On Voting and Facility Location

12/18/2015 ∙ by Michal Feldman, et al. ∙ Tel Aviv University 0

We study mechanisms for candidate selection that seek to minimize the social cost, where voters and candidates are associated with points in some underlying metric space. The social cost of a candidate is the sum of its distances to each voter. Some of our work assumes that these points can be modeled on a real line, but other results of ours are more general. A question closely related to candidate selection is that of minimizing the sum of distances for facility location. The difference is that in our setting there is a fixed set of candidates, whereas the large body of work on facility location seems to consider every point in the metric space to be a possible candidate. This gives rise to three types of mechanisms which differ in the granularity of their input space (voting, ranking and location mechanisms). We study the relationships between these three classes of mechanisms. While it may seem that Black's 1948 median algorithm is optimal for candidate selection on the line, this is not the case. We give matching upper and lower bounds for a variety of settings. In particular, when candidates and voters are on the line, our universally truthful spike mechanism gives a [tight] approximation of two. When assessing candidate selection mechanisms, we seek several desirable properties: (a) efficiency (minimizing the social cost) (b) truthfulness (dominant strategy incentive compatibility) and (c) simplicity (a smaller input space). We quantify the effect that truthfulness and simplicity impose on the efficiency.

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1 Introduction

The Hotelling-Downs model ([10], [17]) used to study political strategies, assumes that individual voters occupy some point along the real line. Non-principled political parties (or ice cream vendors) strategically position themselves at a point along the left-right axis (or along a beach) so as to garner the greatest number of supporters (clients). Implicitly, voters will vote for the closest candidate.

We consider an analogous problem to the Hotelling-Downs model, where candidates are principled (i.e., non-strategic) whereas the voters have preferences but may misrepresent them in order to achieve what is a better outcome from their perspective. In this model, in which both voters and candidates are represented by points in the metric space, a closer candidate is preferable to one further away.

Examples for candidate selection:

  • A municipality plans to erect a public library on a street, and every resident seeks to be as close as possible to the proposed library. However, the new library can only be built on suitable locations (the candidates).

  • Social choice issues in which the distance is not physical: there is a set of policies ranging from left to right, and several political candidates stand for election, each one advocating a different policy. Every voter is associated with a point along the real line. An example of a collective decision problem which does not revolve around the political sphere yet may also fit this setting is the task of determining the temperature of an air conditioner in a room, where each individual has a different ideal point along the scale of temperatures (a line). There are many additional settings of relevant candidate selection problems, e.g., in the realms of recommendation systems, electronic commerce and computational economics. While our results do not necessarily apply to all social choice settings, there are many such problems for which they do apply (whether in entirety or partially).

Assuming quasi-linear utilities, and allowing payments — then the well known Vickrey-Clarke-Groves (VCG) mechanism is truthful and can achieve the optimal social cost (see, e.g., [20]). However, in many real-life situations we restrict the use of money due to ethical, legal or other considerations, e.g, in democratic elections and in the examples previously mentioned.

We study deterministic truthful mechanisms with no payments for the candidate selection problem. In such mechanisms, no agent can benefit from misreporting her location, regardless of the reported locations of the other agents. Such mechanisms are also known as dominant strategy incentive compatible mechanisms. We also consider randomized truthful mechanisms, both universally truthful (ex-post Nash) and truthful in expectation.

Given a set of candidate and voter locations, it is polytime to find the candidate that minimizes the social cost.

When restricted to deterministic truthful mechanisms, we show that the optimal candidate cannot be selected in the general case. Moreover, we show that the cost may be as bad as three times the optimal cost (matching lower and upper bounds). When considering randomized mechanisms on the line, the approximation factor drops to two (matching upper and lower bounds).

There are other reasons that an optimal candidate may not be chosen. In particular, this depends on the amount of information the agents supply to the mechanism. We formulate three different types of mechanisms, based on the information each agent submits to the mechanism—

  • Voting mechanisms, in which each agent casts a vote for her favorite candidate.

  • Ranking mechanisms, in which each agent states her ordinal preferences over the candidates.

  • Location mechanisms, in which each agent sends her exact position.

Clearly, knowing the true location of an agent allows one to infer the ranking preferred by that agent, which in turn allows one to infer the favorite candidate of the agent (up to tie-breaking).

In almost all previous work on the facility location problem every point in the metric space was considered to be a candidate, therefore there was no difference between these three mechanism types.

The social choice literature mostly considers ranking mechanisms. Recognize that Arrow’s impossibility theorem does not hold when assuming the preferences are single-peaked.

The more information an agent transmits, the mechanism has more tools to devise an accurate solution. Albeit, this information comes at a cost, since it might disclose more private information which the agents wish to keep confidential. Furthermore, behavioral economists have long argued that the agents cannot obtain the full information pertaining to their utility, or that obtaining this information requires a high cognitive cost. Additionally, sending more information also casts a higher burden on the mechanism. For all of these reasons deploying a simple mechanism 111We use the term “simplicity” in the perspective of the voters, which have a smaller action space, i.e, less options to choose from. Upon receiving the input, the mechanism itself can act in an arbitrarily complex fashion. which requires less information from agents is advantageous, and generally there is a trade-off between the accuracy of a mechanism to its simplicity. Indeed, in practice many election schemes use voting mechanisms rather than ranking mechanisms, largely due to these desiderata.

1.1 Our Contributions

In the paper, we show the following:

  • In Section 3 we formulate a framework of reductions that compare the various mechanism types. We utilize this framework to show the relations (equivalence or strict containment) between the three classes of truthful mechanisms – voting, ranking and location (see Figure 1). Furthermore, we show that for the case of two candidates, the set of truthful in expectation location mechanisms is equivalent to the set of truthful in expectation voting mechanisms. These results provide a significant step towards a full characterization of truthful mechanisms at large.

    Median

    Voting

    Ranking Location
    (a) Deterministic truthful mechanisms

    Spike

    UT Voting

    TIE Voting

    TIE Ranking

    TIE Location
    (b) Randomized mechanisms

    Figure 1: The relationships between classes of mechanisms in candidate elections (Theorem 1):
    For deterministic truthful mechanisms, the set of ranking mechanisms strictly contains the set of voting mechanisms, yet the set of location mechanisms is equivalent to the set of ranking mechanisms. The lower bound on the social cost for any truthful location mechanism on the line is 3, and it is matched by an upper bound by a voting mechanism - the median mechanism.
    In the randomized case, there is a hierarchy of strict containment in the following order - truthful in expectation (TIE) location mechanisms, TIE ranking mechanisms, TIE voting mechanisms and universally truthful (UT) voting mechanisms. The lower bound on the social cost of any location mechanism on the line is 2. We introduce the spike mechanism, which is a universally truthful voting mechanism, which achieves a matching upper bound of 2.
    Refer to Section 3 for formal definitions of equivalence and strict containment in our setting.
  • In Section 4 we define a family of universally truthful voting mechanisms on the line called weighted percentile voting (WPV) mechanisms, which choose the

    ’th vote with some predetermined probability

    . We introduce the spike

    mechanism, which is a WPV mechanism that carefully crafts the probability distribution to account for misreports by any agent - whether they are near the center or close to the extremes (see Figure

    2). We then use backwards induction to show that spike achieves an approximation ratio of two (Theorem 8).

    Figure 2:

    The density function of the spike mechanism, which gives rise to the mechanism’s name (the cumulative distribution function is written explicitly in Definition

    4). In this example there are 10000 agents and 4 candidates. The 4 candidates, when ordered from left to right, receive 2000, 2000, 3000 and 3000 votes respectively. The votes are arranged in ascending order (with ties broken arbitrarily), and the graph depicts probability of each vote being chosen – votes are chosen with higher probability when they are closer to the 50th percentile. The area beneath the graph represents the probability that each candidate will be elected, e.g., the probability of choosing the second candidate () is the integral of the function between 2000 and 4000.
  • In Section 5 we show additional bounds for randomized mechanisms – On the line there is a lower bound of two, even for location mechanisms, which shows that the result for spike is tight. Furthermore, when combining this understanding with the results of Section 3, it can be concluded that two is also the tight approximation ratio for truthful in expectation mechanisms (voting, ranking or location) and for universally truthful voting mechanisms.

    We move on to show bounds for randomized mechanisms for more general metric spaces222We do not present results for deterministic mechanisms in general metric spaces, since in these cases the incentive compatibility constraints take a significant toll on the approximation ratio – according to Anshelevich et al. [3] in the non-strategic setting it is possible to reach a constant ratio in any metric space, while due to the characterization of Schummer and Vohra [24] there exist metric spaces in which the approximation ratio is even in the continuous model. (see Figure 3). An immediate result is that the random dictator mechanism achieves an upper bound of three for any metric space. Theorem 14 shows a lower bound of for any voting mechanism in by using a counterexample based on a regular simplex. This is enough to conclude that on an arbitrary metric space, the bound of three is tight for any voting mechanism. Theorem 16 displays a lower bound of for any ranking mechanism in (which also holds in any higher dimension Euclidean space ).

    Strategic Non-Strategic Voting (low information) LB (Thm. 14) (Lemma 17) UB (Lemma 18) (Lemma 18) Ranking LB (Thm. 16) (Lemma 17) UB (Lemma 18) (Lemma 18) Location (high information) LB (Obs. 15) UB (Lemma 18)

    Figure 3: Summary of our results for randomized mechanisms in . The columns correspond to the truthfulness constraints (in the strategic setting the mechanism must be truthful in expectation), whereas the rows correspond to the information constraints (voting, ranking or location mechanisms). The rows are further divided to show the lower bound (LB) and upper bound (UB) of each such cell. Note that for non-strategic location mechanisms the result is always optimal by definition, since there are neither information nor strategic constraints.
    In the strategic setting there is a difference between the lower bounds for voting, ranking and location mechanisms.
    Most of the results here are rather straightforward, except for the upper bounds of and of , which are more involved.
  • In Section 6 we present deterministic bounds on the line – there is a lower bound of three, which is met by a matching upper bound due to the median mechanism. All the results on the line, deterministic or randomized, are displayed in the table in Figure 4.

    Recognize the following surprising phenomenon apparent in Figure 4. In both deterministic and randomized cases, any constraint in either information or truthfulness, yields the same ratio as taking the both of these constraints simultaneously — when insisting on truthful mechanisms (in the strategic case), there is no trade-off between high and low information settings, and one can enjoy the benefits of minimal information mechanisms (voting mechanisms) without incurring any additional cost to the approximation ratio; Similarly, when deciding to reduce the information requirements to anything less than location mechanisms, it is possible to devise a truthful [voting] mechanism, without increasing the approximation ratio.

    Deterministic   Randomized Strategic Non-Strategic   Strategic Non-Strategic Voting 3 3   2 2 (low information) (LB:Lm. 19, (LB:Thm. 3 in [3],   (LB:Lm. 17, (LB:Lm. 15, UB:Lm. 21) UB:Lm. 21)   UB:Thm. 8 ) UB:Thm.8) Ranking 3 3   2 2 (LB:Lm. 19, (LB: Thm. 3 in [3],   (LB:Lm. 17, (LB:Obs. 15, UB:Lm. 21) UB: Lm. 21)   UB:Thm. 8 ) UB:Lm. 21) Location 3 1   2 1 (high information) (LB:Lm. 19,   (LB:Lm. 17, UB:Lm. 21)   UB:Thm. 8 )

    Figure 4: The approximation ratios of mechanisms on the line () in various settings. All the results in the table are tight. In particular the upper bound in strategic case is due to the spike mechanism.

1.2 Related Work

Voting systems have been a domain of prolific research for decades. The seminal Gibbard-Satterthwaite theorem [15] shows that if the rankings of agents can be arbitrary and the amount of candidates is greater than two, then the only onto truthful mechanisms are dictatorships. However, if there are limitations on the rankings, then the impossibility theorem of Gibbard-Satterthwaite does not hold. In many cases the rankings can be limited to single-peaked preferences, a notion used as early as 1948 by Black [4]. In 1980 Moulin showed a complete characterization of truthful deterministic mechanisms for single-peaked preferences [19]. Schummer and Vohra [24] extended this characterization to cycles and general graphs.

There has been extensive work describing various candidate selection mechanisms, which have been generally divided to 3 main types ([6], [26]) — scoring rules (e.g., plurality, Borda, anti-plurality, range voting, cumulative), Condorcet extensions (e.g., Copeland, maxmin, Dodgson, Young, ranked pairs), or other mechanisms (e.g., single transferable vote, Bucklin). Some work on social choice also made use of randomized voting schemes, for instance in order to improve the results of the mechanisms [21] or to make manipulation computationally hard ([8] pages 632-633). Most of these mechanisms have no assumptions on the preferences of the agents, and are rankings mechanisms (i.e., they receive the ordinal preferences of the voters as input). Since in these circumstances the Gibbard-Satterthwaite impossibility theorem holds, the mechanisms are typically not truthful. While the importance of truthfulness and of simple mechanisms (with less options for each voter) has been acknowledged, to the best of our knowledge there has not been a formal framework for reduction or an assessment of the relationships between the different types of mechanisms.

Since in the lack of cardinal costs no global objective functions can be measured (e.g, the social cost), the focus of many of the aforementioned mechanisms is on achieving some desirable axiomatic properties. Nonetheless, the use of utilitarianism in the realm of social choice has firm and ancient roots (see, e.g, a 1952 paper by Fleming [12] and a 1955 work by Harsanyi [16]). Moreover, a new line of work commenced in recent years regarding distortion, which also refers to the utilitarian goal of minimizing social cost (partly due emergence of new domains such as recommender systems and e-commerce, as previously noted). The term was coined in 2006 by Procaccia and Rosenschein [22], and it was later used, for instance, by Boutilier et. al. [5]. Recently, Anshelevich et al. [3] assessed the distortion of several voting rules, and provided lower bounds on them. In [3], the distortion is the worst case ratio between the social cost of the candidate elected and the social cost of the optimal candidate, over any ranking profile (that is, preference profile) in any metric space. The distortion is a quite similar to the approximation ratio used in this paper, but it differs in two key properties –

  • Most importantly, the source of the imperfection depicted by the distortion is the mechanism’s lack of information (the mechanism has access to the ordinal ranking of the agents, but not to their exact location, that is - not to their full cardinal utilities). In this paper, the approximation ratio is greater than one both because of this information deficiency (in the cases or ranking and voting mechanisms), and because of incentive compatibility constraints. In this sense, we can quantify the cost of limited information as well as the cost of truthfulness in various settings.

  • The distortion calculates the worst-case ratio in any metric space, whereas the approximation ratio is sometimes calculated over a specific metric space.

Anshelevich et al. show a deterministic lower bound of 3 on the distortion, and they prove that two mechanisms (social choice functions), Copeland and Uncovered Set, achieve a distortion of 5.

Procaccia and Tennenholtz introduced game theoretical aspects to the facility location problem. As mentioned before, their setting is similar to the one in this paper, except that the location of the facility is not restricted to a set of candidates, but instead can be located at any point on the line. This model was extended by these authors and by others in many different ways. The metric space researched spanned from a line ([14], [23]) to a circle ([1], [2]), a tree ([1], [11]) or a general graph ([1]). There are many papers regarding building several facilities (or electing a committee of candidates), where the cost of an agent is her distance to the closest facility ([13], [14], [18], [23]). As opposed to the voting scenario, the goal of the vast majority of these papers was to optimize over some global target function, and the most popular target functions were the utilitarian (social cost) and egalitarian (the maximal cost of an agent) (see, e.g, [1], [14], [23]), but there were also works regarding additional target functions like the norm (the sum of the squared distances of the agents, see [11]). Some papers consider “obnoxious facility location” — a setting in which agents want to be as far away as possible from the facility, e.g., when selecting a location for a central garbage dump ([7]).

When the outcome is constrained to a set of candidates, the facility location literature is far less extensive. In this setting, a recent paper by Sui and Boutilier defines a set of deterministic mechanisms which is GSP on the line and -GSP on [25]. The paper does not show bounds on global objectives such as the social cost.

Dokow et al. [9] characterize deterministic truthful mechanisms on the discrete line and the discrete circle. They move on to give a lower bound on the social cost for large circles, and to the best of our knowledge this is the only result regarding assessment of the social cost in a constrained setting. It is worthy to note the model in [9] has 2 major properties which differ from the one in this paper: (a) The discrete constraints of the locations apply to the agents as well as to the candidates, so all agents are located precisely on some candidate; (b) The distance between any two neighboring candidates must be constant (for instance: 1,2,3,4,…).

2 Model

Let be a set of agents, where each agent is located at some point . We refer to the location of agent as agent ’s type. Let be the location profile of the agents. There exists a fixed set of candidates . Each candidate , is located at point , and this location is publicly known. The agents and candidates are located on some metric space. A significant part of the paper deals with specific metric spaces, and these will be specifically noted. In the parts where the metric space is , it is assumed that the agents and the candidates are both numbered in ascending order based on their locations (otherwise they could be renamed in this manner).

A deterministic mechanism , is a function which maps an action profile to a candidate, that is: . We consider three classes of mechanisms that differ in the input they accept, i.e., in the action space of the agents:

  • Voting mechanisms, in which each agent casts a vote for a candidate, that is: .

  • Ranking mechanisms, in which every agent reports ordinal preferences over all the candidates. The notation indicates a preference of candidate over candidate (or is indifferent between the two). In ranking mechanisms , where is the set of all permutations of the set of candidates . These mechanisms are sometimes referred to in the literature as social choice functions.

  • Location mechanisms, in which every agent reports their location, that is is some point in the metric space.

Given a joint action profile , the cost of point is its distance to the facility, that is: . For agent located at point , we refer to as the cost of agent . The goal of each agent is to minimize her cost.

Truthful mechanisms are usually defined in the context of direct revelation mechanisms. Since in ranking and voting mechanisms the action space does not coincide with the type space, we extend this notion in the following trivial manner for these cases as well. For an agent in location and for any mechanism (location, ranking or voting), let be the set of true actions of this agent — the actions which convey the real preferences of this agent. For instance, in voting mechanisms is the set of candidates closest to , which we refer to as the favorite candidates of (this might be a set since there may be ties). An agent reporting is said to be reporting truthfully, and an action profile in which all agents report truthfully is called a truthful profile. The set of truthful profiles is denoted . A truthful mechanism is one in which no agent can suffer from reporting truthfully, regardless of the actions of the other agents:

A randomized mechanism is a mapping from an action profile to a distribution over the candidates, that is: . The cost of agent is the expected cost of this agent according to the probability distribution returned by the mechanism, that is: .

Two different notions of randomized truthful mechanisms have been studied in the literature, and we extend them naturally based on our definitions of truthful reports:

  • Truthful in expectation (TIE) mechanisms — where the expected cost of an agent reporting truthfully is never higher than any other action. That is: , , , : . In these mechanisms the agent may regret her action ex-post for some of the instances.

  • Universally truthful mechanisms are mechanisms which can be expressed as a probability distribution over deterministic truthful mechanisms. In these mechanisms an agent never regrets reporting truthfully, even after the random outcome is unraveled.

Clearly, every universally truthful mechanism is truthful in expectation mechanisms, but not necessarily vice versa. Throughout the paper, in the randomized setting we use the term “truthful” to refer to truthful in expectation mechanisms, unless otherwise stated.

The social cost of a mechanism is the sum of the agents’ costs. For a location profile and an action profile the social cost is: . The cost of a candidate is the cost of the mechanism which locates the facility on that candidate, that is: . Given a location profile , the optimal mechanism, denoted , is one which chooses a candidate that minimizes the social cost (). For the sake of consistency, when there are several optimal candidates, we refer to the leftmost among them as . For any truthful in expectation mechanism (including universally truthful mechanisms), the social cost of given a location profile is the maximal social cost it yields by any truthful action profile , that is: . The approximation ratio of a truthful in expectation mechanism is the maximal ratio for any location profile , between social cost of given and the optimal social cost given : .

We make use of several terms which are relevant for voting mechanisms:

  • The location of a vote (some candidate) is denoted .

  • When the network is the line, then there is an inherent order of the votes, and therefore it is possible to make use of percentiles. A percentile mechanism is a voting mechanism that elects the ’th percentile vote (for example, the mechanism which chooses the leftmost vote is the percentile mechanism).

  • A weighted percentile voting (WPV) mechanism locates the facility on the ’th percentile vote with some probability , where does not depend on the action profile . For example, “random dictator” is a WPV mechanism which chooses any vote with probability .

In voting mechanisms, the set of candidates induces a partition of the line in the following manner — the voting zone of candidate , denoted , is the set of points whose favorite candidate is : . The voting zones are bounded by voting borders. For example, when the metric space is , there are borders, which are the midpoints between two consecutive candidates: (see Figure 5). When the metric space is , the voting zones create a Voronoi diagram. A candidate which has at least one agent in their zone is called active.

In ranking mechanisms, induces a partition which divides the line into ranking zones. All points in some ranking zone share some ranking . In this case, we say that the ranking is consistent with ranking zone . The ranking zones bounded by ranking borders. For example, when the metric space is the ranking borders are the midpoints between any two candidates: .

Figure 5: Illustration of candidates (white circles), agents (black circles), voting borders and voting zones when the metric space is .
For example, in this case the favorite candidate of is : . The voting borders divide the distance between two consecutive candidates exactly in half, for example: .

3 Classes of Mechanisms

In this section we go over the containment hierarchy of various classes of truthful mechanisms (e.g., Figure 1). We start with some intuition, then defining some necessary terms, and finally present the main theorem of this section.

Intuitively, for any mechanism , there exists a mechanism which receives a “richer” input than , and acts identically to . For instance, for some arbitrary voting mechanism , there obviously exists a ranking mechanism which disregards all of the preferences except the top choice of each agent, and behaves essentially just like does.

We generalize this notion in the following informal definition — a mechanism (whether location/ranking/or voting) is said to be reducible to a mechanism (location/ranking/or voting) if for every location profile and true reports, the output of is identical to the output of (a formal definition, which is based on simulating , is deferred to appendix A.1).

As pointed out, it is clear that every voting mechanism is reducible to some ranking mechanism (or some location mechanism ). In these cases, if is truthful then so is , since only uses the information which is inputted to , so any misreports to which would not change the input of do not affect the outcome at all. Note that the same reasoning also shows that every ranking mechanism is reducible to some location mechanism, and that any voting mechanism is reducible to some location mechanism.

On the other hand, it is not true that every ranking mechanism is reducible to some voting mechanism. Somewhat surprisingly, we will soon show that when we restrict ourselves to deterministic truthful mechanism this does hold, that is — every deterministic truthful ranking mechanism is reducible to some deterministic truthful voting mechanism.

Two sets of mechanisms, and , are said to be equivalent if every is reducible to some , and every is reducible to some .

A set of mechanisms is said to be strictly contained in a set of mechanisms if every mechanism is reducible to some mechanism , yet not every mechanism is reducible to some mechanism . This is a slight abuse of terminology since the sets and may be disjoint, as their input space may be different.

The following theorem shows several claims regarding relations (equivalence or strict containment) between sets of truthful mechanisms. Notice that not only does this theorem show the hierarchy of the different classes, but it also provides notions relevant to characterization of truthful mechanisms. For instance, the second claim proves that no mechanism can use any information regarding the location of the agents beyond their ranking, while maintaining truthfulness. In addition, in the claims showing strict containment, the examples in the proofs portray the expressiveness that the additional information gives the mechanism.

Theorem 1.

The following claims hold in the Euclidean metric space (for any ):

  1. The set of truthful deterministic ranking mechanisms strictly contains the set of truthful deterministic voting mechanisms.

  2. The set of truthful deterministic location mechanisms is equivalent to the set of truthful deterministic ranking mechanisms.

  3. The set of truthful in expectation randomized ranking mechanisms strictly contains the set of truthful in expectation randomized voting mechanisms.

  4. The set of truthful in expectation randomized location mechanisms strictly contains the set of truthful in expectation randomized ranking mechanisms.

  5. The set of truthful in expectation randomized voting mechanisms strictly contains the set of universally truthful randomized voting mechanisms.

  6. When there are two candidates, the set of truthful in expectation randomized location mechanisms is equivalent to the set of truthful in expectation randomized voting mechanisms.

For the ease of readability, we defer the proof of this theorem to the appendix (see A.1).

4 Spike Mechanism

In the next sections we will prove that both the median mechanism and the random dictator mechanism achieve an approximation ratio of three on . However, the cause for this ratio in these two cases is different - for median it is due to an instance which is bad for the median agent, while for random dictator it is due to a bad instance for an agent in one of the extremes. It is therefore desirable to devise a mechanism which is resistant to bad instances of any agent. The spike mechanism stems from this intuition.

This section contains foundations needed for the introduction of the spike mechanism, the definition of the mechanism, and the theorem showing that spike achieves an approximation ratio of 2. In the entirety of this section, the metric space is and the mechanisms are voting mechanisms.

We start by showing a basic lemma regarding WPV mechanisms. Recall that these are voting mechanisms which choose the ’th percentile vote with a predetermined probability .

Lemma 2.

Any weighted percentile voting mechanism is universally truthful.

The proof is straightforward and deferred to appendix A.2.

Definition 3 (Cumulative count of a candidate).

Given a voting profile , the cumulative count of candidate is the amount of agents who voted for any candidate who is not located to the right of , that is: .

Definition 4 (Spike Mechanism).

The spike mechanism is a WPV mechanism 333Spike is a WPV mechanism since it chooses the ’th percentile vote with some probability , and does not depend on the reports. that for each voting profile , chooses candidate according to the following cumulative distribution function:

The mechanism is named after the shape of the density function it creates (see Figure 2). Recognize that the result of the mechanism depends on the amount of votes that each candidate received and on the order of the candidate along the line, but not on the distances between the candidates.

Observation 5.

Spike defines a symmetric distribution, that is: .

Proof.

Let , then:

We now define a few terms needed for the proof of the approximation ratio. Recall that is uniquely defined for a location profile , since ties are broken in favor of the leftmost candidate. We denote the set of borders by .

Definition 6 (Tight profile of , see Figure 6(a)).

Given a location profile , the profile is said to be the tight profile of if it moves all agents who are not on a border as close as possible to within their zones, that is:

(a) The original profile .

(b) The tight profile of . The arrows represent the movements between the two profiles.

Figure 6: Tight profile
Definition 7 (Left-compressed profile of , see Figure 7(a)).

Given a tight location profile , a left-compressed profile of moves all the agents on the leftmost border to their neighboring border on the right, if this border is left of . Formally: let the location of the leftmost agent be , then the left-compressed profile of is:

Note that the left compressed profile of a tight profile is also a tight profile. The right-compressed profile of is defined in a completely symmetrical fashion.

(a) , the original tight profile.

(b) The left-compressed profile of (after moves to ).

Figure 7: Left-compressed profile

After compressing location profiles, there are likely to be locations in which there are many agents. We therefore use the following notation: the location profile is written as , which means that for each , there are agents located at .

We now use these definitions to prove the main result of this section:

Theorem 8.

The spike mechanism is universally truthful, and it achieves an approximation ratio of 2 on .

Proof.

Spike is a WPV mechanism, so according to Lemma 2 it is universally truthful.

The analysis of the approximation ratio is more involved and is based on backwards induction which follows these steps (see Figure 8):

  1. Figure 8(a): Start with a general location profile , and compute its optimal candidate, .

  2. Figure 8(b): Let be the tight profile of . We show that the transition from to cannot decrease the approximation ratio (Lemma 9).

  3. Figure 8(c): Let be the left-compression of . We show that if the ratio of is not higher than 2, then so is the ratio of (Lemma 10).

  4. Figure 8(d): Repeat left and right compressions until we can no longer compress. At this stage, the profile is tight with at most 3 active candidates, and we note this profile . We show that the approximation ratio of is not above 2 (Lemma 11).

(a) Initial state: A general location profile

(b) : The tight profile of , using the notation in which there are agents at point .

(c) : The left-compressed profile of (moves agents from to ).

(d) : The final profile after reapplying left and right compressions on repeatedly. The active candidates are denoted , and , the amount of agents as , and respectively, and we scale the distances to and .

Figure 8: Illustration of the proof of Theorem 8:

Proving these steps is sufficient to complete the proof of the theorem, since in Lemma 11 we show that the ratio of is not higher than 2 (the base case). According to Lemma 10, this implies that the ratio of (prior to all of the compressions) is also not higher than 2 (the induction steps). Since the ratio of the is not higher than that of , this means that the approximation ratio of is not above 2, as needed.

Notice that throughout this process remains the optimal candidate, since it was optimal in the original profile , and in each step all agents move towards it, so the cost of any other candidate can decrease by no more than what the cost of decreases.

Truthful reports to the spike mechanism (like any other voting mechanism) are not necessarily unique since an agent who is located on a border can report either of the two candidates closest to her. For these cases of ties, we show that the worst-case ratio always occurs when the agents vote for the candidate located farther away from (Lemma 30, whose formal definition and proof are deferred to the appendix A.2).

We now present the lemmas formally. The proofs of the lemmas, which are given in the appendix, prove the backwards induction and conclude the proof of the theorem at large.

Lemma 9.

Let be an arbitrary location profile, let be the tight profile of and let be an arbitrary WPV mechanism. Then the approximation ratio of given is not lower than that of given :

The previous lemma holds for any WPV mechanism, in particular for spike.

Lemma 10.

Let be a tight location profile, let be the left-compressed profile of and let be the spike mechanism. Then if the approximation ratio of given is not higher than 2, then so is that of given :

Since the cumulative function defining the spike mechanism is symmetrical, the claim can be trivially extended to right-compressions as well.

After reapplying compressions on both sides, the resulting profile has agents in three locations at most (see Figure 8(d)). The last lemma in the proof states that in this final stage, the ratio is not higher than 2:

Lemma 11.

Let be a tight location profile in which there are at most 3 active candidates: . The ratio of the given is not higher than 2: .

5 Additional Results for Randomized Mechanisms

5.1 Lower Bounds

In this section show lower bounds of randomized mechanisms in different settings. When the network is , we present a lower bound of 2 for any truthful in expectation mechanism, even if it is a location mechanism. By the hierarchy presented in Theorem 1, this lower bound trivially holds for truthful in expectation ranking and voting mechanisms as well (see Figure 1). Ergo, spike is optimal over all truthful in expectation mechanisms.

Additionally, we show a lower bound of 2 for any randomized ranking mechanism, even when the mechanism need not be truthful (in the non-strategic setting). These results prove that the approximation ratio achieved by spike is tight. For more general metric spaces the lower bound changes — In , we show a lower bound for any truthful voting mechanism of . We also present a lower bound of for any truthful ranking mechanism in (this bound also holds for , for any ).

We start by proving a helpful lemma. Informally, the lemma states that when there is an agent located on a border (and can therefore submit several different truthful actions to a ranking or voting mechanism), her cost should not change under any truthful report she submits.

Lemma 12.

For any truthful in expectation ranking mechanism in any metric space, let be the border between ranking zones . Let be the rankings consistent with respectively. Let agent be located on this border, that is: .
Then the cost at point remains the same whether the agent reports or , that is:

The proof is given in appendix A.3.

Observation 13.

The previous lemma also holds for voting mechanisms.

The proof of the observation follows the exact same lines as the proof of the lemma.

Theorem 14.

In the dimensional real space , any truthful in expectation voting mechanism has an approximation ratio of at least .

Proof.

Let there be candidates, located on the vertices of a regular simplex (all vertices are equally distanced from one another). Let there be agents, and let be an arbitrary truthful in expectation voting mechanism.

Let be the profile in which each agent is located precisely on candidate . Therefore is the only truthful voting profile for . Denote the probability of choosing candidate as , that is: . Clearly there exists some candidate which is chosen by with probability at least . Assume without loss of generality that this candidate is , that is: .

We move on to define another location profile, , which is also consistent with the voting profile . Let be the regular simplex in which candidates are on the vertices. Let be the point with equal distance to all vertices in ( is a regular simplex, so such a point necessarily exists). Denote this distance as . However, this distance is different from the distance from to : . Let be the profile in which there are agents at and one agent at (see Figure 9).

According to Observation 13, the cost of an agent at point should not change under any truthful vote, that is for any vote . In particular, this holds when any agent on point votes for candidate . We make use of this observation several times by changing the votes for each of the points at to , one at a time, such that the final voting profile is ( agents vote for , one agent votes for ).. Due to the observation, the cost of point must remain the same throughout these transitions, that is: .

Therefore:

Denote the midpoint between and as . Without loss of generality, scale the distances such that . Examine the following location profile , which is also consistent with the voting profile . In this case the cost of , which is the optimal candidate, is: . The cost of is . Therefore the approximation ratio of is at least:

Figure 9: An illustration of the proof of Theorem 14 for the case of .
The three candidates are on the vertices of an equilateral triangle (a regular simplex with 3 vertices). The lines within the triangle denote the borders between a pair of candidates. The simplex is the line between , and its midpoint is .
These three figures, from left to right, show the dynamics of the proof (location profiles , and respectively). The key observation is that the agents at point are at equal distance to all candidates except , therefore the transition does not change the probability of to be chosen.

Observation 15.

Any truthful in expectation location mechanism has an approximation ratio of at least 2, even on the line.

Proof.

Let there be two candidates on the line. According to Theorem 1, any truthful in expectation location mechanism is equivalent to a voting mechanism. One can apply the same proof as in Theorem 14 for the case of (in this case, both point and point are the midpoint between the two candidates), to achieve a lower bound of . ∎

Theorem 16.

In , any truthful in expectation ranking mechanism has an approximation ratio of at least .

Proof.

Let there be 3 candidates located such that they form an equilateral triangle, and let be a truthful in expectation ranking mechanism. Let be the following ranking profile:

Let be some location profile consistent with (see Figure 10). Denote . From symmetry, there exists some candidate chosen with probability at least . Assume without loss of generality that this is candidate , that is: .

Let be a point such that , and , where . Let . Let and let . Notice that is consistent with both and , therefore according to Lemma 12 the cost at should remain the same for :

Let be a point such that , and , where . Let . Let and let . According to Lemma 12 the cost at should remain the same for :

Let be the midpoint between , and let