DeepAI

# Awareness of Voter Passion Greatly Improves the Distortion of Metric Social Choice

We develop new voting mechanisms for the case when voters and candidates are located in an arbitrary unknown metric space, and the goal is to choose a candidate minimizing social cost: the total distance from the voters to this candidate. Previous work has often assumed that only ordinal preferences of the voters are known (instead of their true costs), and focused on minimizing distortion: the quality of the chosen candidate as compared with the best possible candidate. In this paper, we instead assume that a (very small) amount of information is known about the voter preference strengths, not just about their ordinal preferences. We provide mechanisms with much better distortion when this extra information is known as compared to mechanisms which use only ordinal information. We quantify tradeoffs between the amount of information known about preference strengths and the achievable distortion. We further provide advice about which type of information about preference strengths seems to be the most useful. Finally, we conclude by quantifying the ideal candidate distortion, which compares the quality of the chosen outcome with the best possible candidate that could ever exist, instead of only the best candidate that is actually in the running.

• 9 publications
• 12 publications
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05/08/2018

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## 1 Introduction

One often hears about ‘where candidates stand’ on issues, calling to mind a spatial model of preferences in social choice [4, 21, 23, 26, 29]. In proximity-based spatial models, voters’ preferences over candidates are derived from their distances to each of the candidates in some issue space. In particular, we consider voters and candidates which lie in an arbitrary unknown metric space. Our work follows a recent line of research in social choice which considers this setting [1, 2, 3, 8, 13, 14, 15, 17, 19, 20, 22, 27, 30]. The distance between each voter and the winning candidate is interpreted as the cost to that voter. Naturally, one of the main goals is to select the candidate which minimizes the total Social Cost, i.e., the sum of costs to the voters.

The crucial observation in the work cited above is that the actual costs of the voters for the selection of each candidate (i.e., the distances in the metric space) are often unknown or difficult to obtain [9]. Instead, it is more reasonable to assume that voters only report ordinal preferences: orderings over the candidates which are induced by, and consistent with, latent individual costs. Because of this, past research has often focused on optimizing distortion: the worst-case ratio between the winning candidate selected by a voting rule aware of only ordinal preferences, and the best available candidate which minimizes the overall social cost. Many insights were obtained for this setting, including that there are deterministic voting rules which obtain a distortion of at most a small constant (5 in [1], and more recently 4.236 in [25]), and that no deterministic rule can obtain a distortion of better than 3 given access to only ordinal information.111We focus on deterministic mechanisms in this paper; see Related Work for discussion of why.

The fundamental assumption and motivation in the above work is that the strength or intensity of voter preferences is not possible to obtain, and thus we must do the best we can with only ordinal preferences. And indeed, knowing the exact strength of voter preferences is usually impossible. In many settings, however, some cardinal information about the ardor of voter preferences is readily available or obtainable, and is often used to affect outcomes and make better collective decisions. For example, a decision in a meeting may be decided in favor of a minority position if those in the minority are significantly more adamant or passionate about the issue than the apathetic majority, as revealed during discussion or debate. In political campaigns, the amounts of monetary donations, activists attending rallies, and other measures of “grass-root support” can cause a candidate to become a de-facto front-runner even before an official election or primary is ever held. Because of this, in this paper we ask the question: “How much can the quality of selected candidates be improved if we know some small amount of information about the strength of voter preferences?”

There are many different approaches modeling, measuring, eliciting, and aggregating the strength or intensity of voter preferences [10, 16]. Such measures can be done through survey techniques, measuring the total amount of monetary contributions, amounts of excitement and time people spend volunteering or advocating for particular issues, etc (see Related Work). All such measures are by their very nature imprecise. And yet while it is unreasonable to assume that exact strength of preference is known for every voter, it is certainly possible to obtain insights such as “there are many more voters who are passionate about candidate A as compared to candidate B”, or quantify the approximate amount of extreme preference strengths as opposed to the voters who are mostly indifferent. As we show in this paper, even such a small amount of information about aggregate preference strengths or the amount of passionate voters can greatly improve distortion, and allow mechanisms which provably result in outcomes which are close to optimal. In fact, knowing only a single additional bit of information for each voter (i.e., do they prefer A to B strongly, or not strongly?) is enough to greatly improve distortion.

#### Model and Notation

As in previous work on metric distortion, we have a set of voters and a set of candidates (or alternatives) . These voters and candidates correspond to points in an arbitrary (unknown) metric space . The voter preferences over the candidates are induced by the underlying metric, i.e., voters prefer candidates who are closer to them. Voter prefers candidate over candidate (i.e., ) only if . Moreover, we assume that the strengths of voter preferences are induced by these latent distances. If prefers over , then the strength of this preference is . The cost to voter if candidate is elected is , and the goal is to select the candidate minimizing the Social Cost: .

In previous work on metric distortion only the ordinal preferences were known, i.e., whether or . In this paper, however, we assume that we are also given some information about the preference strengths as well. Note that knowing these values still does not tell us how compares with for , only how strongly each voter feels when comparing different candidates. In fact, while even knowing the exact preference strengths of all the voters is not enough to be able to select the optimum candidate (as we show in this paper), knowing just one bit of information about (such as whether for a threshold ) is enough to create mechanisms with much better distortion.

For a given voting rule and instance , let be the winning candidate selected by and let be the best available candidate (the one minimizing the Social Cost). Then, the distortion of winning candidate is defined as

 δI=SC(PI)SC(ZI)

The distortion of a voting rule is defined its behavior on a worst-case instance:

 δ=maxIδI=maxISC(PI)SC(ZI)

#### Our Contributions

What type of knowledge of the strengths of voter preferences is most useful and advantageous? What voting mechanisms should be used in order to minimize distortion if you have access to more information than only ordinal preferences? If you could gather data about voter preferences in different ways, what should you aim for in order to reduce distortion? These are some of the questions which we attempt to illuminate in this paper.

In this work, we study the possible distortion with different levels of voter preference strength information. A summary of our results is shown in Table 1. We begin with the setting in which we are given the voters’ ordinal preferences, as well as a threshold of voter preference strength. In other words, for any two candidates and , we know the number of voters who prefer to , as well as how many of them prefer to by at least a factor of (i.e., ). Based on only this information about the voter preferences (and the fact that the voters and candidates are embedded in some arbitrary unknown metric space), we are able to provide new voting mechanisms with much better distortion than possible when only knowing ordinal preferences. For the case that there are only two candidates, we provide a mechanism which achieves provably best possible distortion of , as shown in Figure 1. For the setting with more than two candidates, we get a distortion of as shown in Figure 2. Note that when , we get a distortion of 5. A recently paper shows a deterministic algorithm that gives a distortion of 4.236. We believe our result can be improved using similar mechanisms to start the curve in Figure 2 from 4.236.

From Figure 1 and 2, we can see that the distortion is minimized when in both settings. With only voter preferences being known, the best known deterministic distortion bounds are 3 for two candidates [1], and 4.236 for multiple candidates [25]. Interestingly, if we are also allowed to a choose a threshold , our results indicate that the optimal thing to do is to differentiate between candidates with lots of supporters who prefer them at least times to other candidates, and candidates which have few such supporters. By obtaining this information, we can improve the quality of the chosen candidate from a 3-approximation to only a 1.83 approximation (for 2 candidates), and from a 4.236-approximation to a 3.35-approximation (for candidates). This is a huge improvement obtained with relatively little extra cost in information gathering.

In Section 5 we consider the case when we only know the preferences of voters who feel strongly about their choice (prefer to by at least times), but do not know the preferences of voters who are relatively indifferent. We show that knowing how many voters feel strongly about a candidate is actually more important than knowing the ordinal preferences of all voters when attempting to minimize distortion: for example if we have we can obtain a distortion of 2 as well, even if we don’t know the preferences of all voters.

We then consider a more general case in Section 6. Suppose we have different thresholds , and voters report the largest threshold which their preference strength exceeds for each pair of candidates. As gets larger, the information about preference strengths gets less coarse; for most settings it would be realistic to assume that is small, but we provide a result which is as general as possible. With this information, we give a mechanism achieving the provably best distortion of in the two candidates setting, and a distortion of in the multiple candidates setting. Note that knowing all the preference strengths exactly is still not enough to always be able to choose the optimum candidate: the preference strengths are relative (“I like A twice as much as B”) as opposed to absolute. We never obtain information about how the costs of different voters compare to each other, the only thing we know is that the voters lie in a metric space. In fact, when we know the exact preference strengths of every voter, we obtain a distortion bound of in the two candidates setting, and a distortion of in the multiple candidates setting. Moreover, we prove that even knowing the exact preference strengths, it is not possible to obtain distortion better than in the worst case.

##### Ideal Candidate Distortion

In addition to forming mechanisms with small distortion, we also have a secondary goal in this paper. Rather than only comparing the winning candidate to the best available candidate, we can also measure them against the ideal conceivable candidate who may not be an available option to vote upon. is the point in the metric space which minimizes social cost; it is the absolute best consensus of the voters, and it would be wonderful if that point corresponded to a candidate, but that may not be the case (i.e., may not be in ). We introduce the notion of ideal candidate distortion as follows, where is any instance and is the winner that our mechanism selects for instance :

 Δ=SC(PI)SC(Z∗I)

As we show, while the ideal candidate distortion is unbounded in general, for many simple voting rules it can be bounded as a function of the distortion of the winning candidate (). Intuitively, the distortion can only be high when the best available candidate (best in ) is close to being the ideal possible candidate (best in the entire metric space).

A summary of our results on this topic is shown in Table 2. These results imply that if we are only given ordinal preferences, as in most previous work, and use certain mechanisms like the Copeland voting mechanism, then either the selected candidate is much closer to the best candidate in the running than the worst-case distortion bound indicates (say within factor of instead of the worst-case of 5 for the Copeland mechanism), or the selected candidate is not far from the ideal candidate, i.e., the best candidate that could ever exist (say within factor of 6 if ). So in the case when distortion is high, we at least can comfort ourselves with the fact that the selected candidate is not too far away from the best possible candidate that could ever exist, not just from the best candidate in the running.

## 2 Related Work and Discussion

The concept of distortion was introduced by [28] as a measure of efficiency for ordinal social choice functions (see also [1, 9] for discussion). Since then, two main approaches have emerged for analyzing the distortion of various voting mechanisms. One is assuming that the underlying unknown utilities or costs are normalized in some way, as in e.g., [5, 6, 7, 9, 11, 12]. The second approach, which we take here, assumes all voters and candidates are points in a metric space [1, 2, 3, 8, 13, 14, 15, 17, 19, 20, 22, 27, 30]. In particular, when the latent numerical costs that induce voter preferences over a set of candidates obey the triangle inequality, it is known that simple deterministic voting rules yield distortion which is always at most a small constant (5 for the well-known Copeland mechanism [1], and recently 4.236 for a more sophisticated, yet elegant, mechanism [25]). While [1] showed that no deterministic mechanism can always produce distortion better than 3, closing this gap remains an open question.

Randomized vs Deterministic Mechanisms   In this paper we restrict our attention to deterministic social choice rules, instead of randomized ones as in e.g., [2, 11, 17, 22]

, for several reasons. First, consider looking at our mechanisms from a social choice perspective, i.e., as voting rules that need to be adopted by organizations and used in practice. People are far more resistant to adopting randomized voting protocols. This is because an election with a non-trivial probability of producing a terrible outcome is usually considered undesirable, even if the

expected outcomes are good. There are many exceptions to this, of course, but nevertheless deterministic mechanisms are easier to convince people to adopt. Second, consider looking at our mechanisms from the point of view of approximation algorithms, i.e., as algorithms which attempt to produce an approximately-optimal solution given a limited amount of information. For traditional randomized approximation algorithms with guarantees on the quality of the expected outcome it is possible to run the algorithm several times, take the best of the results, and be relatively sure that you have achieved an outcome close to the expectation. In this setting of limited information, however, we cannot know the “true” cost of a candidate even after a randomized mechanism chooses it, and thus cannot take the best outcome after several runs. Therefore, unless stronger approximation guarantees are given than simply bounds on the expectation, it is quite likely that the outcome of a randomized algorithm in our setting would be far from the expected value. While randomized algorithms are certainly worthy of study even in our setting, and many interesting questions about them exist, we choose to focus only on deterministic algorithms in this paper.

Attempts to exploit preference strength information have led to various approaches for modeling, eliciting, measuring, and aggregating people’s preference intensities in a variety of fields, including Likert scales, semantic differential scales, sliders, constant sum paired comparisons, graded pair comparisons, response times, willingness to pay, vote buying, and many others (see [10, 16, 18] for summaries). In our work we specifically consider only a small amount of coarse information about preference strengths, since obtaining detailed information is extremely difficult. Intuitively, any rule used to aggregate preference strengths must ask under what circumstances an ‘apathetic majority’ should win over a more passionate minority [31], and we provide a partial answer to this question when the objective is to minimize distortion.

Perhaps most related to our work is that of [2] which introduced the concept of decisiveness. Using our notation, [2] proves bounds on distortion under the assumption that every voter has a preference strength at least between their top and second-favorite candidates. We, on the other hand, do not require that voters have any specific preference strength between any of their alternatives, and provide general mechanisms and distortion bounds based on knowing a bit more about voters (arbitrary) preference strengths. In other words, while [2] limits the possible space of voter preferences and locations in the metric space, we instead allow those to be completely arbitrary, but assume that we are given slightly more information about them.

In our model, when voter preference strength is less than the smallest threshold (), they effectively abstain because their preferred candidate is unknown, and so any reasonable weighted majority rule must assign them a weight of 0. Therefore, our work also bears resemblance to literature on voter abstentions in spatial voting (see [19] and references therein). While there are major technical differences in our model and that of [19], at a high level the model of [19] is similar to a special case of ours with only two candidates and a single threshold on preference strengths (and no knowledge of voter preferences otherwise), which we analyze in Section 5.

Finally, in this paper we assume that the preference strengths given to our algorithms are truthful, i.e., that the voters do not lie. While it would certainly be interesting and important to consider the case where voters may not be truthful (as in e.g., [7, 17]), for many settings with preference strengths it is actually more reasonable to expect voters to be truthful than for settings with only ordinal votes. This is because preference strengths are often signaled passively (e.g., average response times to surveys) or expressing this intensity comes at a cost (e.g., time commitments, activism, or monetary contributions and payments). Even in debates and committees where a member signals their strong preference for A over B, this member is putting their reputation on the line in doing so, and so may not want to do this unless their preference is actually that strong, in order to not look foolish or inconsistent in the future.

## 3 Preliminaries and Lower Bounds

In our model we have a set of voters and a set of candidates . These voters and candidates correspond to points in an arbitrary metric space , so for any three points the triangle inequality holds: . We assume that voters’ preferences over the candidates are induced by the underlying metric, and that voters are truthful (i.e., non-strategic). That is, voters prefer candidates who are closer to them. Voter prefers candidate over candidate only if . Moreover, we assume that the strength of voters’ preferences are induced by these latent distances. If prefers over , then the strength of this preferences is . When it is clear we are referring to 2 candidates and , we will drop the superscript.

Given a set of preference strength thresholds , voters report the largest threshold which their preference strength exceeds for each pair of candidates. We let and and and . For convenience, we say and . When we know the preferred candidate of every voter. When we let denote the set of voters with preference strength strictly less than whose preferred candidate is unknown. When , we know the exact preference strength of every voter for every pair of candidates.

We consider cost to voter if candidate is elected as and the Social Cost is the sum of the costs to all of the individual agents, . We would like to select the candidate with the minimum social cost. However, preference strength information is insufficient for any mechanism to guarantee selection of the best available candidate. Therefore, our primary goal is study and design mechanisms which minimize distortion (), the worst-case approximation ratio between the social cost of the candidate we select and the best available candidate over all possible instances, as defined in the Introduction.

### 3.1 Lower Bounds on Distortion with Preference Strengths

Here, we provide lower bounds on the minimum distortion any deterministic mechanism can achieve given only preference strength information. First, note that even if all exact preference strengths were known to us, we still would not be able to choose the optimum candidate: knowing the relative strength of preference for every voter is not the same thing as knowing their exact distances to every candidate (i.e., we would only know and not and themselves).

###### Theorem 1.

No deterministic mechanism with only preference strength information can achieve a worst-case distortion less than .

###### Proof.

The example used is in 1D, where candidates and are represented by points on a line. We normalize the distances so that is at location 0 and is at location 1. Suppose half the voters prefer with strength , and the other half prefer with strength . Since this is the only information known to the mechanism, the mechanism must tie-break in some arbitrary way (if tie-breaking is undesirable, we can have one extra voter prefer , which will result in distortion arbitrarily close to instead of exactly ). Thus without loss of generality, we let be the winner over .

Suppose the true location of the voters is as follows. Half of the voters are located at and the other half are located at . All voters have a preference strength of . If there are voters, the candidates have social costs and . Thus, if wins we have a lower bound on distortion of . ∎

Of course it is unrealistic to expect to know the exact preference strengths of all the voters. Below we give a general lower bound for the best distortion possible given knowledge of certain preference thresholds.

###### Theorem 2.

When given knowledge of fixed thresholds, no deterministic mechanism can always achieve a distortion less than

###### Proof.

The proof follows from the following 3 lemmas. The examples used for these lemmas are all in 1D, where candidates and are represented by points on a line. We normalize the distances so that is at location 0 and is at location 1 and use to denote an infinitesimal quantity. Without loss of generality, we let be the winner over . Recall that we have defined and for convenience.

###### Lemma 3.

If we have a set of thresholds of which the smallest is , no deterministic mechanism can always achieve a distortion less than .

###### Proof.

Suppose all voters are located at position . All voters therefore have preference strength less than , so and the preferred candidates of the voters are unknown. If wins over due to tie-breaking, as this yields a lower bound on distortion of . ∎

###### Lemma 4.

If we have a set of thresholds of which the largest is , no deterministic mechanism can always achieve a distortion less than .

###### Proof.

Suppose half of the voters are located on top of at position 1 and the other half of voters are located at , so . If wins over due to tie-breaking, as this yields a lower bound on distortion of . ∎

###### Lemma 5.

If we have a set of thresholds, of which two consecutive thresholds are and where , no deterministic mechanism can achieve a distortion less than .

###### Proof.

Suppose half of the voters are located at position and the other half are located at position . Once again, the mechanism must choose randomly between the candidates because . Therefore, if wins over due to tie-breaking, as , this yields a lower bound on distortion of . ∎

The combination of the three preceding lemmas guarantees the lower bound of Theorem 2. ∎

## 4 Adding the knowledge of a single threshold τ to ordinal preferences

### 4.1 Distortion with Two Candidates

In this section we begin by analyzing the case with only two possible candidates. In the section that follows, we use these results to form mechanisms with small distortion for multiple candidates. Suppose there are two candidates and . We are given the users’ ordinal preferences, and a strength threshold , i.e., for every voter we only know two bits of information: whether they prefer or , and whether their preference is strong () or weak (). Note that our results still hold if we only have this knowledge in aggregate, i.e., if for both and we know approximately how many people prefer to strongly versus weakly, and vice versa.

Notice that preference strengths tell us little about the true underlying distances for voters with weak preference strengths, because the preference strength of a voter almost directly between and who is very close to both can have the same preference strength as a voter who is very distant from both candidates. However, if a voter’s preference strength is large, we know they must be fairly close to one of the candidates - and it is these passionate voters who contribute most to distortion.

###### Weighted Majority Rule 1.

Given voters’ preferences and a threshold for two candidates, if , assign weight to all the voters with preference strengths and weight to all the voters with preference strengths . If , assign weight to all the voters with preference strengths and weight to all the voters with preference strengths . Choose the candidate by a weighted majority vote.

The following theorem shows that the above voting rule produces much better distortion than anything possible from knowing only the ordinal preferences. Moreover, due to the lower bounds in the previous section, this is the best distortion possible (apply Theorem 2 with and ).

###### Theorem 6.

With 2 candidates in a metric space, if we know voters’ preferences and a strength threshold , Weighted Majority Rule 1 has a distortion of at most .

###### Proof.

Denote the set of voters prefer with preference strengths as , and with preference strengths as . Also denote the set of voters prefer with preference strengths as , and with preference strengths as . Without loss of generality, suppose we choose as the winner by our weighted majority rule. It means that if , , and for , .

Proof Sketch and Main Idea: For all voters, consider their individual ratio of , regardless of which candidate they prefer. For voters who prefer this is their preference strength, and for voters who prefer this is the reciprocal of their preference strength. If for all voters this was less than , then clearly we have a distortion of at most by just summing them up. However, for some voters this ratio is higher and for others it is lower. If we think of charging to , we should charge the voters for whom this ratio is lower to the voters for whom this ratio is higher. Clearly, for any voters who prefer this ratio is less than 1 and so it is less than . For voters who prefer , some voters with weak preferences will allow us to save charge while others with stronger preferences will use up the extra charge. However, charging the voters to other voters seems quite difficult in this setting. The main new technique in our proof is to use as a sort of numeraire or store of value. We first perform the charging for all voters for whom this ratio is small, and we use to quantify how much extra charge is saved. We then show that this quantity of charge stored in terms of is sufficient to expend the charge from the remaining voters, yielding a distortion at most .

We first show some lemmas to bound by and for every voter .

, for any , .

###### Proof.

, . By the triangle inequality,

 d(P,Q)≤d(i,P)+d(i,Q)≤1τd(i,Q)+d(i,Q)=1+ττd(i,Q)

Thus . ,

 d(i,P) ≤1τd(i,Q) =δd(i,Q)−(δ−1τ)d(i,Q) ≤δd(i,Q)−δτ−1τ+1d(P,Q)

, for any , .

###### Proof.

, by the triangle inequality,

 d(P,Q)≤d(i,P)+d(i,Q)≤d(i,Q)+d(i,Q)=2d(i,Q)

Thus . ,

 d(i,P) ≤d(i,Q) =δd(i,Q)−(δ−1)d(i,Q) ≤δd(i,Q)−δ−12d(P,Q)

, for any , .

, for any , .

###### Proof.

First consider the case that .

, . Also, by the triangle inequality, . By a linear combination of these two inequalities,

 d(j,P) ≤δ−1τ−1τd(j,Q)+(1−δ−1τ−1)(d(j,Q)+d(P,Q)) ≤δd(j,Q)+τ−δτ−1d(P,Q)

Then consider the case that .

, . By the triangle inequality,

 d(P,Q)≤d(j,P)+d(j,Q)≤τd(j,Q)+d(j,Q)=(1+τ)d(j,Q)

Thus . ,

 d(j,P) ≤τd(j,Q) =δd(j,Q)−(δ−τ)d(j,Q) ≤δd(j,Q)−δ−ττ+1d(P,Q)

, .

###### Proof.

This lemma follows directly by the triangle inequality. ∎

Using the four lemmas above, sum up for all voters, for any ,

 ∑i∈A1d(i,P)+∑i∈A2d(i,P)+∑j∈B1d(j,P)+∑j∈B2d(j,P) ≤δ∑i∈A1d(i,Q)−|A1|δ−12d(P,Q)+δ∑i∈A2d(i,Q)−|A2|δτ−1τ+1d(P,Q) +δ∑j∈B1d(j,Q)−|B1|δ−ττ+1d(P,Q)+∑j∈B2d(j,Q)+|B2|d(P,Q) ≤δ∑id(i,Q)+(−|A1|δ−12−|A2|δτ−1τ+1−|B1|δ−ττ+1+|B2|)d(P,Q) (1)

Similarly, for any ,

 ∑id(i,P)≤δ∑id(i,Q)+(−|A1|δ−12−|A2|δτ−1τ+1+|B1|τ−δτ−1+|B2|)d(P,Q) (2)

Now we prove Theorem 6 by considering two cases: and .

Case 1, , and

We prove the distortion is at most in this case. Set . Note that when , . By inequality 2, if we can prove , then .

When ,

 −|A1|δ−12−|A2|δτ−1τ+1+|B1|τ−δτ−1+|B2| =−τ−1τ+1|A1|−3τ2−2τ−1(τ+1)2|A2|+τ−1τ+1|B1|+|B2| ≤−τ−1τ+1|A1|−|A2|+τ−1τ+1|B1|+|B2| ≤0

The second to last line follows because when . The last line follows because .

Case 2, , and

We prove the distortion is at most in this case. Set . Furthermore, we consider two subcases that and .

Case 2.1,

When and , it is easy to show that . By inequality 2, if we can prove , then .

When ,

 −|A1|δ−12−|A2|δτ−1τ+1+|B1|τ−δτ−1+|B2| =−1τ|A1|−|A2|+τ2−τ−2τ(τ−1)|B1|+|B2| ≤−1τ|A1|−|A2|+1τ|B1|+|B2| ≤0

The second to last line follows because when . The last line follows because .

Case 2.2,

Because and , it is easy to show that . By inequality 4.1, if we can prove , then . When ,

 −|A1|δ−12−|A2|δτ−1τ+1−|B1|δ−ττ+1+|B2| =−1τ|A1|−|A2|+τ−2τ|B1|+|B2| ≤−1τ|A1|−|A2|+1τ|B1|+|B2| ≤0

The second to last line follows because when . The last line follows because .

Thus, we have shown that the distortion is at most when , and at most when . Note that when , and when . Thus, the distortion of the weighted majority rule in this setting is . ∎

Note that Weighted Majority Rule 1 is not the only rule that gives the optimal distortion for two candidates. Consider the following simpler rule:

###### Weighted Majority Rule 2.

Given voters’ preferences and a threshold for two candidates, assign weight to all the voters with preference strengths and weight to all the voters with preference strengths .

This rule gives the same distortion as Weighted Majority Rule 1 for two candidates, as we prove below. When extending these rules to more than 2 candidates, however, Weighted Majority Rule 1 allows us to form better mechanisms, thus sacrificing a small amount of simplicity for an improvement in distortion. We discuss this in the next section.

###### Theorem 11.

Weighted Majority Rule 2 has a distortion of at most .

###### Proof.

Denote the set of voters prefer with preference strengths as , and with preference strengths as . Also denote the set of voters prefer with preference strengths as , and with preference strengths as . Without loss of generality, suppose we choose as the winner by Weighted Majority Rule 2. Thus, .

Similar to the proof of Theorem 6, we discuss three cases based on different values of .

Case 1,

Set . Because , it is easy to show that . By inequality 4.1, if we can prove , then . When ,

 −|A1|δ−12−|A2|δτ−1τ+1−|B1|δ−ττ+1+|B2| =−1τ|A1|−|A2|+τ−2τ|B1|+|B2| ≤−τ−1τ+1|A1|−|A2|+τ−1τ+1|B1|+|B2| ≤0

The second to last line follows because and when . The last line follows because .

Case 2,

Set . When , it is easy to show that . By inequality 2, if we can prove , then .

When ,

 −|A1|δ−12−|A2|δτ−1τ+1+|B1|τ−δτ−1+|B2| =−1τ|A1|−|A2|+τ2−τ−2τ(τ−1)|B1|+|B2| ≤−τ−1τ+1|A1|−|A2|+τ−1τ+1|B1|+|B2| ≤0

The second to last line follows because and when . The last line follows because .

Case 3,

Set . Note that when , . By inequality 2, if we can prove , then .

When ,

 −|A1|δ−12−|A2|δτ−1τ+1+|B1|τ−δτ−1+|B2| =−τ−1τ+1|A1|−3τ2−2τ−1(τ+1)2|A2|+τ−1τ+1|B1|+|B2| ≤−τ−1τ+1|A1|−|A2|+τ−1τ+1|B1|+|B2| ≤0

The second to last line follows because