Title: Metric Distortion Under Probabilistic Voting
Venue: NeurIPS 2024
Accepted: no
Keywords: Social choice

Metric Distortion

Probabilistic voting
Abstract: Metric distortion in social choice provides a framework for assessing how well voting rules minimize social cost in scenarios where voters and candidates exist in a shared metric space, with voters submitting rankings and the rule outputting a single winner. We expand this framework to include probabilistic voting. Our extension encompasses a broad range of probability functions, including widely studied models like Plackett-Luce (PL) and Bradley-Terry, and a novel "pairwise quantal voting" model inspired by quantal response theory.

We demonstrate that distortion results under probabilistic voting better correspond with conventional intuitions regarding popular voting rules such as Plurality, Copeland, and Random Dictator (RD) than those under deterministic voting. For example, in the PL model with candidate strength inversely proportional to the square of their metric distance, we show that Copeland's distortion is at most 2, whereas that of RD is $\Omega(\sqrt{m})$ in large elections, where $m$ is the number of candidates. This contrasts sharply with the classical model, where RD beats Copeland with a distortion of 3 versus 5 [1].

Abstract

Metric distortion in social choice provides a framework for assessing how well voting rules minimize social cost in scenarios where voters and candidates exist in a shared metric space, with voters submitting rankings and the rule outputting a single winner. We expand this framework to include probabilistic voting. Our extension encompasses a broad range of probability functions, including widely studied models like Plackett-Luce (PL) and Bradley-Terry, and a novel "pairwise quantal voting" model inspired by quantal response theory. We demonstrate that distortion results under probabilistic voting better correspond with conventional intuitions regarding popular voting rules such as Plurality, Copeland, and Random Dictator (RD) than those under deterministic voting. For example, in the PL model with candidate strength inversely proportional to the square of their metric distance, we show that Copeland's distortion is at most 2, whereas that of RD is Ω( √ m ) in large elections, where m is the number of candidates. This contrasts sharply with the classical model, where RD beats Copeland with a distortion of 3 versus 5 [1].

1 Introduction

Societies must make decisions collectively; different agents often have conflicting interests, and the choice of the mechanism used for combining everyone's opinions often makes a big difference to the outcome. The machine learning community has applied social choice principles for AI alignment [2, 3], algorithmic fairness [4, 5], and preference modelling [6, 7]. Over the last century, there has been increasing interest in using computational tools to analyse and design voting rules [8-11]. One prominent framework for evaluating voting rules is that of distortion [12], where the voting rule has access to only the ordinal preferences of the voters. However, the figure of merit is the sum of all voters' cardinal utilities (or costs). The distortion of a voting rule is the worst-case ratio of the cost of the alternative it selects and the cost of the truly optimal alternative.

An additional assumption is imposed in metric distortion [1] - that the voters and candidates all lie in a shared (unknown) metric space, and costs are given by distances (thus satisfying non-negativity and triangular inequality). This model is a generalization of a commonly studied spatial model of voting in the Economics literature [13, 14], and has a natural interpretation of voters liking candidates with a similar ideological position across many dimensions. While metric distortion is a powerful framework and has led to the discovery and re-discovery of interesting voting rules (e.g. Plurality Veto [15] and the study of Maximal Lotteries [16] for metric distortion by Charikar et al. [17]), its outcomes sometimes do not correspond with traditional wisdom around popular voting rules. For example, the overly simple Random Dictator (RD) rule (where the winner is the top choice of a uniform randomly selected voter) beats the Copeland rule (which satisfies the Condorcet Criterion [10] and other desirable properties) with a metric distortion of 3 versus 5 [1].

While not yet adopted in the metric distortion framework, there is a mature line of work on 36 Probabilistic voting (PV) [18-20]. Here, the focus is on the behavioural modelling of voters and 37 accounting for the randomness of their votes. Two sources of this randomness often cited in the 38 literature are the boundedness of the voters' rationality and the noise in their estimates of candidates' 39 positions. A popular model for this behaviour is based on the Quantal Response Theory [20]. Another 40 closely related line of work is on Random Utility Models (RUMs) [21-23] in social choice where 41

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the hypothesis is that the candidates have ground-truth strengths. Voters make noisy observations of these strengths and vote accordingly. We adopt these models of voting behaviour and study it within the metric distortion framework. The questions we ask are:

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1.1 Preliminaries and Notation

Let N be a set of n voters and A be the set of m candidates. Let S be the set of total orders on A . Each voter i ∈ N has a preference ranking σ i ∈ S . A vote profile is a set of preference rankings σ N = ( σ 1 , ..., σ n ) ∈ S n for all voters. The tuple ( N , A , σ N ) defines an instance of an election. Let ∆( A ) denote the set of all probability distributions over the set of candidates.

Definition 1 (Voting Rule) . A voting rule f : S n → ∆( A ) takes a vote profile σ N and outputs a probability distribution p over the alternatives.

For deterministic voting rules, we overload notation by saying that the rule's output is a candidate and not a distribution. We now define some voting rules [10]. Let I denote the indicator function.

Random Dictator Rule: Select a voter uniformly at random and output their top choice, i.e., RD ( σ N ) = p such that p j = 1 n ∑ i ∈N I ( σ i, 1 = j ) .

Plurality Rule: Choose the candidate who is the top choice of the most voters, i.e., PLU ( σ N ) = arg max j ∈A ∑ i ∈N I ( σ i, 1 = j ) . Ties are broken arbitrarily.

Copeland Rule: Choose the candidate who wins the most pairwise comparisons, i.e., COP ( σ N ) = arg max j ∈A ∑ j ′ ∈A\{ j } I (∑ i ∈N I ( j ≻ σ i j ′ ) > n 2 ) . Ties are broken arbitrarily.

Distance function d : ( N ∪ A ) 2 → R ≥ 0 satisfies triangular inequality ( d ( x, y ) ≤ d ( x, z ) + d ( z, y ) ) and symmetry ( d ( x, y ) = d ( y, x ) ). The distance between voter i ∈ N and candidate j ∈ A is also referred to as the cost of j for i . We consider the most commonly studied social cost function, which is the sum of the costs of all voters. SC ( j, d ) := ∑ i ∈N d ( i, j ) .

In deterministic voting, the preference ranking σ i of voter i is consistent with the distances. That is, d ( i, j ) > d ( i, j ′ ) = ⇒ j ′ ≻ σ i j for all voters i ∈ N and candidates j, j ′ ∈ A . Let ρ ( σ N ) be the set of distance functions d consistent with vote profile σ N . The metric distortion of a voting rule is:

1.2 Our Contributions

We extend the study of metric distortion to probabilistic voting (Definition 4). This extension is useful since voters, in practice, have been shown to vote randomly [20]. We define axiomatic properties of models of probabilistic voting which are suitable for studying metric distortion. These are scalefreeness with distances (Axiom 1), pairwise order probabilities being independent of other candidates (Axiom 2), and strict monotonicity of pairwise order probabilities in distances (Axiom 3).

All our results apply to a broad class of models of probabilistic models, as explained in § 2. We provide distortion bounds for all n ≥ 3 and m ≥ 2 , which are most salient in the limit n →∞ . For large elections ( m fixed, n →∞ ), we provide matching upper and lower bounds on the distortion of Plurality, an upper bound for Copeland, and a lower bound for RD. The distortion of plurality grows linearly in m . The distortion upper bound of Copeland is constant. The distortion lower bound for RD increases sublinearly in m where this rate depends on the probabilistic model. Crucially, our results match those in deterministic voting in the limit where the randomness goes to zero.

The technique is as follows. For the problem of maximizing the distortion, we establish a critical 83 threshold of the expected fraction of votes on pairwise comparisons on all edges on a directed path 84 from a winner to the 'true optimal" candidate for Copeland and Plurality. This path is one or two hops 85 for Copeland and one for Plurality. We then formulate a linear-fractional program which incorporates 86 this critical threshold. We linearize this program via the sub-level sets technique [24], and find a 87 feasible solution of the dual problem. Concentration inequalities on this solution provide an upper 88

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1.3 Related Work

Metric distortion Anshelevich et al. [1] initiated the study of metric distortion and showed that any deterministic voting rule has a distortion of at least 3 and that Copeland has a distortion of 5. The Plurality Veto Rule attains the optimal distortion of 3 [15]. Charikar and Ramakrishnan [25] showed that any randomized voting rule has a distortion of at least 2 . 112 . Charikar et al. [17] gave a randomized voting rule with a distortion of at most 2.753. Anshelevich et al. [26] gave a useful survey on distortion in social choice.

Distortion with Additional Information Abramowitz et al. [27] showed that deterministic voting rules achieve a distortion of 2 when voters provide preference strengths as ratios of distances. Amanatidis et al. [28] demonstrated that even a few queries from each voter can significantly improve distortion in non-metric settings. Anshelevich et al. [29] examined threshold approval voting, where voters approve candidates with utilities above a threshold. Our work relates to these studies since in probabilistic voting, the likelihood of a voter switching the order of two candidates depends on the relative strength of their preference, often resulting in lower distortion than deterministic methods.

Probabilisitc voting and random utility models (RUMs) Hinich [30] showed that the celebrated Median Voter Theorem of [31] does not hold under probabilistic voting. Classical work has focused on studying the equilibrium positions of voters and/or candidates in game-theoretic models of probabilistic voting [20, 32-35]. McKelvey and Patty [20] adopt the quantal response model, a popular way to model agents' bounded rationality.

RUMs have mostly been studied in social choice [21, 23, 36] with the hypothesis that candidates have universal ground-truth strengths, which voters make noisy observations of. Our model is the same as RUM regarding the voters' behaviour; however, voters have independent costs from candidates. The Plackett-Luce (PL) model [37, 38] has been widely studied in social choice [39-41]. For probabilities on pairwise orders, PL reduces to the Bradley-Terry (BT) model [42]. These probabilities are proportional to candidates' strengths (which we define as the inverse of powers of costs).

The widely studied Mallows model [43], based on Condorcet [44], flips the order of each candidate pair (relative to a ground truth ranking) with a constant probability p ∈ (0 , 1 2 ) [45, 46]. The process is repeated if a linear order is not attained. In the context of metric distortion, a limitation of this model is that it doesn't account for the relative distance of candidates to the voter. For a comprehensive review of RUM models, see Marden [47]. Critchlow et al. [48] does an axiomatic study of RUM models; our axioms are grounded in metric distortion and are distinct from theirs.

Recently, there has been significant interest in smoothed analysis [49] of social choice. Here a small amount of randomness is added to problem instances and its effect is studied on the satisfiability of axioms [50-53] and the computational complexity of voting rules [54-56]. Baumeister et al. [50] term this model as being 'towards reality,' highlighting the need to study the randomness in the election instance generation processes. Unlike smoothed analysis where the voter and candidate positions are randomized, we consider these positions fixed, but the submitted votes are random given these positions. The technical difference appears in the benchmark (the 'optimal" outcome in the denominator of the distortion is unchanged in our framework and changes in smoothed analysis).

2 Axioms and Model

Under probabilistic voting, the submitted preferences may no longer be consistent with the underlying distances. For a distribution P ( d ) over σ N , let q P ( d ) ( i, j, j ′ ) denote the induced marginal probability that voter i ranks candidate j higher than j ′ . We focus on these marginal probabilities on pairwise orders and provide axioms for classifying which q P ( d ) ( · ) are suitable for studying distortion.

Axiom 1 (Scale-Freeness (SF)) . The probability q P ( d ) ( · ) must be invariant to scaling of d . That is, for any tuple ( i, j, j ′ ) and any constant κ > 0 , we must have q P ( d ) ( i, j, j ′ ) = q P ( κd ) ( i, j, j ′ ) .

Note that the metric distortion (Definition 2) for deterministic voting is scale-free. We want to retain the same property in the probabilistic model as well. Conceptually, one may think of the voter's preferences as being a function of the relative (and not absolute) distances to the candidates.

Axiom 2 (Independence of Other Candidates (IOC)) . The probability q P ( d ) ( i, j, j ′ ) must be independent of the distance of voter i to all 'other' candidates, i.e., those in A\{ j, j ′ } .

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This axiom extends Luce's choice axioms [38], defined for selecting the top choice, to entire rankings. IOC is reminiscent of the independence of irrelevant alternatives axiom for voting rules.

Axiom 3 (Strict Monotonicity (SM)) . For every tuple ( i, j, j ′ ) , for fixed distance d ( i, j ) > 0 , the probability q P ( d ) ( i, j, j ′ ) must be strictly increasing in d ( i, j ′ ) at all but at most finitely many points.

The monotonicity in follows since

This axiom is natural.

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Here h ( k, ϕ ) = k (1 -ϕ k ) . Whereas r j and r j ′ are the positions of j and j ′ in the ground-truth (noiseless) ranking, and the constant ϕ is a dispersion parameter. Observe that this model fails Axiom 2 since it depends on the number of candidates between j and j ′ in the noiseless ranking. It also fails Axiom 3 since it does not depend on the exact distances but only on the order of the distances.

Plackett-Luce Model: The PL model [37, 38] is 'sequential' in the following way. For each voter i ∈ N , each candidate j ∈ A has a 'strength' s i,j . In most of the literature on RUMs, a common assumption is that s i,j is the same for all voters i . However, we choose this more general model to make it useful in the context of metric distortion. The voter chooses their top choice with probability proportional to the strengths. Similarly, for every subsequent rank, they choose a candidate from among the remaining ones with probabilities proportional to their strengths. In terms of the pairwise order probabilities, the PL model reduces to the Bradley-Terry (BT) model [42], that is:

Prima facie, in the metric distortion framework, any decreasing function of distance d ( i, j ) would be a natural choice for s i,j . However, not all such functions satisfy Axiom 1. The exponential function is a popular choice in the literature employing BT or PL models. However, in general, e -d ( i,j ) e -d ( i,j ) + e -d ( i,j ′ ) = e -2 d ( i,j ) e -2 d ( i,j ) + e -2 d ( i,j ′ ) , thus failing the Scale-Freeness Axiom 1.

̸

On the other hand, observe that all functions s = d -θ for any θ ∈ (0 , ∞ ) satisfy our axioms. We use the regime θ ∈ (1 , ∞ ) for technical simplicity in this work.

We also define the following class of functions 'PQV' for q P ( d ) ( · ) motivated by Quantal Response Theory [58] and its use in probabilistic voting [20]. Observe that PQV satisfies all our axioms.

Definition 3 (Pairwise Quantal Voting (PQV)) . Let the relative preference r ( i, j, j ′ ) be the ratio of distances, d ( i,j ′ ) d ( i,j ) . For constant λ > 0 , PQV is as follows: q P ( d ) ( i, j, j ′ ) = e -λ/r ( i,j,j ′ ) e -λr ( i,j,j ′ ) + e -λ/r ( i,j,j ′ ) .

We now define a general class of functions for pairwise order probabilities in terms of the relative preference (ratio of distances) r . Let G be a class of functions such that any G ∋ g : [0 , ∞ ) ∪{∞} → [0 , 1] has the following properties.

1. g is continuous and twice-differentiable.

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There ∃ c ∈ [0 , ∞ ) s.t. g ′′ ( r ) > 0 ∀ r ∈ (0 , c ) i.e. g is convex in the open interval (0 , c ) .

Observe that PL (with g ( r ) = r θ 1+ r θ , θ > 1 ) and PQV (with g ( r ) = e -λ/r e -λr + e -λ/r , λ > 0 ) are in G . Construction of distributions (if any exists) on rankings σ N which generate pairwise order

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probabilities q P ( d ) ( i, j, j ′ ) = g ( d ( i,j ′ ) d ( i,j ) ) according to PQV is left for future work. We do not need it 177 for our technical derivations. For PL, these distributions are known from prior work [40]. 178

We assume g ∈ G in the rest of the paper. Let M ( N ∪ A ) denote the set of valid distance functions 179 on ( N , A ) . For any g and d ∈ M ( N ∪ A ) let ˆ P ( g ) ( d ) denote the set of probability distributions on 180 σ N for which the marginal pairwise order probabilities are g ( d ( i,j ′ ) d ( i,j ) ) . That is, 181

We assume that all voters vote independently of each other. We now define metric distortion under 182 probabilistic voting as a function of g for a given m and n . 183

Definition 4 (Metric Distortion under Probabilistic Voting) .

DIST ( g ) ( f ) = sup n,m DIST ( g ) ( f, n, m ) by supremizing over all possible n and m . 184

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The expectation is both over the randomness in the votes and the voting rule f .

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Observe that the distortion is a supremum over all distributions in ˆ P ( g ) ( d ) . Since we focus on large elections (with large n and relatively small m ), we define DIST ( g ) as a function of m and n .

As in Fig. 1, consider the 1-d Euclidean space with candidate X at the origin and Y at 1 . Observe that g ( x 1 -x ) and g ( x 1+ x ) denote the probability that a voter located at a distance x from X votes for Y when the voter is to the left and right of X respectively. Interestingly, this 1-d intuition extends well for general metric spaces. Towards this, we define the following functions.

Lemma 1. g MID ( x ) x and g OUT ( x ) x have unique local maxima in (0 , 1) and (0 , ∞ ) respectively.

Denote the unique maximisers of g MID ( x ) x and g OUT ( x ) x by x ∗ MID and x ∗ OUT respectively.

For simplifying notation, in the rest of the work, we use ˆ g MID for g MID ( x ∗ MID ) x ∗ MID and ˆ g OUT for g OUT ( x ∗ OUT ) x ∗ OUT .

In the analysis in the rest of the paper, we will see ˆ g MID and ˆ g OUT appear many times, so we note these 195 quantities for the PL and PQV models here. For the PL model with θ = 2 , ˆ g MID = √ 2+1 2 ≈ 1 . 21 and 196 ˆ g OUT = √ 2 -1 2 ≈ 0 . 21 . When θ = 4 , ˆ g MID ≈ 1 . 42 and ˆ g OUT ≈ 0 . 06 . When θ → ∞ , ˆ g MID → 2 and 197 ˆ g OUT → 0 . This limit is where PL resembles deterministic voting. 198

For PQV with λ = 1 , ˆ g MID ≈ 1 . 25 and ˆ g OUT = 0 . 18 . When λ →∞ , ˆ g MID → 2 and ˆ g OUT → 0 . 199

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3 Distortion of Plurality Rule Under Probabilistic Voting

In this section, we give upper and lower bounds on the distortion of the Plurality rule [59] (PLU).In the limit the number of voters n →∞ ('large election"), our upper and lower bounds match and are linear in the number of candidates m. Let B represent the candidate that minimizes the social cost (referred to as 'best'), and let { A j } j ∈ [ m -1] denote the set of other candidates.

3.1 Upper bound on the distortion of Plurarity(PLU)

Theorem 1. For every ϵ > 0 and m ≥ 2 and n ≥ m 2 we have

Further, lim n →∞ DIST ( g ) ( PLU , n, m ) ≤ max( m ˆ g MID -1 , m ˆ g OUT +1) . 207

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To prove this theorem, we first give a lemma which upper bounds SC ( W,d ) SC ( B,d ) under the constraint that the expected number of voters that rank candidate W over B is given by α . This ratio will be useful to bound the contribution of non-optimal candidate W to the distortion of PLU. We state an optimization problem (7) below, which would be required to bound the ratio as a function of α .

Lemma 2. For any two candidates W,B ∈ A which satisfy n ∑ i =1 /a80 [ W ≻ i B ] = α , we have 212

Our proof is via Lemmas 3 and 4. Lemma 3 shows that we can bound the ratio of social costs by the 213 inverse of the optimum value of E α and Lemma 4 gives a lower bound on the optimum value of E α . 214

Lemma 3. For any two candidates W,B ∈ A satisfying ∑ n i =1 [ W ≻ i B ] = α , we have 215

Proof. b i and w i in (7) represent the distances d ( i, B ) and d ( i, W ) . The last constraint is the triangle 216 inequality i.e. | d ( i, B ) -d ( i, W ) | ≤ d ( B,W ) ≤ | d ( i, B ) + d ( i, W ) | for every voter i ∈ N . 217

Consider the following linearized version of (7). 218

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Our proof uses Lemma 5 and is by solving a linearized version of (7) in (10). This is done by introducing an extra non-negative parameter µ ≤ 1 . Note that it is sufficient to consider µ ≤ 1 since opt ( E α ) ≤ 1 because B minimises the social cost by definition. We find the smallest µ ∈ (0 , 1) such that its objective is non-negative.

Lemma 5. If opt ( E µ,α ) ≥ 0 , then opt ( E α ) ≥ µ .

The first part follows since scaling each term by a constant r satisfies the constraints and also yields the same objective. And thus we may replace the constraints by max i | w i -b i | ≤ 1 and min i ( w i + b i ) ≥ 1 in equation (10). Further, the objective function is linearized as ( ∑ n i =1 b i ) -µ ( ∑ n i =1 w i ) .

The proof of the second part is technical and has been moved to Appendix B. It involves introducing a Lagrangian multiplier λ and demonstrating that the objective function is non-negative for a suitably chosen λ . To establish this, we show that minimising the Lagrangian over the boundaries of the constraint set given by | b i -w i | = 1 and b i + w i = 1 is sufficient. This requires a careful analysis.

The main technique used in proving Theorem 1 involves considering two cases for every non-optimal candidate A j : one where the expected number of voters ranking candidate A j above B (call it α j ) exceeds a threshold of n m -n ϵ +1 / 2 m and one where it does not. In the first case, the ratio of social costs of A j and B is bounded using Lemma 2 that naturally gives a bound on contribution of candidate A j to the distortion. In the later case, we use Chernoff bound to bound the probability of A j being the winner and multiply it with the ratio of social costs of A j and B to bound the distortion. The proof of Theorem 1 is in Appendix C.

3.2 Lower bound on the distortion of Plurality

We now present a lower bound on the distortion of PLU for any m in the limit n tends to infinity. This lower bound matches the upper bound of Theorem 1 in the limit. A full proof is in Appendix D. Note that the proof has an adversarially chosen distribution over the rankings subject to the marginals on pairwise relationships satisfying g (as in the definition of distortion under probabilistic voting 4). This lower bound does not apply to the PL model, which has a specific distribution over rankings.

Proof Sketch. The proof is by an example in an Euclidean metric space in R 3 . One candidate 'C" is at (1 , 0 , 0) . The other m -1 candidates are 'good" and are equidistantly placed on a circle of radius ϵ on the y -z plane centred at (0 , 0 , 0) . We call them G := { G 1 , G 2 , . . . , G m -1 } .

We present sketches of two constructions below for every ϵ, ζ > 0 .

Construction 1 : Let q MID := g ( √ ( x ∗ MID ) 2 + ϵ 2 1 -x ∗ MID ) and a MID := 1 m -1 ( 1 -1+ ζ mq MID ) . Each of the m -1 candidates in G has ⌊ a MID n ⌋ voters overlapping with it. The remaining voters (we call them 'ambivalent') are placed at ( x ∗ MID , 0 , 0) . Clearly, each voter overlapping with a candidate votes for it as the most preferred candidate with probability one. Each of the ambivalent voters votes as follows.

- With probability q MID , vote for candidate C as the top choice and uniformly randomly permute the other candidates in the rest of the vote.

- With probability 1 -q MID , vote for candidate C as the last choice and uniformly randomly permute the other candidates in the rest of the vote.

We show that the probability that C wins tends to 1 as n →∞ and the distortion is m ˆ g MID -1 .

Construction 2 : We give a construction where the locations of the candidates are identical as in Construction 1, and some voters are located with the 'good" candidates. The ambivalent voters are at ( -x ∗ OUT , 0 , 0) . We show that P [ C wins ] tends to 1 as n →∞ and the distortion is m ˆ g OUT +1 .

This result establishes that the distortion of Plurality is bound to increase linearly with m even under probabilistic voting, and is therefore not a good choice when m is even moderately large.

4 Distortion of Copeland Rule Under Probabilistic Voting 265

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We now bound the distortion of the Copeland voting rule. We say that candidate W defeats candidate Y if more than half of the voters rank W above Y .

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Theorem 3. For every ϵ > 0 , m ≥ 2 and n ≥ 4 , we have

For every m ≥ 2 , we have lim n →∞ DIST ( g ) ( COP , n, m ) ≤ max ( (2ˆ g MID -1) 2 , (2ˆ g OUT +1) 2 ) .

Proof Sketch. A Copeland winner belongs to the uncovered set in the tournament graph, as demonstrated in [1, Theorem 15]. Recall that B denotes the candidate with the least social cost. For a Copeland winner W , either W defeats B or it defeats a candidate Y who defeats B .

We now consider two exhaustive cases on candidate A j and define event E j for every j ∈ [ m -1] by computing the expected fraction of votes on pairwise comparisons. The event E j denotes the existence of an at-most two hop directed path from a candidate A j to candidate B for Copeland such that the expected fraction of votes on all edges along that path exceed n -n (1 / 2+ ϵ ) .

2 2

If E j holds true, we upper bound the ratio of social cost of candidate A j and social cost of candidate B using Lemma 2 which in-turn would give a bound on the distortion. Otherwise, we use union bound and Chernoff's bound to upper bound the probability of A j being the winner. Multiplying the probability bound with the ratio of social costs (one obtained from Lemma 2) leads to a bound on the distortion. A detailed proof is in Appendix E.

5 Distortion of Random Dictator Rule Under Probabilistic Voting

We first give an upper bound on the distortion of RD; the proof is in Appendix F.

Theorem 4. DIST ( g ) ( RD , m, n ) ≤ ( m -1)ˆ g MID +1 .

We now give a lower bound on the distortion of RD. We do this by constructing an example.

Proof. We have a 1-D Euclidean construction. Let B be at 0 and all other candidates A\{ B } be at 1. m -1 voters are at 0 and one voter V is at ˜ x = g -1 ( 1 m -1 ) / (1 + g -1 ( 1 m -1 )) .

The ranking for V is generated as follows: pick a candidate from A\{ B } as the top rank uniformly at random. Keep B on the second rank. Permute the remaining candidates uniformly at random for the remaining ranks. Observe that the marginal pairwise order probabilities are consistent with the distance of V from B and each candidate in A\{ B } . In particular g ( ˜ x 1 -˜ x ) = 1 m -1 . The distortion for this instance is P [ B wins ] · 1+ P [ B loses ] · n -˜ x ˜ x = n -1 n + 1 n n -˜ x ˜ x = 1+ 1 ˜ x -2 n = 2+ 1 g -1 ( 1 m -1 ) -2 n .

However, note that this result does not apply to the PL model! This is because the PL model has a specific distribution on the rankings. In contrast, the above result is obtained by choosing an adversarial distribution on rankings subject to the constraint that its marginals on pairwise relations

d

A k

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∑

(

i,A

∈A

j

d

(

)

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θ

i,A

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)

-

θ

following result for the PL model. A proof via a similar construction as Theorem 5 is in Appendix G.

Theorem 6. Let DIST θ PL ( RD , m, n ) denote the distortion when the voters' rankings are generated per the PL model with parameter θ . We have lim n →∞ DIST θ PL ( RD , m, n ) ≥ 1 + ( m -1) 1 /θ 2 .

are given by

g

. In the PL model,

P

[

A

is top-ranked in

σ

] =

[45]. We have the

6 Numerical Evaluations 303

1

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64

in PL model

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Distortion

Plurarity: m=3

RD: m=3

Plurarity: m=6

RD: m=6

Plurarity: m=12

RD: m=12

Copeland

0.5

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32

in PQV model

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Distortion

Plurarity: m=3

RD: m=3

Plurarity: m=6

RD: m=6

Plurarity: m=12

RD: m=12

Copeland

Recall that higher values of θ and λ correspond to lower randomness. From Figure 2, we observe that under sufficient randomness, the more intricate voting rule Copeland outshines the simpler rule RD, which only looks at a voter's top choice. Moreover, its distortion is independent of m in the limit n →∞ . This is in sharp contrast to RD, where the distortion is Ω( m 1 /θ ) in the PL model, a sharp rate of increase in m for low values of θ. The distortion of Plurality increases linearly in m .

An important observation is regarding the asymptotics when θ or λ increases. The distortion of RD converges to its value under deterministic voting, i.e., 3. The distortion of Plurality also converges to 2 m -1 , the same as in deterministic voting. Since our bound on Copeland is not tight, it converges to 9 rather than 5 . So far, in the study of metric distortion, the social choice community has looked only at these asymptotic; here, we present insights available from looking at the 'complete' picture. Interestingly, the distortion of RD increases with randomness, whereas that of Copeland decreases up to a certain point and then increases again. The reason for the increases in the high randomness regime is that the votes become too noisy to reveal the best candidate any more.

Since these plots have no abrupt transitions, this figure hints that smoothened analysis [52] (typically done with small amounts of noise) is unlikely to give any new insights regarding metric distortion.

7 Discussion and Future Work

We extend the metric distortion framework in social choice in an important way - by capturing the bounded rationality and randomness in voters' behaviour. Consideration of this randomness shows that, in general, the original metric distortion framework is too pessimistic on important voting rules, most notably on Copeland. On the other hand, the simplistic voting rule Random Dictator, which attains a distortion of 3 (at least as good as any deterministic rule [1]), is not so good when we look at the full picture - its distortion increases with the number of candidates in our model. Our framework opens up opportunities to revisit the metric distortion problem with a closer-to-reality view of voters. It may hopefully lead to the development of new voting rules that consider the randomness of voters' behaviour. For example, Liu and Moitra [46] take a learning theory approach to design voting rules under the assumption of random voting per the Mallows model. However, technical analysis in our framework may be challenging because of the interplay of the geometric structure of voters' positions and the probabilistic nature of their votes.

Future Work An interesting extension would be to other tournament graph-based voting rules (weighted or unweighted). Our techniques are well-suited for this class of rules since it is based on the expected weights of the edges of the tournament graph. Closing the gap for the distortion of Copeland would be useful for getting deeper insights. Another open problem is the characterization of the set of distributions on rankings that induce the pairwise probabilities per PQV.

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A Proof of Lemma 1

Lemma (Restatement of Lemma 1) . g MID ( x ) x and g OUT ( x ) x have unique local maxima in (0 , 1) and (0 , ∞ ) respectively.

To prove Lemma 1, we first state and prove Lemma 6 which shows that g MID ( x ) and g OUT ( x ) change from convex to concave in intervals (0 , 1) and (0 , ∞ ) respectively.

Lemma 6. · There ∃ c 1 ∈ [0 , 1] s.t. g MID ( x ) is convex in [0 , c 1 ] and concave in [ c 1 , 1] .

· There ∃ c 2 ∈ [0 , ∞ ) s.t. g OUT ( x ) is convex in [0 , c 2 ] and concave in [ c 2 , ∞ ) .

Proof. Observe that g ′′ ( x ) < 0 for x ≥ 1 .

Now we show that there cannot exist two distinct c 1 , c 2 ∈ (0 , 1) such that g ′′ MID ( c 1 ) = 0 and g ′′ MID ( c 2 ) = 0 . We prove this statement by contradiction assuming the contrary which implies that g ′′ MID ( x ) must have changed its sign twice. However, since g ′ ( x 1 -x ) > 0 we must have g ′′ ( x 1 -x ) changing its sign twice which is a contradiction since g ′′ ( r ) > 0 for r ∈ (0 , c ) and g ′′ ( r ) < 0 for r ∈ ( c, ∞ ) .

Using Lemma 6, we now prove Lemma 1 showing the existence of unique maximas of g MID ( x ) x and g OUT ( x ) x .

Proof of Lemma 1. Recall from Lemma 6 that g MID ( x ) is convex in [0 , c 1 ] and concave in [ c 1 , 1] .

Since the first derivative equals zero at every local maxima, we must have xg ′ MID ( x ) -g ( x ) = 0 for any local maxima x . We now argue that such a maxima cannot exist in [0 , c 1 ] . Suppose such a maxima exists in that case, we must have g ′ MID ( x ) = g MID ( x ) -g MID (0) x -0 for some x ∈ (0 , c 1 ) . Applying LMVT in the interval [0 , x ] 1 , we must have some t ∈ (0 , x ) s.t. g ′ MID ( t ) = g MID ( x ) -g MID (0) x -0 , thus implying g ′ MID ( x ) = g ′ ( t ) contradicting the fact that g ′ MID ( r ) is strictly increasing in [0 , c 1 ] .

We now argue that this local maxima x ∗ is unique. Suppose we have two distinct local maximas at x 1 , x 2 ∈ ( c 1 , ∞ ) and thus, we have x 1 g ′ MID ( x 1 ) -g MID ( x 1 ) = 0 and x 2 g ′ MID ( x 2 ) -g MID ( x 2 ) = 0 . Rolle's theorem would imply that there exists t ∈ ( x 1 , x 2 ) 2 s.t. tg ′′ MID ( t ) = 0 which is a contradiction since g ′′ MID ( x ) < 0 in ( c 1 , ∞ ) .

Similarly, we can prove the result on the existence and uniqueness of maxima of the function g ( x x +1 ) x .

B Proof of Lemma 5

Lemma (Restatement of Lemma 5) . If opt ( E µ,α ) ≥ 0 , then opt ( E α ) ≥ µ .

Proof. To lower bound the optimal value of E µ,α , we first pre-multiply the first constraint by λ (and substitute 511 b i w i = r i ∀ i ∈ [ n ] ) and thus define, 512

Further, we define the set which satisfies the last two constraints in E µ,α by C as

From the theory of Lagrangian, we have the following

Now for a fixed λ > 0 , we minimise F ( r , b , λ ) over ( r , b ) ∈ C . Observe that for every i ∈ [ n ] , it is sufficient to minimise h ( r i , b i ) defined as follows.

Observe that the constraints in C can be written as b i ≥ r i 1+ r i and b i ≤ r i | 1 -r i | .

Observe that for a given r i , the function h ( r i , b i ) is monotonic in b i and thus the optimum point must lie on the boundary and first optimize over b i (1 + 1 /r i ) = 1 (call it C MID i ) and | b i (1 -1 /r i ) | = 1 (call it C OUT i ) respectively.

Recall from Lemma 6 that there exists c 1 , c 2 s.t. g MID ( x ) is convex in (0 , c 1 ) and concave in ( c 1 , 1) and g OUT ( x ) is convex in (0 , c 2 ) and concave in ( c 2 , ∞ ) .

Minimisation of h ( r i , b i ) over b i (1 + 1 /r i ) = 1 .

We first substitute 1 /r i = 1 /b i -1 in the function and thus, can write the function h ( b i ) = b i ( µ + 1) -µ -λ ( g ( b i 1 -b i ) -α n ) = b i ( µ +1) -µ -λ ( g MID ( b i ) -α n ) .

Observe that on optimizing over b i , we obtain two local minima, one at b i = 0 and the other at b i = ˜ x MID ( λ ) ∈ ( c 1 , ∞ ) where ˜ x MID ( λ ) satisfies the following equations if λ ≥ 1+ µ g ′ MID ( c 1 ) . Otherwise, we have a unique minima at b i = 0 . 3

Observe ˜ x MID ( λ ) > c 1 since g MID is concave only in [ c 1 , 1] . Also observe that since g ′ MID ( x ) is monotonically increasing, ˜ x MID ( λ ) is monotonically increasing in λ .

Minimisation of h ( r i , b i ) over b i | (1 -1 /r i ) | = 1 .

On substituting, we write the function

( a ) follows from the fact that g ( r ) + g (1 /r ) = 1 .

Since the second function has only a single minima at b i = 1 , it is sufficient to consider only the first function in the case r i ≥ 1 .

Observe that on optimizing over b i , we obtain two local minima one at b i = 0 and one at b i = ˜ x OUT ( λ ) ∈ ( c 2 , ∞ ) where ˜ x OUT ( λ ) satisfies the equations if λ ≥ 1 -µ g ′ OUT ( c 2 ) . Otherwise, we have a unique minima at b i = 0 . 4

Thus, we have ˜ x OUT ( λ ) > c 2 since g OUT is concave only in [ c 2 , ∞ ) . Also observe that since g ′ OUT ( x ) is monotonic, 536 ˜ x OUT ( λ ) is monotonic in λ . 537

513

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Since, this argument is true for every i ∈ [ n ] , we obtain 538

Since x ∗ MID is the local maximiser of g MID ( x ) x , we have 539

For the purpose of this analysis, we define two functions δ MID ( λ ) and δ OUT ( λ ) below.

540

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549

We also define

Recall that we aim to show opt ( E µ,α ) ≥ 0 when µ = µ ∗ and thus substitute µ = µ ∗ in every subsequent equation. Observe that it is sufficient to show δ MID ( λ ∗ ) and δ OUT ( λ ∗ ) are non-negative since this would imply that max λ ≥ 0 min ( r , b ) ∈C F ( r , b , λ ) is non-negative.

We now consider the following two exhaustive cases.

· Case 1: ˆ g MID -α n > ˆ g OUT + α n . Observe from Equation (15),

( d ) follows from the fact that both ˜ x MID ( λ ∗ ) and x ∗ MID exceed c 1 and g ′ MID ( x ) is monotonically decreasing for x ≥ c 1 .

( b ) follows from g ′ MID (˜ x MID ( λ ∗ )) = 1+ µ ∗ λ ∗ 5 as stated in Equation (15). 550

( c ) follows from ˜ x MID ( λ ∗ ) = x ∗ MID (in Equation (24)) and g MID ( x ∗ MID ) = x ∗ MID ˆ g MID (in Equation (19)) and the fact that 551 µ ∗ = λ ∗ α n . Now consider, 552

( d ) follows from the fact that x ∗ OUT is the local maximiser of g OUT ( x ) /x , 553

( e ) follows from the fact that ˆ g MID -α n > ˆ g OUT + α n in Case 1. 554

( g ) follows from the definition of λ ∗ and that µ = λ ∗ α n . 555

( h ) follows from the constraint in (17). 556

Similarly,

( i ) follows from the fact that g ′ OUT ( x ) is monotonically decreasing in x in [ c 2 , ∞ ) . 557

( j ) follows from g ′ OUT (˜ x OUT ( λ )) = 1 -µ λ as stated in Equation (17), and 558

( k ) follows from the following reasons: 559

-Observe that xg ′ OUT ( x ) -g OUT ( x ) is monotonically decreasing in [ c 2 , ∞ ) as g OUT is concave in this region. However, since x ∗ OUT ≥ ˜ x OUT ( λ ∗ ) ≥ c 2 , we have

-λ ∗ = µ ∗ n α follows from the definition of λ ∗ . 560

Thus, using (25) and (26) we show that for the chosen value of λ ∗ = µ ∗ n α , we have 561

min ( r , w ) ∈C F ( r , w , λ ∗ ) ≥ 0 implying from (13) that opt ( E µ,α ) ≥ 0 . 562

· Case 2: ˆ g MID -α n ≤ ˆ g OUT + α n 563

Choosing λ ∗ = µ ∗ n α , we can prove opt ( E µ,α ) ≥ 0 in a very similar manner whenever µ = µ ∗ . 564

C Proof of Theorem 1 565

Theorem (Restatement of Theorem 1) . For every ϵ > 0 and m ≥ 2 and n ≥ m 2 we have 566

Further, lim n →∞ DIST ( g ) ( PLU , n, m ) ≤ max( m ˆ g MID -1 , m ˆ g OUT +1) . 567

Proof. Recall that candidate B ∈ A minimises the social cost. The other candidates are denoted by { A j } j ∈ [ m -1] . 568

For every j ∈ [ m -1] , we now bound the probability of A j being the winner. This event implies that at least n m voters choose A j as the top preference, implying that the same voters rank A j over B . Further, we now define Bernoulli random variables { Y i,j } n i =1 each denoting the event that voter i ranks candidate A j over B . Recall from Equation 3, Y i,j ∼ Bern ( g ( d ( i,B ) d ( i,A j ) )) . Therefore,

Let α j be the expectation of the random variable ∑ n i =1 Y i,j i.e. the expected number of voters ranking A j over B . 573

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Now we use Chernoff bounds on the sum of Bernoulli random variable for every j ∈ [ m -1] when α j ≤ n m -n (1 / 2+ ϵ ) m 574 to bound the probability of A j being the winner. 575

( a ) follows from applying the Chernoff bound. We restate the bound from [60] below. 576

Suppose X 1 , X 2 , . . . , X n be independent Bernoulli random variables with P ( X i ) = µ i for every i ∈ [ n ] and µ := 577 ∑ n i =1 µ i , then we have 578

( c ) holds since xe -x is increasing in (0 , 1) and because α n/m -1 ≤ 1 and α ≤ n m -n ( 1 2 + ϵ ) m , the maxima is attained at 579 α = n m -n ( 1 2 + ϵ ) m . ( d ) holds since log(1 + x ) ≤ 2 x 2+ x for -1 < x ≤ 0 . 580

Let S := { j ∈ [ m -1] : α j < n m -n (1 / 2+ ϵ ) m } i.e. S denotes the indices of candidates with α j less than n m -n (1 / 2+ ϵ ) m . 581 Now using Lemma 2 and α j ≥ n m -n (1 / 2+ ϵ ) m for every j ∈ [ m -1] \ S , we have 582

We now have 583

DIST ( g ) ( PLU , n, m )

( a ) follows from the following observations. 584

· Apply Lemma 2 to bound SC ( A j ,d ) SC ( B,d ) . Since α j ≤ n m -n (1 / 2+ ϵ ) m ∀ j ∈ S , apply Equation (31) to bound /a80 [ A j wins ] . 585 586

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( b ) follows from the fact that | S | ≤ m -1 , max( a, b ) ≤ a + b , and applying Equation (40).

D Proof of Theorem 2

Theorem (Restatement of Theorem 2) . For every m ≥ 2 , lim n →∞ DIST ( g ) ( PLU , n, m ) ≥ max( m ˆ g MID -1 , m ˆ g OUT +1) .

Proof. The proof is by an example in an Euclidean metric space in R 3 . One candidate 'C" is at (1 , 0 , 0) . The other m -1 candidates are 'good" and are equidistantly placed on a circle of radius ϵ on the y -z plane centred at (0 , 0 , 0) . We call them G := { G 1 , G 2 , . . . , G m -1 } .

We present two constructions below for every ϵ, ζ > 0 .

Construction 1 : Let q MID := g ( ( x ∗ MID ) 2 + ϵ 2 1 -x ∗ MID ) and a MID := 1 m -1 ( 1 -1+ ζ mq MID ) . Each of the m -1 candidates in G has ⌊ a MID n ⌋ voters overlapping with it. The remaining voters (we call them 'ambivalent') are placed at ( x ∗ MID , 0 , 0) . Clearly, each voter overlapping with a candidate votes for it as the most preferred candidate with probability one. Each of the ambivalent voters votes as follows.

- With probability q MID , vote for candidate C as the top choice and uniformly randomly permute the other candidates in the rest of the vote.

- With probability 1 -q MID , vote for candidate C as the last choice and uniformly randomly permute the other candidates in the rest of the vote.

Observe that this satisfies the pairwise probability criterion in Equation 3. Since lim n →∞ ⌊ an ⌋ /n = a and that the distance of a candidate in G from any non-ambivalent voter is at most 2 ϵ , we have that for every j ∈ [ m -1] ,

√

Clearly every candidate in G minimises the social cost and now we show that lim n →∞ P [ C wins ] = 1 . 605

Let Bernoulli random variables { Y i } n i =1 denote the events that voter i ∈ N ranks candidate C at the top. Here, ∑ n i =1 P [ Y i = 1] = q MID ( n -( m -1) ⌊ a MID n ⌋ ) and thus

By the law of large numbers, we have that /a80 [ ∑ i Y i ≥ n m ] = 1 as n →∞ . Since every candidate in G is equally likely 606 to win, the event ∑ i Y i ≥ n m implies the event that C is the winner and thus, lim n →∞ P [ C wins ] = 1 . Thus, 607

Construction 2: Let q OUT := g ( √ ( x ∗ OUT ) 2 + ϵ 2 1+ x ∗ OUT ) and a OUT := 1 m -1 ( 1 -1+ ζ mq OUT ) . Each candidate in G has ⌊ a OUT n ⌋ voters 608 overlapping with it, and the remaining 'ambivalent" voters are at ( -x ∗ , 0 , 0) . 609

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OUT

Clearly, each voter overlapping with a candidate votes for it as the most preferred candidate with probability one. Each of the ambivalent voters votes as follows.

With probability q OUT , vote for candidate C as the top choice and uniformly randomly permute the other candidates in 612 the rest of the vote. 613

614

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With probability 1 -q OUT , vote for candidate C as the last choice and uniformly randomly permute the other candidates in the rest of the vote.

This satisfies the pairwise probability criterion in Equation 3. For every j ∈ [ m -1] , 616

Clearly, every candidate in G minimises the social cost. Now, we show that lim n →∞ P [ C wins ] = 1 . 617

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LEt Bernoulli random variables { Y i } n i =1 denote the events that voter i ∈ N ranks candidate C at the top. We have ∑ n i =1 P [ Y i = 1] = q MID ( n -( m -1) ⌊ an ⌋ ) and thus, lim n →∞ ∑ n i =1 P [ Y i =1] n = 1+ ζ m . Applying the law of large numbers, we get that /a80 [ ∑ i Y i ≥ n m ] = 1 as n tends to ∞ . However since every candidate in G is equally likely to win, the event ∑ i Y i ≥ n m corresponds to the event that C is the winner and thus, lim n →∞ P [ C wins ] = 1 . Therefore we have,

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On applying the limit ϵ, ζ → 0 and substituting for q MID and q OUT , we get the desired lower bound by combining the results from the two constructions.

E Proof of Theorem 3

Theorem 7. Restatement of Theorem 3 For every ϵ > 0 , m ≥ 2 and n ≥ 4 , we have 625

For every m ≥ 2 , we have lim n →∞ DIST ( g ) ( COP , n, m ) ≤ max ( (2ˆ g MID -1) 2 , (2ˆ g OUT +1) 2 ) . 626

Proof. Recall that B ∈ A minimises the social cost, and { A j } j ∈ [ m -1] denotes the set A\ B . 627

Consider a Copeland winner W . As noted by prior work [1], W must be in the uncovered set of the tournament graph, 628 and one of the following two cases must be true. 629

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· W defeats B .

There exists a candidate Y ∈ A s.t. W defeats Y and Y defeats B .

For every j ∈ [ m -1] , we now bound the probability of A j being the winner. For every j ∈ [ m -1] , we define Bernoulli random variables { Y i,j } n i =1 denoting the event that voter i ranks candidate A j over candidate B . From Equation 3, we have that Y i,j ∼ Bern ( g ( d ( i,A j ) d ( i,B ) )) . For every distinct j, k ∈ [ m -1] , we define Bernoulli random variables { Z i,j,k } n i =1 denoting the event that voter i ranks candidate A j over A k . Z i,j,k ∼ Bern ( g ( d ( i,A k ) d ( i,A j ) ) ) .

Observe that 636

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Let α j denote the expected value of the random variable ∑ n i =1 Y i,j , i.e., the expected number of voters who rank candidate A j over B .

Let β j,k denote the expected value of the random variable ∑ n i =1 Z i,j,k , i.e., the expected number of voters who rank candidate A j over A k .

Similar to Equation (31), we have the following bound: 641

From Equation (31) in the proof of Theorem 1, we have 642

643

Consider two exhaustive cases on candidate A j and define an event E j for every j ∈ [ m -1] . We compute the expected 644 fraction of votes on pairwise comparisons. The event E j denotes the existence of an at-most two hop directed path 645 from a candidate A j to candidate B for Copeland such that the expected fraction of votes on all edges along that path 646 exceed n 2 -n (1 / 2+ ϵ ) 2 . Recall that we only considered one hop path for the case of PLU in the proof of Theorem 1. 647

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If E j holds true, we can directly upper bound the ratio of the social cost of candidate A j to the social cost of candidate B using Lemma 2, which in turn provides a bound on the distortion. If E j does not hold, we apply the union bound and Chernoff's bound to upper bound the probability of A j being the winner. By multiplying this probability bound with the ratio of social costs obtained from Lemma 2, we derive a bound on the distortion.

Define S := { j ∈ [ m -1] : E j is not true } . Furthermore, we define K 1 ( j ) := { j ∈ [ m -1] : α k ≥ β j,k } and K 2 ( j ) := { j ∈ [ m -1] : α k < β j,k } denotes complement of K 1 ( j ) for every j ∈ [ m ] .

From Equations (58) and (59), both of the following conditions 1 and 2 are satisfied for every j ∈ S .

2. For every k ∈ [ m -1] \ { j } , 656

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Furthermore, we define γ j := max ( max k ∈ [ m -1] \{ j } (min( α k , β j,k )) , α j ) .

Since, for every Copeland winner W , it must either defeat B or there exists a Y ∈ A s.t. W defeats Y and Y defeats B . Using union bound for every j ∈ S , we have

The last inequality follows from the definition of γ j . 662

Furthermore from Lemma 2 and the definition of γ j , 6 we have 663

Using Equation (62) and (61) and applying max( a, b ) ≤ a + b , we have 664

6 This follows on splitting SC ( A j ,d ) SC ( B,d ) = SC ( A j ,d ) SC ( A k ,d ) × SC ( A k ,d ) SC ( B,d ) and applying the lemma separately. We further use the fact that 1 γ = min ( min k ∈ [ m -1] \{ j } ( max( 1 α k , 1 β j,k ) ) , 1 α j )

Recall that for every j ∈ [ m -1] \ S , E j is satisfied. Let us further denote 665

Observe that E j being satisfied implies either a) ˆ E j is satisfied or b) ∃ k ∈ [ m -1] \ { j } s.t ˆ E k and ˆ D j,k are satisfied. 666 We consider both cases separately. 667

Suppose ˆ E j is satisfied for some j ∈ [ m -1] \ S . Then we have from Lemma 2, 668

Now we consider case (b) where ˆ E k and ˆ D j,k are both satisfied for some k ∈ [ m -1] \ { j } . From Lemma 2 we have, 669

Now combining Equations (63), (64), and (65), we have for any metric space d ∈ M ( N ∪ A ) , 670

( a ) follows from Equation (61) and the fact that ∑ j ∈ S ( P [ A j wins ] SC ( A j ,d ) SC ( B,d ) ) + P [ B wins ] ≤ max ( max j ∈ S SC ( A j ,d ) SC ( B,d ) , 1 ) . 671 ( b ) follows from combining Equations (63), (64), and (65). 672

F Proof of Theorem 4 673

Theorem (Restatement of Theorem 4) . DIST ( g ) ( RD , m, n ) ≤ ( m -1)ˆ g MID +1 . 674

Proof. The probability of voter i voting for candidate W as its top candidate is upper bounded by g ( d ( i,B ) d ( i,W ) ) which is 675 the probability that W is ranked over B . Therefore, under RD, the probability of W winning satisfies: 676

Recall that we define the set of candidates in A\ B as { A 1 , A 2 , . . . , A m -1 } . In the rest of the analysis we denote 677 d ( i, A j ) by y i,j (for all j ∈ [ m -1] ) and d ( i, B ) by b i for every i ∈ [ n ] . We also denote d ( B,A j ) by z j for every 678

j ∈ [ m -1] . Now for every metric d , we bound the distortion as follows. 679

( a ) follows from Equation (66). 680

( d ) follows from the fact that y i,j -b i ≤ z j which follows from triangle inequality. 681

( e ) follows from the following arguments by considering two cases namely b i z j ≤ 1 and b i z j ≥ 1 . 682

When b i z j ≤ 1 and thus, y i,j z j ≥ 1 -b i z j from triangle inequality. Similarly, we have y i,j z j ≥ b i z j -1 when b i z j ≥ 1 . Thus, 683

The last inequality follows from the fact that g ( x x -1 ) x ≤ 1 when x ≥ 1 . Further, we have ˆ g MID ≥ 1 for all valid g . 684

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G Proof of Theorem 6

Theorem (Restatement of Theorem 6) . Let DIST θ PL ( RD , m, n ) denote the distortion when the voters' rankings are generated per the PL model with parameter θ . We have lim n →∞ DIST θ PL ( RD , m, n ) ≥ 1 + ( m -1) 1 /θ 2 .

Proof. We have a 1-D Euclidean construction. Let B be at 0 and all other candidates A\{ B } be at 1. m -1 voters are at 0 , and one voter is at t. We will set t later by optimizing for the distortion.

The distortion for this instance is P [ B wins ] · 1 + P [ B loses ] · n -t t = n -1 n + 1 n t -θ t -θ +( m -1)(1 -t ) -θ + 1 n ( m -1)(1 -t ) -θ t -θ +( m -1)(1 -t ) -θ n -t t . We drop the terms which are O (1 /n ) to obtain 1 + ( m -1)(1 -t ) -θ t ( t -θ +( m -1)(1 -t ) -θ ) . This simplifies to 1 + ( m -1) t θ -1 (1 -t ) θ +( m -1) t θ . This is lower bounded by 1 + ( m -1) t θ -1 1+( m -1) t θ . Setting t = ( m -1) -1 /θ , we obtain a distortion lower bound of 1 + ( m -1) 1 /θ 2 .

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nMetric distortion is a framework to evaluate the \"accuracy\" of social choice rules, by considering a worst-case candidate and voter embedding in a metric space, and by assuming that reported votes are derived from distances in the metric space. So far, votes were assumed to be a deterministic function of the distances. The paper investigates the case where they are probabilistic functions of the distances, in the asymptotic limit of a large number of voters. The key finding is that this inverts the evaluation of some voting rules, in particular Copeland and Random Dictatorship, for highly noisy voting.\n\nSTRENGTHS:\nThe model introduced by the authors is an insightful generalization of previous work which, remarkably, provides a markedly different view on social choice rules. Given the growing importance of social choice in machine learning, as well as accounting for noisy inputs while considering embedded vector spaces, I believe that this work scores high in significance.\n\nAdditionally, the analysis is quite thorough, with matching lower/upper asymptotic bounds for Plurality upper bound for Copeland and lower/upper bounds for Random Dictatorship.\n\nThe paper is also fairly well written.\n\nWEAKNESSES:\nMy main concern is Lemma 3. The proof seems to argue that the constraints $\\forall i, | b_i - w_i | \\leq b_i + w_i$ (i.e. inequality constraint on $(i, W, B)$), but the optimization problem (7) has a constraint $\\max_i | w_i - b_i | \\leq \\min_i b_i + w_i$. It is not clear to me why these constraints would be equivalent. Note that the latter implies the former set of constraints. Thus if $\\mathcal E_{\\alpha}'$ was defined with all voter-wise constraints, then it would be a minimum over a smaller set, and thus $opt(\\mathcal E_\\alpha') \\geq opt(\\mathcal E_\\alpha)$. Since Lemma 3's proof actually says $\\frac{SC(W, d)}{SC(B, d)} \\leq 1 / opt(\\mathcal E'_\\alpha)$, using this inequality seems to imply the actual Lemma 3. Am I reading this correctly?\n\nIt is disappointing that the upper-bound for Copeland. It would be helpful if the authors can point out where the argument gets loose.\n\nQUESTIONS:\nCould the authors clarify the proof of Lemma 3? I will happily increase my score given a convincing response.\n\nCan the authors provide insight into why the Copeland upper bound is not tight?\n\nMore anecdotically, I wonder why \"Metric Distortion\" is restricted to the distortion of ranking-based algorithms. I feel that the concept is more general than that. Essentially, it is the worst social cost ratio of any voting rule, when executed on votes derived from distances in the metric space, right? In particular, it could be studied even for the social-cost minimizing geometric median, but given, e.g., noisy inputs or strategic votes? I believe that this is an interesting research venue, given that machine learning provides many tools to embed candidate features and voters' preferences in a vector space.\n\nLIMITATIONS:\nThe paper stated its results very clearly and factually. I have no concerns about unaddressed limitations.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper extends the framework of metric distortion, measuring how well voting rules minimize the social cost in a given metric space, to probabilistic voting scenarios where the preferences of voters are drawn from a probability distribution defined by the relative distances between candidates and each voter's ground truth position in the metric space.\n\nThe authors base their analysis on three different axioms that the induced marginal probabilities of relative preferences must verify, namely *scale-freeness*, *independence of other candidates* and *strict monotonicity.* They define a general class of marginal probabilities that verify the three axioms, and show that it encompasses the widely used *Plackett-Luce* model.\n\nThey then provide upper and lower bounds for the distortion of the *Plurality* rule, both linear in the number of candidates and matching asymptotically when the number of voters grows to infinity.\n\nThey then provide a upper-bound for the distortion of *Copeland* rule and show that it is independent of the number of candidates in the limit of a large number of voters.\n\nMoreover, they give upper and (non-matching) lower-bounds for the distortion of *Random Dictator.*\n\nThey finally compare their results under both the *Plackett-Luce* and *Pairwise Quantal Voting* models (the latter being inspired form Quantal Response Theory), and show that the classical bounds of the metric distortion literature are recovered in the limit of vanishing randomness (although not for Copeland rule, hinting at a loose analysis).\n\nSTRENGTHS:\nThis paper, in extending the metric distortion analysis to probabilistic voting models perhaps closer to reality, is rather original and proposes a more optimistic view of metric distortion where randomized dictator is beatable in a worst case distortion sense.\nThe paper is fairly well written and the proofs seem correct.\n\nIt is an interesting idea, with an interesting result.\n\nWEAKNESSES:\nThe main problem of this paper is that it is not a good match to NeurIPS. This paper would work very well at an algorithms conference like SODA or ICALP, or a CS econ conference like EC (okay, probably one tier down like WINE), or possibly even at the those AI conferences that have a history in social choice theory like AAAI or IJCAI. And I also understand that NeurIPS has accepted such papers in the past. However, is it really a good fit for NeurIPS 2024? \n\n- There is no mention of the proof of Lemma 1 being in Appendix A.\n- In the proof of Theorem 2, $\\zeta$ is hardly introduced (also not in the Appendix).\n- Formatting may be improved in place: e.g. Equations 6, 7, 10, or Theorem 3.\n- l.312 \"converges to 9 instead of 5\". This part is not very clear, reminding the general bound in the deterministic case would improve readability.\n- Typos:\n\t- l.220 \"and is by solving\".\n\t- Equation 24 showcases $(d)$ instead of $(a)$.\n\t- Equation 34: $\\geq$ should be $=$.\n\t- l.618 \"LEt\".\n\t- l.641 should probably be deleted (equivalent to l.642).\n\t- l.660 weird grammar.\n\t- Footnote 6: missing index $\\gamma_j$.\n\t- l.670: missing $(\\hat{g}_{MID} +\\hat{g}_{OUT}  )^2$ in inequalities $(a)$ and $(b).$ Furthermore, Equation 63 is used in $(a)$ rather than in $(b)$.\n\nQUESTIONS:\nn/a\n\nLIMITATIONS:\n- The existence of distributions on rankings that generate pairwise order marginals of the form described in the paper is assumed and left for future work.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the problem of metric distortion in single-winner elections. The key assumption is that the voters' preferences are not exactly compatible with the metric space, but they rather agree with it with a certain probability. The authors propose several axioms that formalize the requirements for the probability distribution for it to make sense in the context of distortion. Then they provide upper bounds of distortion in the probabilitic setting for Plurality, Random Dictator and Copeland (in case of the first two rules, they provide also lower bounds).\n\nSTRENGTHS:\nThis work is the first one to combine probabilistic voting with metric distortion, hence the novelty is clear. The paper is overall of good quality, the axioms proposed in their work make sense to me, and the results are sound. The research direction introduced in this paper can be continued in further follow-up papers.\n\nWEAKNESSES:\nThe paper could have been more clearly written --- for example, the formal notation should be introduced at the beginning of Section 2   (together the model) rather than in the middle of the introduction.\n\nBesides, I think that Axiom 2 (Independence of Other Candidates) could have been better motivated. I can imagine that it was crucial to obtain the authors' results, but it seems rather natural to me that  in the real-life scenarios that motivated the research, the presence of additional candidates can impact the voter's probability of ranking one candidate over another.\n\nAnother weakness is that the authors only consider three rules, and the analysis of only two of them is complete --- many important rules (like Borda or PluralityVeto) are not considered at all. This could raise a question whether this amount of technical contribution is enough for a top conference like NeurIPS.\n\nQUESTIONS:\n-\n\nLIMITATIONS:\nThe authors adequately addressed the limitations and there are no potential negative societal impact of their work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-14"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers metric distortion in probabilistic models of voting. In the metric distortion framework the voters and alternatives are embedded in a metric space, and given the ranked preferences the goal is to find an alternative with low distortion. In this setting these rankings come from a probabilistic model.\n\nIn the first part of the paper, there axioms are introduced and authors show which axiom is satisfied by which probabilisitic model. In the second half of the paper the goal is to find the distortion of Plurality and Copeland rules for a specific class of probabilistic models. The results show matching upper and lower bound of $m$ for plurality and constant upper bound for Copeland.\n\nSTRENGTHS:\nDefining a model for distortion in stochastic models is a useful idea in future analysis of voting systems.\nThe idea of the paper is novel and it uses novel techniques in the second half.\nI like the definition of the three axioms. I find them natural and easy to understand.\n\nWEAKNESSES:\nMy main concern is about the presentation. The preliminaries section is incomplete. The definition of distortion is hard to understand for a general audience and you made it harder by just putting the formula there. You have to add a description in words and give some intuition on why this definition makes sense. \n\nIt's not clear how the probabilistic model works and how you define distortion on it until section 3 where you define it for a specific class. You have to formally define your probabilistic approach in Section 1.1 and also define distortion in this model. Not knowing the exact definition makes following the first paragraph of section 2 really hard. Before reading the rest of the paper I didn't understand why $P$ is a function of $d$ or why the preferences may not be consistent with the distances.\n\nI think you have to add more intuition on the probabilisitic models. For instance you mention ground-truth in the definition of Mallows model but you have to explain in more details that how this model distributes the probabilities based on the distance to this ground-truth. The same explanations are needed for PQV.\n\nMy understanding is that the analysis that you provide works for any member of $G$, but currently the only members for which we have the final bound are PL and PQV. Is that true? If so is there any other interesting member of this class?\n\nQUESTIONS:\nSee above.\n\nLIMITATIONS:\n-", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-15"}