. Recently, some models for locating multiple heterogenous facilities have also been studied [23, 11, 3]. Wada et al. [29] considered the agents who dynamically arrive and depart. Some research also considered facility location on grids [25, 9] and cycles [1, 2, 8]. Melo et al. [16] overviewed applications in practical decision making.

Over the last decade, false-name-proofness has been scrutinized in various mechanism design problems [30, 4, 26, 28], as a refinement of truthfulness for such open and anonymous environments, as the internet. Bu [6] clarified a connection between false-name-proofness and population monotonicity in general social choice. Lesca et al. [14] also addressed FNP SCFs that are associated with monetary compensation. Sonoda et al. [24] considered the case of locating two homogeneous facilities. Ono et al. [20] studied some discrete structures, but focused on randomized SCFs.

One of the most similar works to this paper is Nehama et al. [18], 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., -grid for arbitrary ), but do not cover any other graphs considered in this paper, such as larger (hyper-)grid graphs and cycle graphs with lengths over three.

Locating a public bad has also been widely studied in both economics and computer science fields. Manjunath [15] characterized truthful SCFs on an interval. Lahiri et al. [13] studied the model for locating two public bads. Feigenbaum and Sethuraman [10] considered the cases where single-peaked and single-dipped preferences coexist. Nevertheless, all of these works just focused on truthful SCFs. To the best of our knowledge, this paper is the very first work on FNP facility location with single-dipped preferences.

3 Preliminaries

Let be an undirected, connected graph, defined by the set of vertices and the set of edges. The distance function is such that for any , , where is the shortest path between and . We say that a graph has another graph as a distance-preserving induced subgraph if has as an induced subgraph and for any pair and their corresponding vertices , holds.

In this paper we focus on three classes of graphs, namely tree, cycle, and hypergrid. A tree graph is an undirected, connected and acyclic graph. A special case of tree graphs is called as a path graph, in which only two vertices have a degree of one and all the others have a degree of two. Indeed, tree graphs are a simplest generalization of path graphs, so that most of the properties of path graphs, such as the uniqueness of the shortest path between two vertices, carries over to tree graphs. A cycle graph is an undirected and connected graph that only consists of a single cycle. When a cycle graph has vertices, we refer to it as , and its vertices are labeled, in a counter-clockwise order, from to .

A hypergrid graph is a Cartesian product of more than one path graphs. When a hypergrid is a Cartesian product of path graphs, we call it a -dimensional (-D, in short) grid. In this paper, a 2-D grid is sometimes represented by the number of vertices on each path, as -grid. In a given -D grid graph, each vertex is represented as a -tuple . Note that the -D -grid is a cycle graph .

Let be the set of potential agents, and let be a set of participating agents. Each agent has a type . When agent has type , agent is located on vertex . Let denote a profile of the agents’ types, and let denote their profile without ’s. Given , let be the set of vertices on which at least one agent is located, i.e., . Furthermore, given and vertex , let be the profile obtained by removing all the agents at the vertex from . By definition, .

Given and , let be the preference of the agent located on vertex over the set of alternatives, where and indicate the strict and indifferent parts of , respectively. A preference is single-peaked (resp. single-dipped) under if, for any , if and only if (resp. ), and if and only if . That is, an agent located on strictly prefers alternative , which is strictly closer to (resp. farther from) than other alternative , and is indifferent between these alternatives when they are the same distance from . By definition, for each possible type , the single-peaked (resp. single-dipped) preference is unique.

A (deterministic) SCF is a mapping from the set of possible profiles to the set of vertices. Since each agent may pretend to be multiple agents in our model, an SCF must be defined for different-sized profiles. To describe this feature, we define an SCF as a family of functions, where each is a mapping from to . When a set of agents participates, the SCF uses function to determine the outcome. The function takes profile of types jointly reported by as an input, and returns as an outcome. We denote as if it is clear from the context. We further assume that an SCF is anonymous, i.e., for any input and its permutation , holds.

We are now ready to define the two desirable properties of SCFs: false-name-proofness and Pareto efficiency.

Definition 1.

An SCF is false-name-proof (FNP) if for any , , , , , , and , it holds that .

The set indicates the set of identities added by for the manipulation. The property coincides with the canonical truthfulness when , i.e., agent only uses one identity.

Definition 2.

An alternative Pareto dominates under if for all and for some . An SCF is Pareto efficient (PE) if for any and , no alternative Pareto dominates .

Given , let indicate the set of all alternatives that are not Pareto dominated by any alternative.

In general, the following theorem holds, which has recently been provided by the authors’ another paper [19] and justifies to focus on a special class of FNP and PE SCFs. An SCF is said to ignore duplicate ballots (or satisfies IDB in short) if for any pair , implies .

Theorem 1 (Okada et al. [19]).

Assume there is an FNP and PE SCF that does not satisfy IDB. Then, there is also an FNP and PE SCF that satisfies IDB.

Therefore, in what follows, we focus on FNP and PE SCFs that also satisfies IDB.

4 Single-Peaked Preferences

In this section, we focus on single-peaked preferences, i.e., every agent prefers to have the facility closer to her. It is already known that for any tree graph, and thus for any path graph, an FNP and PE SCF exists.

Theorem 2 (Nehama et al. [18]).

Assume that agents’ preferences are single-peaked. For any tree graph, there is an FNP and PE SCF.

An example of such an SCF is the target rule [12], originally proposed for an interval, i.e., a continuous line such as . It is shown that the target rule is FNP and PE for any tree metric [27]. Almost the same proof works for any tree graph.

In the following two subsections, we investigate the existence of such SCFs for cycle and hypergrid graphs. The two lemmata presented below are useful to prove the impossiblity results for single-peaked preferences.

Lemma 1.

Let be an arbitrary graph. Assume that agents’ preferences are single-peaked under and there is no truthful and PE SCF for . Then, for any graph that contains as a distance-preserving induced subgraph, there is no truthful and PE SCF.

Proof.

Since there is no such SCF for , we can find a sequence of profiles s.t. at least one agent benefits by a manipulation. Since has as a distance-preserving induced subgraph, and any PE alternative for profiles inside is located in due to single-peakedness, exactly the same benefit is guaranteed for an agent by a manipulation in . ∎

Lemma 2.

Let be an arbitrary graph. Assume that agents’ preferences are single-peaked under a graph . Then, for any FNP SCF , any and any , .

Proof.

Assume that . Since , there is some agent located at , who incurs the cost of zero when is reported and is still present when all the agents at are removed. Since , such an agent incurs the cost of more than zero when all the agents at are removed. Thus, the agent located at has an incentive to add identities at , which contradicts false-name-proofness. ∎

4.1 Single-Peaked Preferences on Cycles

In this section, we show that, under single-peaked preferences, there is an FNP and PE SCF for if and only if .

The if part is is informally mentioned in Nehama et al. [18], but the formal proof is not given. To show the existence of such SCFs, we first define a class of SCFs, called sequential Pareto rules (SPRs). Given cycle , an SPR has an ordering of all the alternatives in . For a given input , it sequentially checks, in the order specified by , whether the first (second, third, and so on) alternative is PE, and terminates when it finds a PE one. By definition, any SPR is automatically PE.

For a continuous circle, any truthful and PE SCF is dictatorial [22]. Since choosing such a dictator in a non-manipulable manner, when there is uncertainty on identities, is quite difficult, FNP and PE SCFs are not likely to exist for a continuous circle. Our results in this section thus demonstrate the power of the discretization of the alternative space; by discretizing the set of alternatives so that at most five alternatives exist along with a cycle, we can avoid falling into the impossibility.

Dokow et al. [8] showed that any truthful and onto SCF is nearly dictatorial for a cycle with . In this paper we clarify a stricter threshold on such an impossibility when agents can pretend to be multiple agents; FNP SCFs exist for a cycle if and only if .

Theorem 3.

Let be a cycle graph s.t. . When preferences are single-peaked, there is an FNP and PE SCF.

Proof.

It is obvious that any SPR is FNP for . For , the SPR with the ordering is FNP. Finally, for , the SPR with the ordering is FNP, which was also informally mentioned in [18]. These rules are described in Fig. 1. ∎

Figure 1: SPRs for , , and . The arrows indicate the associated ordering .
Theorem 4.

Let be a cycle graph s.t. . When preferences are single-peaked, there is no FNP and PE SCF.

The impossibility for is important because we will use it in the proof of Theorem 6 in the next subsection.

Proof.

Assume that an FNP and PE SCF exists for any , say, for even and

for odd

, and w.l.o.g. that for any profile s.t. .

For : Consider a profile s.t.  (see the left cycle of Fig. 2). Since is FNP and PE, holds. Let be a profile s.t. , i.e., the antipodal to the vertex and its two neighbors. Lemma 2 implies . We then consider another profile s.t. . From symmetry and Lemma 2, , which contradicts . Almost the same argument holds for any larger even .

For : Consider profile s.t.  (see the right cycle of Fig. 2). Since is FNP and PE, must be either or . Assume w.l.o.g. that . From Lemma 2, removing either or does not change the outcome, i.e., for s.t. , .

Then consider another profile s.t. . Since is FNP and PE, ; otherwise agents located on would add fake identities so that the outcome changes to . Similarly, ; otherwise, it must be the case that , which yields a contradiction. Thus, .

However, implies ; otherwise agents located on would add fake identities so that the outcome changes to . This also yields a contradiction. Almost the same argument holds for any larger odd . ∎

Figure 2: Profiles for Theorem 4. The intersection of two profiles corresponds to .

One might think that an SPR associated with any possible ordering is FNP. However, the following example shows that the ordering must be carefully chosen to guarantee false-name-proofness (and truthfulness as well). Characterizing FNP and PE SCFs for a given cycle graph remains open.

Example 1.

Consider and an SPR associated with ordering . Assume that there are three agents, whose types are . Since , is chosen as an outcome when all the agents reports truthfully, where the agent located at incurs the cost of . However, she can benefit by reporting as her type, since , and thus reduces her cost to .

4.2 Single-Peaked Preferences on Hypergrids

The facility location on a hypergrid graph is a reasonable simplification of multi-criteria voting [25], where each candidate has a pledge for each criteria, such as taxation and diplomacy, that is embeddable on a hypergrid. Each voter then has the most/least preferred point on the hypergrid.

In this section, we completely clarify under which condition on a given hypergrid graph an FNP and PE SCF exists when agents’ preferences are single-peaked.

It is already known that, when preferences are single-peaked, an FNP and PE SCF exists for any -grid [18]. Our main contribution in this subsection, Theorem 5, complements their result; no such SCF exists for any other 2-D grid. Theorem 6 further shows that this impossibility carries over into any -D grid with .

Theorem 5.

Let be an -grid, where . When preferences are single-peaked, there is no FNP and PE SCF.

Proof.

Lemma 4 below shows that, for the -grid, there is no FNP and PE SCF. Since any -grid , for arbitrary , contains the -grid graph as a distance-preserving induced subgraph, the impossiblity carries over into according to Lemma 1. ∎

Lemma 3.

Let be the -grid, where the set of vertices . Assume that agents’ preferences are single-peaked under and there is an FNP and PE SCF . Then, for any s.t. , must be one of the four corners of , i.e., .

Proof.

Asssume w.l.o.g. that . We construct a profile s.t. . Since is FNP, . We also construct another profile s.t. . Since is FNP, . Finally, let be the profile constructed by removing all the agents located on , , and . By applying Lemma 2 to those profiles, we obtain both and , which yield a contradiction. ∎

Figure 3: Type profiles for Lemma 4. On each gray vertex there is some agent, and the vertex with must be chosen under the profile. The proof derives a contradiction on .
Lemma 4.

Let be the 2-D -grid. When preferences are single-peaked, there is no FNP and PE SCF.

Proof.

Assume that an FNP and PE SCF exists for the -grid. From Lemmata 2 and 3, for any s.t. , holds. From symmetry, assume w.l.o.g. that (see the top-left grid in Fig. 3).

We now remove all the agents located at from the above profile , and refer to them as . Since is FNP and PE, . Here, let be the profile that further removes all the agents located at , , , and from . Note that , and thus holds by the same argument. We also consider another profile, , which is obtained by removing all the agents at , , and from . Note that , and by the same argument.

Then we construct by removing all the agents in the vertices except for , , and from . Since is reachable from both and , Lemma 2 implies and , which yields a contradiction. ∎

Theorem 6.

Let be an arbitrary -D grid. When preferences are single-peaked, there is no FNP and PE SCF.

Proof.

We can easily observe that a three-dimensional -grid, a.k.a. the binary cube, contains as a distance-preserving induced subgraph. As we showed in Theorem 4 in the previous section, there is no FNP and PE SCF for . Therefore, by Lemma 1, no such SCF also exists for the -grid. Any other larger grid (possibly of more than three dimensions) contains the three-dimensional -grid, and thus the impossibility is carried over by Lemma 1. ∎

5 Single-Dipped Preferences

5.1 Single-Dipped Preferences on Trees

For the case of a public bad, where agents’ preferences are single-dipped, we can find an FNP and PE SCF.

Theorem 7.

Let be an arbitrary tree graph. When preferences are single-dipped, there is an FNP and PE SCF.

Figure 4: The red edges construct the longest path . For the profile where some agent exists on each of the gray vertices, the SCF in the proof of Theorem 2 returns as an outcome.
Proof.

Consider the SCF described as follows. First, choose an arbitrary longest path of a given tree, whose extremes are called and . Then, return as an outcome if at least one agent strictly prefers to ; otherwise return as an outcome.

For each agent , either or is one of the most preferred alternative; otherwise, the path from the most preferred point of to one of the two extremes is longer than . In Fig. 4, the agents at the bottom left gray vertex most prefer , while agents at the middle or top-right gray vertices most prefer . It is therefore obvious that the above SCF is PE, since either or is the most preferred alternative for each agent, and the choice between and is made by a unanimous voting, guaranteeing that the chosen alternative is the most preferred for at least one agent. Furthermore, such a unanimous voting over two alternatives is obviously FNP. ∎

5.2 Single-Dipped Preferences on Cycles

We next consider locating a public bad on a cycle. Single-dipped preferences resemble single-peaked preferences for cycle graphs, especially for sufficiently large ones. Actually, in this subsection we provide almost the same results with the case of single-peaked preferences.

Theorem 8.

Let be a cycle graph s.t. . When preferences are single-dipped, there is an FNP and PE SCF.

Proof.

For , it is easy to see that any SPR is FNP. For , the domain of single-dipped preferences coincides with the domain of single-peaked preferences, since the point diagonal from a dip point can be considered as a peak point. Therefore, the SPR with ordering is FNP, as shown in Theorem 3. Finally, for , the SPR with ordering is FNP. ∎

Theorem 9.

Let be a cycle graph s.t. . When preferences are single-dipped, there is no FNP and PE SCF.

Proof.

The identical proof of Theorem 4 applies for any even , since a single-dipped preference over a cycle of even length, with a dip point , coincides with the single-peaked one with the peak point that is antipodal to .

We therefore focus on odd . Assume that an FNP and PE SCF exists for, say, , and w.l.o.g. that for any s.t. .

Consider a profile s.t. . Since is FNP and PE, must be either or ; otherwise some agent has incentive to add fake identities. Furthermore, for the profile s.t. , holds. Therefore, holds; otherwise the agent located at has incentive to add fake identity on , which moves the facility to either or .

On the other hand, for another profile s.t. , must be either or due to symmetry. Therefore, for the above , must hold, which contradicts the condition of . Almost the same argument holds for any larger odd number . ∎

6 Conclusions

We tackled whether there exists an FNP and PE SCF for the facility location problem under a given graph. We gave complete answers for path, tree, and cycle graphs, regardless whether the preferences are single-peaked or single-dipped. For hypergrid graphs, an open problem remains for single-dipped preferences. When such SCFs exist, completely characterizing their class of such SCFs is crucial future work. Investigating randomized SCFs is another interesting direction.

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