On the Distortion of Voting with Multiple Representative Candidates
aa r X i v : . [ c s . G T ] N ov On the Distortion of Votingwith Multiple Representative Candidates
Yu ChengDuke University Shaddin DughmiUniversity of Southern CaliforniaDavid KempeUniversity of Southern California
Abstract
We study positional voting rules when candidates and voters are embedded in a commonmetric space, and cardinal preferences are naturally given by distances in the metric space. Ina positional voting rule, each candidate receives a score from each ballot based on the ballot’srank order; the candidate with the highest total score wins the election. The cost of a candidateis his sum of distances to all voters, and the distortion of an election is the ratio between thecost of the elected candidate and the cost of the optimum candidate. We consider the case whencandidates are representative of the population, in the sense that they are drawn i.i.d. from thepopulation of the voters, and analyze the expected distortion of positional voting rules.Our main result is a clean and tight characterization of positional voting rules that haveconstant expected distortion (independent of the number of candidates and the metric space).Our characterization result immediately implies constant expected distortion for Borda Countand elections in which each voter approves a constant fraction of all candidates. On the otherhand, we obtain super-constant expected distortion for Plurality, Veto, and approving a con-stant number of candidates. These results contrast with previous results on voting with metricpreferences: When the candidates are chosen adversarially, all of the preceding voting ruleshave distortion linear in the number of candidates or voters. Thus, the model of representativecandidates allows us to distinguish voting rules which seem equally bad in the worst case.
In light of the classic impossibility results for axiomatic approaches to social choice [4] and voting[18, 28], a fruitful approach has been to treat voting as an implicit optimization problem of findingthe “best” candidate for the population in aggregate [9, 11, 24, 25]. Using this approach, votingsystems can be compared based on how much they distort the outcome, in the sense of leadingto the election of suboptimal candidates. A particularly natural optimization objective is the sum of distances between voters and the chosen candidate in a suitable metric space [1, 2, 3, 19].The underlying assumption is that the closer a candidate is to a voter, the more similar theirpositions on key questions are. Because proximity implies that the voter would benefit from thecandidate’s election, voters will rank candidates by increasing distance, a model known as single-peaked preferences [7, 15, 8, 23, 22, 6, 27, 5].Even in the absence of strategic voting, voting systems can lead to high distortion in this setting,1ecause they typically allow only for communication of ordinal preferences , i.e., rankings of candi-dates [10]. In a beautiful piece of recent work, Anshelevich et al. [2] showed that this approach candraw very clear distinctions between voting systems: some voting systems (in particular, Copelandand related systems) have distortion bounded by a small constant, while most others (includingPlurality, Veto, k -approval, and Borda Count) have unbounded distortion, growing linearly in thenumber of voters or candidates.The examples giving bad distortion typically have the property that the candidates are not“representative” of the voters. Anshelevich et al. [2] show more positive results when there areno near-ties for first place in any voter’s ranking. Cheng et al. [12] propose instead a model ofrepresentativeness in which the candidates are drawn randomly from the population of voters;under this model, they show smaller constant distortion bounds than the worst-case bounds formajority voting with n = 2 candidates. Cheng et al. [12] left as an open question the analysis ofthe distortion of voting systems for n ≥ positional voting systems with n ≥ position in the voter’s ordering. The mapfrom positions to scores is known as the scoring rule of the voting system, and for n candidates is afunction g n : { , . . . , n − } → R ≥ . The total score of a candidate is the sum of scores he obtainsfrom all voters, and the winner is the candidate with maximum total score. The most well-knownexplicitly positional voting system is Borda Count [13], in which g n ( i ) = n − i for all i . Many othersystems are naturally cast in this framework, including Plurality (in which voters give 1 point totheir first choice only) and Veto (in which voters give 1 point to all but their last choice).In analyzing positional voting systems, we assume that voters are not strategic, i.e., they reporttheir true ranking of candidates based on proximity in the metric space. This is in keeping withthe line of work on analyzing the distortion of social choice functions, and avoids issues of game-theoretic modeling and equilibrium existence or selection (see, e.g., [16]) which are not our focus.As our main contribution, we characterize when a positional voting system is guaranteed tohave constant distortion, regardless of the underlying metric space of voters and candidates, andregardless of the number n of candidates that are drawn from the voter distribution. The char-acterization relies almost entirely on the “limit voting system.” By normalizing both the scoresand the candidate index to lie in [0 ,
1] (we associate the i th out of n candidates with his quantile in − ∈ [0 , g of the scoring functions g n as n → ∞ .Our main result (Corollary 3.2 in Section 3) states the following: (1) If g is not constant onthe open interval (0 , g is a constant otherthan 1 on the open interval (0 , g ≡ , g n to g matters,and a precise characterization is given by Theorem 3.1.As direct applications of our main result, we obtain that Borda Count and k -approval for k = Θ( n ) representative candidates have constant distortion; on the other hand, Plurality, Veto,the Nauru Dowdall method (see Section 2), and k -approval for k = O (1) have super-constant Of course, it is also highly questionable that voters would be able to quantify distances in a metric spacesufficiently accurately, in particular given that the metric space is primarily a modeling tool rather than an actualconcrete object. For consistency, we always use male pronouns for candidates and female pronouns for voters. O (1)-approval, even with representative candidates, is Ω( n ).Our results provide interesting contrasts to the results of Anshelevich et al. [2]. Under adver-sarial candidates, all of the above-mentioned voting rules have distortion Ω( n ); the focus on repre-sentative candidates allowed us to distinguish the performance of Borda Count and Θ( n )-approvalfrom that of the other voting systems. Thus, an analysis in terms of representative candidatesallows us to draw distinctions between voting systems which in a worst-case setting seem to beequally bad.As a by-product of the proof of our main theorem, in Lemma 3.3, we show that every votingsystem (positional or otherwise) has distortion O ( n ) with representative candidates. Combinedwith the lower bound alluded to above, this exactly pins down the distortion of Plurality, Veto, and O (1)-approval with representative candidates to Θ( n ). For Veto, this result also contrasts with theworst-case bound of Anshelevich et al. [2], which showed that the distortion can grow unboundedlyeven for n = 3 candidates. The voters/candidates are embedded in a closed metric space (Ω , d ), where d ω,ω ′ is the distancebetween points ω, ω ′ ∈ Ω. The distance captures the dissimilarity in opinions between voters (andcandidates) — the closer two voters or candidates are, the more similar they are. The distribution ofvoters in Ω is denoted by the (measurable) density function q ω . We allow for q to have point masses. Unless there is no risk of confusion, we will be careful to distinguish between a location ω ∈ Ω and aspecific voter j or candidate i at that location. We apply d equally to locations/voters/candidates.We frequently use the standard notion of a ball B ( ω, r ) := { ω ′ | d ω,ω ′ ≤ r } in a metric space.For balls (and other sets) B , we write q B := R ω ∈ B q ω d ω .An election is run between n ≥ n candidates are assumed to be representative of the population, in the sense that their locations aredrawn i.i.d. from the distribution q of voters.Each voter ranks the n candidates i by non-decreasing distance from herself in (Ω , d ). Ties arebroken arbitrarily, but consistently , meaning that all voters at the same location have the sameranking. We denote the ranking of a voter j or a location ω over candidates i by π j ( i ) or π ω ( i ).The distance-based ranking assumption means that π ω ( i ) < π ω ( i ′ ) implies that d ω,i ≤ d ω,i ′ and d ω,i < d ω,i ′ implies that π ω ( i ) < π ω ( i ′ ). As mentioned in the introduction, we assume that votersare not strategic; i.e., they express their true ranking of candidates based on proximity in the metricspace. Since the continuum model allows for point masses, it subsumes finite sets of voters. Changing all our results tofinite or countable voter sets is merely cosmetic. Our results do not depend on specific tie breaking rules. .2 Social Cost and Distortion Candidates are “better” if they are closer to voters on average. The social cost of a candidate (orlocation) i is c i = Z ω d ω,i q ω d ω. The socially optimal candidate among the set C of candidates running is denoted by o ( C ) :=argmin i ∈ C c i . The overall optimal location is denoted by b o ∈ argmin ω ∈ Ω c ω , which is any 1-medianof the metric space. (If there are multiple optimal locations, consider one of them fixed arbitrarily.)The argmin always exists, because the metric space is assumed to be closed, and the cost function iscontinuous and bounded below by 0. Note that it is not necessary that there be any voters locatedat b o .Based on the votes, a voting system will determine a winner w ( C ) for the set C of candidates,who will often be different from o ( C ). The distortion measures how much worse the winner is thanthe optimum D ( C ) = c w ( C ) c o ( C ) . We are interested in the expected distortion of positional voting systems under i.i.d. random can-didates, i.e., E C i.i.d. ∼ q [ D ( C )] . Our distortion bounds are achieved by lower-bounding c o ( C ) ≥ c b o . A particularly useful quantityin this context is the fraction of voters outside a ball of radius r around b o , which we denote by H ( r ) := 1 − q B ( b o,r ) . The following lemma captures some useful simple facts that we use: Lemma 2.1.
1. For any candidate or location i , c i ≤ c b o + d i, b o . (1)
2. The cost of any candidate or location i can be written as c i = Z ∞ (1 − q B ( i,r ) )d r. (2)
3. For all r ≥ , the cost of the optimum location b o is lower-bounded by c b o ≥ rH ( r ) . (3) Proof.
1. The proof of the first inequality simply applies the triangle inequality under theintegral: c i = Z ω d ω,i q ω d ω ≤ Z ω ( d ω, b o + d i, b o ) q ω d ω = c b o + d i, b o .
2. For the second equation, observe that c i = E ω ∼ q [ d i,ω ], and the expectation of any non-negative random variable X can be rewritten as E [ X ] = R ∞ Pr[ X ≥ x ]d x .4. For the third inequality, we apply the previous part with i = b o , and lower bound Z ∞ Pr ω ∼ q [ d b o,ω ≥ x ]d x = Z r Pr ω ∼ q [ d b o,ω ≥ x ]d x + Z ∞ r Pr ω ∼ q [ d b o,ω ≥ x ]d x ≥ Z r Pr ω ∼ q [ d b o,ω ≥ r ]d x + Z ∞ r x = r · H ( r ) . We are interested in positional voting systems . Such systems are based on scoring rules : votersgive a ranking of candidates, and with each position is associated a score.
Definition 2.1 (Scoring Rule) . A scoring rule for n candidates is a non-increasing function g n : { , . . . , n − } → [0 , with g n (0) = 1 and g n ( n −
1) = 0 . Definition 2.2 (Positional Voting System) . A positional voting system is a sequence of scoringrules g n , one for each number of candidates n = 1 , , . . . . The interpretation of g n is that if voter j puts a candidate i in position π j ( i ) on her ballot, then i obtains g n ( π j ( i )) points from j . The total score of candidate i is then σ ( i ) = Z ω g n ( π ω ( i )) q ω d ω. The winning candidate is one with highest total score, i.e., for a set C of n candidates, w ( C ) ∈ argmax i ∈ C σ ( i ); again, ties are broken arbitrarily, and our results do not depend on tie breaking.The restriction to monotone non-increasing scoring rules is standard when studying positionalvoting systems. One justification is that in any positional voting system violating this restriction,truth-telling is a dominated strategy, rendering such a system uninteresting for most practicalpurposes. Given this restriction, the assumption that g n (0) = 1 and g n ( n −
1) = 0 is without lossof generality, because a score-based rule is invariant under affine transformations.Next, we want to capture the notion that a positional voting system is “consistent” as we varythe number of candidates n . Intuitively, we want to exclude contrived voting systems such as “Ifthe number of candidates is even, then use Borda Count; otherwise use Plurality voting.” This iscaptured by the following definition. Definition 2.3 (Consistency) . Let V be a positional voting system with scoring rules { g n | n ∈ N } .We say that V is consistent if there exists a function g : Q ∩ [0 , → [0 , such that for eachrational quantile x ∈ [0 , and accuracy parameter ǫ > , there exists a threshold n such that g n ( ⌊ x ( n − ⌋ ) ≥ g ( x ) − ǫ and g n ( ⌈ x ( n − ⌉ ) ≤ g ( x ) + ǫ for all n ≥ n . We call g the limit scoringrule of V . Intuitively, this definition says that the sequence of scoring rules g n is consistent with a singlescoring rule g in the limit. Using the fact that g n is monotone non-increasing for each n , it canbe shown that g is also monotone non-increasing. We note that g n converges pointwise to g in aprecise and natural sense. Formally, when x ∈ [0 ,
1] is rational, there exists an infinite sequence ofintegers n with ⌊ x ( n − ⌋ = ⌈ x ( n − ⌉ = x ( n − g ( x ) must equal5he limit of g n ( x ( n − n . Therefore the limit scoring rule g isuniquely defined if it exists.All positional voting systems we are aware of are consistent according to Definition 2.3. Example 2.1.
To illustrate the notion of a consistent positional voting system, consider the fol-lowing examples, encompassing most well-known scoring rules. • In Plurality voting with n candidates, g n (0) = 1 and g n ( k ) = 0 for all k > . The limitscoring rule is g (0) = 1 and g ( x ) = 0 for all x > . • In Veto voting with n candidates, g n ( k ) = 1 for all k < n − and g n ( n −
1) = 0 . The limitscoring rule is g ( x ) = 1 for all x < and g (1) = 0 . • In k -approval voting with constant k , we have g n ( k ′ ) = 1 for k ′ ≤ min( k − , n − , and g n ( k ′ ) = 0 for all other k ′ . The limit scoring rule is g (0) = 1 and g ( x ) = 0 for all x > ,i.e., the same as for Plurality voting. (This relies on k being constant, or more generally, k = o ( n ) .) • In k -approval voting with linear k , there exists a constant γ ∈ (0 , with g n ( k ) = 1 for all k ≤ γn , and g n ( k ) = 0 for all larger k . The limit scoring rule is g ( x ) = 1 for x ≤ γ and g ( x ) = 0 for x > γ . • The Borda voting rule has g n ( k ) = 1 − kn − (after normalization). The limit scoring rule is g ( x ) = 1 − x . • The Dowdall method used in Nauru [17, 26] has g n ( k ) = 1 / ( k + 1) . After normalization, therule becomes g n ( k ) = n − · ( nk +1 − . The limit scoring rule is g (0) = 1 and g ( x ) = 0 for all x > , i.e., the same as for Plurality voting. This is because for every constant quantile x ,the score of the candidate at x is n − (cid:0) x − (cid:1) n →∞ → . In this section, we state and prove our main theorem, characterizing positional voting systems withconstant distortion.
Theorem 3.1.
Let V be a positional voting system with a sequence g n of scoring rules for n =1 , , . . . . Then, V has constant expected distortion if and only if there exist constants n and y ∈ (0 , such that for all n ≥ n , y · ⌈ y ( n − ⌉− X k =0 ( g n ( k ) − g n ( ⌈ y ( n − ⌉ )) > (1 − y ) · n − X k = n −⌈ y ( n − ⌉ (1 − g n ( k )) . (4)We prove Theorem 3.1 in Sections 3.1 (sufficiency) and 3.2 (necessity). Condition (4) is quiteunwieldy. In most cases of practical interest, we can use Corollary 3.2. Corollary 3.2.
Let V be a consistent positional voting system with limit scoring rule g .1. If g is not constant on the open interval (0 , , then V has constant expected distortion. . If g is equal to a constant other than 1 on the open interval (0 , , then V does not haveconstant expected distortion. Corollary 3.2 is proved in Section 4.The constant in Theorem 3.1 and Corollary 3.2 depends on V , but not on the metric spaceor the number of candidates. Corollary 3.2 has the advantage of determining constant expecteddistortion only based on the limit scoring rule g . The only case when it does not apply is when g ( x ) = 1 for all x ∈ [0 , V has constant distortion. Fortunately, Veto voting is the only rule of practicalimportance for which g ( x ) ≡ , k -approval with linear k = Θ( n ), and super-constantexpected distortion for Plurality, k -approval with k = o ( n ), and the Dowdall method.This leaves Veto voting, for which it is easy to apply Theorem 3.1 directly. Because g n ( k ) = 1for all k < n −
1, for any constant y < n , the left-hand side of (4) is 0, whilethe right-hand side is positive. Hence, (4) can never be satisfied for sufficiently large n , implyingsuper-constant expected distortion. The proof easily generalizes to show that when voters can veto o ( n ) candidates, the distortion is super-constant. In this section, we prove that condition (4) suffices for constant distortion. First, because of themonotonicity of g n , if (4) holds for y ∈ (0 , y ′ ∈ [ y, b r large enough so that the ball B ( b o, b r )around the socially optimal location b o contains a very large (but still constant) fraction y of allvoters, such that y satisfies (4). If the number of candidates n is large enough (a large constant),standard Chernoff bounds ensure that as r ≥ b r grows large, most candidates who are running willbe from inside B ( b o, r ). In turn, if many candidates inside B ( b o, r ) are running, all candidates outside B ( b o, r ) are very far down on almost everyone’s ballot, and therefore cannot win. In particular,Inequality (4) implies that the total score of an average candidate in B ( b o, r ) exceeds the maximumpossible total score of a candidate outside B ( b o, r ). This allows us to bound the expected distortionin terms of the cost of b o .The case of small n is much easier, since we can treat n as a constant. In that case, the followinglemma is sufficient. Lemma 3.3. If n candidates are drawn i.i.d. at random from q , the expected distortion is at most n + 1 . Proof.
The proof illustrates some of the key ideas that will be used later in the more technicalproof for a large number of candidates. We want to bound E C (cid:2) c w ( C ) (cid:3) (1) ≤ c b o + E C (cid:2) d b o,w ( C ) (cid:3) = c b o + Z ∞ Pr C [ d b o,w ( C ) ≥ r ]d r. In order for a candidate at distance at least r from b o to win, it is necessary that at least onesuch candidate be running. By a union bound over the n candidates, the probability of this event7s at most Pr C [ d b o,w ( C ) ≥ r ] ≤ nH ( r ), so E C (cid:2) c w ( C ) (cid:3) ≤ c b o + n Z ∞ H ( r )d r (2) = ( n + 1) c b o . Lower-bounding the cost of the optimum candidate from C in terms of the overall best location b o , the expected distortion is E C (cid:20) c w ( C ) c o ( C ) (cid:21) ≤ E C (cid:20) c w ( C ) c b o (cid:21) = 1 c b o E C (cid:2) c w ( C ) (cid:3) ≤ c b o · ( n + 1) c b o = n + 1 . In preparation for the case of large n , we begin with the following technical lemma, which showsthat whenever (4) holds, it will also hold when the terms on the left-hand side are “shifted,” andthe right-hand side can be increased by a factor of 2 (or, for that matter, any constant factor). Lemma 3.4.
Assume that there exist y ∈ (0 , and n such that (4) holds. Then, there exists z ∈ ( , such that for all n ≥ n , all z ≥ z , and all integers ≤ m ≤ (1 − z ) · n , z · ⌈ z · ( n − ⌉− X k =0 ( g n ( m + k ) − g n ( m + ⌈ z · ( n − ⌉ )) > − z ) · n − X k = n −⌈ z · ( n − ⌉ (1 − g n ( k )) . (5)We now flesh out the details of the construction. By Lemma 3.4, there exists z ∈ ( ,
1) and n such that (5) holds for all z ≥ z , all n ≥ n , and all integers 0 ≤ m ≤ (1 − z ) · n . For simplicityof notation, write z := z , and let µ := (1 − ) + · z ∈ ( z,
1) and b n := max( n , − z ). Notice that µ, z, b n only depend on V , but not on the metric space or number of voters.Let b r := inf { r | q B ( b o,r ) ≥ µ } , so that q B ( b o, b r ) ≥ µ , and q { ω | d b o,ω ≥ b r } ≥ − µ . (Both inequalitieshold with equality unless there is a discrete point mass at distance b r from b o .)Consider any r ≥ b r and write T := B ( b o, r ) and S := B ( b o, r ), as depicted in Figure 1. When n candidates are drawn i.i.d. from q , the expected fraction of candidates drawn from outside of S isexactly H ( r ) ≤ − µ . Let E r be the event that more than (1 − z ) n candidates are from outside S .Lemma 3.5 uses Chernoff bounds and the definitions of the parameters to show that E r happenswith sufficiently small probability; Lemma 3.6 then shows that unless E r happens, the distortion isconstant. Lemma 3.5.
Pr[ E r ] ≤ e1 − z · H ( r ) .Proof. By the Chernoff bound Pr[
Z > (1 + δ ) E [ Z ]] < (cid:16) e δ (1+ δ ) δ (cid:17) E [ Z ] , applied with E [ Z ] = H ( r ) · n and δ = − zH ( r ) − >
0, the probability of E r is at mostPr[ E r ] ≤ e − zH ( r ) − ( − zH ( r ) ) − zH ( r ) H ( r ) · n = e − z − H ( r ) ( − zH ( r ) ) − z ! n ≤ (cid:18) e · H ( r )1 − z (cid:19) (1 − z ) · n . Recall that µ := (1 − ) + · z ∈ ( z, r ≥ b r , we have that H ( r ) ≤ − µ = − z e ; inparticular, e · H ( r )1 − z ≤
1, so the probability can be upper-bounded by making the exponent (1 − z ) · n as small as possible. Because n ≥ b n ≥ − z , the exponent is lower-bounded by 1. Thus, we obtainthat the probability of E r is at most e1 − z · H ( r ). 8 b o rS T Figure 1: T = B ( b o, r ) and S = B ( b o, r ). Most of the voters are in S . Lemma 3.6 states thatwhenever most of the candidates are from T , the winner must come from T . The reason is that forany i / ∈ T , even an average candidate in S beats i ; in particular, the best candidate from S mustbeat i . Lemma 3.6.
Whenever E r does not happen, the winner of the election is from B ( b o, r ) . Proof.
Let Z := ⌈ z · ( n − ⌉ . Assume that exactly s ≥ Z out of the n candidates are drawnfrom S . Consider a candidate i / ∈ T . We will compare the average number of points of candidatesin S with the maximum possible number of points of candidate i , and show that the former exceedsthe latter. • Each voter j / ∈ S gives at most one point to i . On the other hand, even if j ranks all of S in thelast s positions, the total number of points assigned by j to S is at least P n − k = n − s g n ( k ). Thedifference between the number of votes to i and the average number of votes to candidates in S is thus at most 1 − s · n − X k = n − s g n ( k ) ! = 1 s · n − X k = n − s (1 − g n ( k )) . Because no more than a 1 − µ fraction of voters are strictly outside S , the total advantage of i over an average candidate in S resulting from such voters is at most∆ i := 1 s · (1 − µ ) · n − X k = n − s (1 − g n ( k )) . • Each voter j ∈ S will rank all candidates in S (who are at distance at most 2 r from her)ahead of all candidates outside T (who are at distance strictly more than 3 r − r = 2 r fromher).Let m ≥ j ranks i in position s + m . Then, i gets g n ( s + m ) points from j .Because j ranks all of S ahead of i , she gives at least P s − k =0 g n ( k + m ) points in total to S .Hence, the difference in the number of points that j gives to an average candidate in S andthe number of votes that j gives to i is at least (cid:16) s · s − X k =0 g n ( k + m ) (cid:17) − g n ( s + m ) = 1 s · s − X k =0 ( g n ( k + m ) − g n ( s + m )) . µ fraction of voters are in S , the total advantage of an average candidatein S resulting from voters in B is at least∆ S := 1 s · µ · s − X k =0 ( g n ( k + m ) − g n ( s + m )) . We show that ∆ S > ∆ i , using condition (5). Because g n is monotone non-increasing, andbecause s ≥ Z , we get that s − X k =0 ( g n ( k + m ) − g n ( s + m )) ≥ Z − X k =0 ( g n ( k + m ) − g n ( Z + m )) (5) > − z ) z · n − X k = n − Z (1 − g n ( k )) . Because z > and g n is monotone, we get that P n − k = n − Z (1 − g n ( k )) ≥ P n − k = n − s (1 − g n ( k )).Hence, ∆ S > s · µ · − zz · n − X k = n − s (1 − g n ( k )) µ ≥ z ≥ s · (1 − µ ) · n − X k = n − s (1 − g n ( k )) = ∆ i . We now wrap up the sufficiency portion of the proof of Theorem 3.1. We distinguish two cases,based on the number of candidates n . If n < b n , then Lemma 3.3 implies an upper bound of n + 1 ≤ b n ≤ max( n , − z ) = O (1) on the expected distortion. Now assume that n ≥ b n . Recallthat b r := inf { r | q B ( b o,r ) ≥ y } . By Lemmas 3.5 and 3.6, for any r ≥ b r , the probability that theelection’s winner is outside B ( b o, r ) is at most e1 − z · H ( r ). The rest of the proof is similar to thatof Lemma 3.3. We again use that E C (cid:2) c w ( C ) (cid:3) ≤ c b o + Z ∞ Pr C [ d b o,w ( C ) ≥ r ]d r, and bound Z ∞ Pr C [ d b o,w ( C ) ≥ r ]d r = Z b r Pr C [ d b o,w ( C ) ≥ r ]d r + Z ∞ b r Pr C [ d b o,w ( C ) ≥ r ]d r Lemmas 3.5, 3.6 ≤ Z b r r + Z ∞ b r e1 − z · H ( r )d r ≤ b r + e1 − z · Z ∞ H ( r )d r (2) = 3 b r + e1 − z · c b o . To upper-bound b r , recall that at least a 1 − µ fraction of voters are outside of B ( b o, b r ) or on theboundary. Therefore, by Inequality (3), c b o ≥ b r · (1 − µ ). Substituting this bound, the expected costof the winning candidate is at most (cid:18) − µ + e1 − z (cid:19) · c b o = (cid:18) − µ (cid:19) · c b o = O ( c b o ) , as y depends only on the voting system V , but not on the metric space or the number of candidates.This completes the proof of sufficiency. 10 .1.1 Proof of Lemma 3.4Proof of Lemma 3.4. Because condition (4) holds for all y ′ > y , we may assume that y ≥ .Define z := + y , and consider any z ≥ z . Fix n ≥ n , and write Y := ⌈ y ( n − ⌉ and Z := ⌈ z ( n − ⌉ . Let m ≤ (1 − z )( n −
1) be arbitrary. We define S := n − X k =0 (1 − g n ( k )) ,S := m − X k =0 ( g n ( k ) − g n ( Y )) ,S := Y − X k = m ( g n ( k ) − g n ( Y )) . By monotonicity of g n , Z − X k =0 ( g n ( m + k ) − g n ( m + Z )) ≥ S ;furthermore, P n − k = n − Z (1 − g n ( k )) ≤ S . Therefore, it suffices to show that S ≤ z − z ) S . Bycondition (4) and monotonicity of g n , and because y ≥ , S + S = Y − X k =0 ( g n ( k ) − g n ( Y )) ≥ − yy n − X k = n − Y (1 − g n ( k )) ≥ − y y S . To upper-bound S in terms of S , we show that the contribution of S to the preceding sumis small, and upper-bound S in terms of S + S . Because S ≤ (1 − z )( n − · (1 − g n ( Y )), usingthe monotonicity of g n , we can write S + S = Y − X k =0 (1 − g n ( k )) + S + n − X k = Y (1 − g n ( k )) ≥ Y − X k = m (1 − g n ( Y )) + n − X k = Y (1 − g n ( Y ))= ( n − m ) · (1 − g n ( Y )) ≥ z · ( n − · (1 − g n ( Y )) ≥ z − z · S . Combining the preceding inequalities, we now obtain that1 − y y · S ≤ − zz ( S + S ) + S = 1 z · S + 1 − zz · S . Solving for S , and using that the definition of z ensures 1 − z ≤ − y , we now bound S ≤ yz (1 − y ) − y (1 − z ) · S ≤ y − z ) · S ≤ z − z ) · S , completing the proof. 11 .2 Necessity Next, we prove that the condition in Theorem 3.1 is also necessary for constant distortion. Weassume that the condition (4) does not hold, i.e., for every y ∈ (0 ,
1) and n , there exists an n ≥ n such that y · ⌈ y · ( n − ⌉− X k =0 ( g n ( k ) − g n ( ⌈ y · ( n − ⌉ )) ≤ (1 − y ) · n − X k = n −⌈ y · ( n − ⌉ (1 − g n ( k )) . (6)We will show that the distortion of V is not bounded by any constant.The high-level idea of the construction is as follows: we define two tightly knit clusters A and B that are far away from each other. A contains a large α fraction of the population, and thusshould in an optimal solution be the one that the winner is chosen from. We will ensure that withprobability at least , the winner instead comes from B . Because B is far from A , most of thepopulation then is far from the chosen candidate, giving much worse cost than optimal.The metrics underlying A and B are as follows: B will essentially provide an “ordering,”meaning that whichever set of candidates is drawn from B , all voters in B (and essentially all in A ) agree on their ordering of the candidates. This will ensure that one candidate from B will geta sufficiently large fraction of first-place votes, and will be ranked highly enough by voters from A ,too. A will be based on a large number M of discrete locations ω . Their pairwise distances arechosen i.i.d.: as a result, the rankings of voters are uniformly random, and there is no consensusamong voters in A on which of their candidates they prefer. Because the vote is thus split, the bestcandidate from B will win instead.The following parameters (whose values are chosen with foresight) will be used to define themetric space. • Let c > c . • Let β ∈ (0 , ) solve the quadratic equation β +13 β · (1 − β ) = 2 c −
1. A solution exists becauseat β = , the left-hand side is < c −
1; it goes to infinity as β →
0, while the right-handside is a positive constant. β is the fraction of voters in the small cluster B . • Let α = 1 − β denote the fraction of voters in the large cluster A . • Let s = ββ be the distance between the clusters B and A . (Each cluster will have diameterat most 2.) • Let b α ≥ + α > α satisfy 4 b α · (1 − b α ) < α · (1 − α ); such an b α exists because the left-hand sidegoes to 0 as b α → b α < A . • Let n = β >
16; this is a lower bound on the number of candidates that ensures that theactual fraction of candidates drawn from A is at most b α with sufficiently high probability. • Let n ≥ n be the n whose existence is guaranteed by the assumption (6) (for y = b α and n ). • Let M = n ; this is the number of discrete locations ω we construct within the larger cluster A . 12e now formally define the metric space consisting of two clusters: Definition 3.1.
The metric space consists of two clusters A and B . A has M discrete locations,and q has a point mass of αM on each such location. The total probability mass on B is q B = 1 − α ,distributed uniformly over the interval [1 , . Locations in B are identified by x ∈ [1 , . Thedistances are defined as follows:1. For each distinct pair ω, ω ′ ∈ A , the distance d ω,ω ′ is drawn independently uniformly atrandom from [1 , .2. For each distinct pair x, x ′ ∈ B of locations, the distance is defined to be d x,x ′ := min( x, x ′ ) .3. Partition B = [1 , into M ! disjoint intervals I π of length /M ! each, one for each permu-tation of the M locations in A . For ω ∈ A and x ∈ I π , let π − ( ω ) be the position of ω in π ,and define the distance between ω and x to be d ω,x = s + x + π − ( ω ) M ! . Proposition 3.7.
Definition 3.1 defines a metric.Proof.
Non-negativity, symmetry, and indiscernibles hold by definition. Because all distances withinclusters are in [1 , ω, ω ′ ∈ A and all pairs x, x ′ ∈ B .Because d ω,x ∈ [ s, s + 1] for all ω ∈ A and x ∈ B , and distances within A or B are at least 1,there can be no shorter path than the direct one between any ω ∈ A and x ∈ B . Therefore, thetriangle inequality is satisfied.Now consider a (random) set C of n candidates, drawn i.i.d. from q . We are interested in theevent that the resulting slate of candidates is highly representative of the voters, in the followingsense. Definition 3.2.
Let C be the (random) set of n candidates drawn from q . Let E be defined as theconjunction of the following:1. For each location ω ∈ A , the set C contains at most one candidate from ω .2. At least a β fraction of candidates in C is drawn from B (and thus at most an b α fraction ofcandidates are from A ).3. At least an α fraction of candidates in C is drawn from A .4. No pair x, x ′ ∈ B ∩ C has | x − x ′ | < M − . Lemma 3.8 uses standard tail bounds to show that E happens with probability at least ; then,Lemma 3.9 shows that whenever E happens, the winner is from B . Lemma 3.8. E happens with probability at least .Proof. We upper-bound the probability of the complement of each of the four constituent sub-events.1. For each of the at most n pairs of candidates, the probability that they are both drawn fromthe same location is at most α/M ≤ /n . By a union bound over all pairs, the probabilitythat any location has at least two pairs is at most 1 /n .13. Let the random variable X be the number of candidates drawn from B . Then, E [ X ] = β · n ,and X is a sum of i.i.d. Bernoulli random variables. By the Hoeffding bound Pr[ X < ( β − ǫ ) n ] ≤ exp( − ǫ n ), with ǫ = β/
2, we obtain that the fraction of candidates from B is toosmall with probability at most exp( − β · n ) ≤ exp( − β · n ) = .3. The proof is essentially identical to the previous case (except because α ≥ β , the bounds areeven stronger), so this event happens with probability at least as well.4. Consider all intervals of [1 ,
2] of length M − , starting at 1+ k ( M − for some k = 0 , , . . . , ( M − −
2. If x, x ′ with | x − x ′ | ≤ M − existed, they would both be contained in at least onesuch interval (because the interval length is twice as long as the distance).For any of the ( M − − I , the probability that a specific pair of candidates is drawnfrom I is at most M − . By a union bound over all (at most n ) pairs of candidates andall intervals, the probability that any pair is drawn from any interval I is at most n ( M − ≤ n .Because n ≥
9, a union bound shows that E happens with probability at least . Lemma 3.9.
Whenever E happens, the winning candidate is from B .Proof. Let b be the actual number of candidates drawn from B , and a = n − b the number ofcandidates drawn from A . Because we assumed that E happened, b ≥ β · n and a ≤ b α · n . Let C A be the set of candidates drawn from A . Under E , C A contains at most one candidate from eachlocation ω ∈ A . As a result, because the random distances within A are distinct with probability1, there will be no ties in the rankings of any voters.Let b ı be the candidate from B with smallest value b x . With probability 1, the x value of b ı isunique. Consider some arbitrary candidate i ∈ C A from location ω ′ . We calculate the contributionsto b ı and i from voters in B and in A separately, and show that b ı beats i . Because this holds forarbitrary i , the candidate b ı or another candidate from B wins.1. We begin with points given out by voters in B . By definition of the distances within B , b ı isranked first by all voters in B .Voters in I π rank the candidates from A according to their order in π . For each orderingof C A , exactly a a ! fraction of permutations induces that ordering. In particular, for each k ∈ , . . . , a , exactly a 1 /a fraction of voters places i in position k + b . Thus, i obtains a totalof (1 − α ) · P n − k = n − a a · g n ( k ) points from voters in B . Overall, b ı obtains an advantage of atleast ∆ B = (1 − α ) · g n (0) − a · n − X k = n − a g n ( k ) ! = (1 − α ) · a · n − X k = n − a (1 − g n ( k )) .
2. Next, we analyze the number of points given out by voters in A . The distance from any voterlocation ω ∈ A to b ı is at most s + b x + nM ! . Under E , no other candidate from B can be at alocation x ≤ b x + M − ; therefore, the distance from any voter location ω ∈ A to any othercandidate x ∈ B is at least d ω,x ≥ s + x ≥ s + b x M − > s + b x nM ! ≥ d ω, b ı ,
14o all voters in A prefer b ı over any other candidate from B . Hence, b ı obtains at least α · g n ( a )points combined from voters in A .To analyze the votes from voters in A for candidates from A , we first notice that E and thedraw of candidates are independent of the distances within A . Hence, even conditioned on E , the distances d ω,ω ′ between locations in A are i.i.d. uniform from [1 , ω ∈ A ranks the candidates in C A in uniformly random order. Furthermore, for twolocations ω = ω ′ , the rankings of C A are independent; the reason is that they are based ondisjoint vectors of distances ( d ω,i ) i ∈ C A , ( d ω ′ ,i ) i ∈ C A . We use this independence to apply tailbounds. Let ω ′ be the location of i . Voters rank i as follows: • Among locations ω without a candidate of their own, in expectation, a 1 /a fraction ofvoters will rank i in position k , for each k = 0 , . . . , a − • Among the a − ω = ω ′ with a candidate of their own, in expectation, a1 / ( a −
1) fraction of voters will rank i in position k , for each k = 1 , . . . , a − • Voters at ω ′ will rank i in position 0.For each k , let the random variable X k be the number of locations that rank i in position k . By the preceding arguments, E [ X k ] = Ma , and X k is a sum of M independent (not i.i.d.)Bernoulli random variables. Hence, by the Hoeffding bound, the probability that more thana a fraction of voters rank i in position k is at most 2 exp( − · a · M ) ≤ − n ). By aunion bound over all candidates i ∈ C A and all values k = 0 , . . . , a −
1, with high probability,for all i and k , the fraction of voters (in A ) ranking i in position k is at most α · a . Becausethe total fraction of voters in A is α , any excess votes for some (early) positions k mustbe compensated by fewer votes for other (late) positions k ′ . Relaxing the constraint thatthe number of votes for each position k must be non-negative, we can upper-bound the totalpoints for i by assuming that each of the positions k = 0 , . . . , a − k = a − i over b ı from votes from A is at most∆ A := α · a − X k =0 a · g n ( k ) + 2 − aa · g n ( a − − g n ( a ) ! = αa · (cid:18) a − X k =0 (cid:0) g n ( k ) − g n ( a − − g n ( a ) (cid:1) + g n ( a − − g n ( a ) (cid:19) g n monotone ≤ αa · a − X k =0 ( g n ( k ) − g n ( a )) . Finally, we can bound 15 A · a ≤ α · a − X k =0 ( g n ( k ) − g n ( a )) α ≤ b α, g n mon. ≤ b α · ⌈ b α ( n − ⌉− X k =0 ( g n ( k ) − g n ( ⌈ b α ( n − ⌉ )) (6) , Def. of n ≤ − b α ) · n − X k = n −⌈ b α ( n − ⌉ (1 − g n ( k )) g n mon. ≤ − b α ) · b α · ( n − a n − X k = n − a (1 − g n ( k )) a ≥ α · n/ ≤ − b α ) · b αα n − X k = n − a (1 − g n ( k )) Def. of b α < (1 − α ) · n − X k = n − a (1 − g n ( k ))= ∆ B · a. Thus, b ı beats all candidates drawn from A , and the winner will be from B .Using the preceding lemmas, the proof of necessity is almost complete. Consider the metricspace with all the parameters as defined above. By Lemmas 3.8 and 3.9, with probability at least , the winner is from B . The social cost of any candidate from B is at least β · − β ) · ( s + 1).On the other hand, the social cost of any candidate from A is at most (1 − β ) · β · ( s + 1) = 3.The distortion in this case is thus at least(1 − β ) · ( s + 1)3 = (2 β + 1) · (1 − β )3 β = 2 c − . In the other case (when E does not occur — this happens with probability at most ), the distortionis at least 1, so that the expected distortion is at least (2 c −
1) + · c . Proof of Corollary 3.2.
For the first part of the corollary, assume that g is not constant on(0 , y ) will be Ω( n ), while the sum on the right-hand side is obviously at most n . By making y a constant close enough to 1, we can dominate the constant from Ω, and thus ensure that theinequality (4) holds. Then, the constant distortion follows from Theorem 3.1.More precisely, let 0 < ℓ < u < g ( ℓ ) > g ( u ). Let δ := g ( ℓ ) − g ( u ) and y := max( u, − δℓ ) ∈ (0 , n be such that for all n ≥ n , we have g n ( ⌊ ℓ · ( n − ⌋ ) ≥ g ( ℓ ) − δ/ , g n ( ⌈ u · ( n − ⌉ ) ≤ g ( u ) + δ/ , ⌊ ℓ · ( n − ⌋ ≥ ℓn , ⌈ y · ( n − ⌉ ≤ yn. n exists by the consistency of V and basic integer arithmetic. Then, for all n ≥ n , y · ⌈ y · ( n − ⌉− X k =0 ( g n ( k ) − g n ( ⌈ y · ( n − ⌉ )) ≥ y · ⌊ ℓ · ( n − ⌋− X k =0 ( g n ( ⌊ ℓ · ( n − ⌋ ) − g n ( ⌈ u · ( n − ⌉ )) ≥ y · ⌊ ℓ · ( n − ⌋− X k =0 ( δ/ ≥ · y · ℓ · n · δ ≥ y · (1 − y ) · n> (1 − y ) · n − X k = n −⌈ y · ( n − ⌉ (1 − g n ( k )) . Because the condition (4) is satisfied, Theorem 3.1 implies constant distortion.For the second part of the corollary, assume that g ( x ) = c < x ∈ (0 , y ∈ (0 , n , the condition (4) is violated.The intuition is that the sum on the right-hand side of (4) consists of terms that will in thelimit be 1 − c >
0, while the left-hand side is a sum in which each term converges to 0. Thus, nevermind how large the constant y < y and 1 − y will eventually not be enough tomake the left-hand side larger than the right-hand side. Making this intuition precise requires somecare: while the functions g n converge to g , we did not assume that they do so uniformly . To dealwith this issue, we will consider consistency with g at two points γ and 1 − γ only (with γ being avery small constant), and use monotonicity of each g n to bound the remaining terms. The terms ofthe sum corresponding to points to the left of γ and to the right of 1 − γ can then not be bounded,but there are few enough of them that we still obtain the desired inequality. More specifically, let γ ∈ (0 ,
1) be a sufficiently small constant such that γ < min( y, − y ) and δ := (1 − y ) · (1 − c ) − γ y − γ > . Such a γ exists, since both the numerator and denominator tend to strictly positive numbersas γ →
0. Recall that g ( x ) = c for all x ∈ (0 , n be such that for all n ≥ n , g n ( ⌊ γ · ( n − ⌋ ) ≤ g n ( ⌈ γ · ( n − ⌉ ) ≤ c + δ,g n ( ⌈ (1 − γ )( n − ⌉ ) ≥ g n ( ⌊ (1 − γ n − ⌋ ) ≥ c − δ, ⌈ y · ( n − ⌉ − ⌊ γ · ( n − ⌋ ≤ y − γ )( n − . Such an n exists by basic integer arithmetic and the consistency of V applied at x = γ/ x = 1 − γ/
2. 17riting Γ = ⌊ γ ( n − ⌋ and Γ ′ = ⌈ (1 − γ )( n − ⌉ , we get y · ⌈ y · ( n − ⌉− X k =0 ( g n ( k ) − g n ( ⌈ y · ( n − ⌉ )) ≤ y · Γ − X k =0 ⌈ y · ( n − ⌉− X k =Γ (cid:0) g n (Γ) − g n (Γ ′ ) (cid:1) ≤ y · ( γ · ( n −
1) + 2 · ( y − γ ) · ( n − · δ )= y · ( n − · ( γ + ( y − γ ) · δ ) . The first inequality uses y < − γ and the monotonicity of g n , and the second inequality usesthe bounds obtained from consistency of g n with respect to g . To bound the right-hand side of (4),(1 − y ) · n − X k = n −⌈ y · ( n − ⌉ (1 − g n ( k )) ≥ (1 − y ) · n − X k = n −⌈ y · ( n − ⌉ (1 − g n (Γ)) ≥ (1 − y ) · y · ( n − · (1 − c − δ ) . The first inequality again used monotonicity of g n , and the second used the bounds obtainedfrom the consistency of g n with respect to g . Canceling the common term y ( n −
1) between theleft-hand side and right-hand side, the right-hand side of (4) is at least as large as the left-handside whenever (1 − y ) · (1 − c − δ ) ≥ γ + ( y − γ ) · δ . Solving for δ , this is equivalent to δ ≤ (1 − y ) · (1 − c ) − γ y − γ , which is exactly ensured by our choice of γ and δ . This completes the proof. When candidates are drawn i.i.d. from the voter distribution, we showed that whether a positionalvoting system V has expected constant distortion can be almost fully characterized by its limitingbehavior. In particular, if the limiting scoring rule is not constant on (0 , V has constantexpected distortion; if the limiting scoring rule is a constant other than 1 on (0 , V has super-constant expected distortion. A more subtle condition depending on the “rate of convergence” tothe limit rule completes the characterization.Our Theorem 3.1 currently does not characterize the order of growth of the distortion. Withsome effort, the proof could likely be adapted to the case where the y in the theorem is a function y ( n ), which would allow us to characterize the rate at which the distortion grows with n .For specific voting systems, the proof of Theorem 3.1 can often be adapted to give tighterbounds. For example, straightforward modifications of the proof can be used to show that thedistortion of k -approval or k -veto (where each voter can veto k candidates) for constant k grow asΩ( n ). This matches the O ( n ) upper bound from Lemma 3.3, giving a tight analysis of the distortionof these voting systems. Similarly, the sufficiency proof can be adapted to show that the distortionof Borda Count is at most 16, for all metric spaces and all n . When the number of candidatesgrows large enough, the expected distortion is in fact bounded by 10.Our results indicate that if one is concerned about systematic, and possibly adversarial, biasin which candidates run for office, randomizing the slate of candidates may be part of a solutionapproach. Such an approach can be considered as a step in the direction of lottocracy and sortition representativeness of office holders, but one of its main drawbacksis the potential lack of competency. As a broader direction for future research, our work heresuggests devising models that capture the tension between these two objectives, and would allowfor the design of hybrid mechanisms that navigate the tradeoff successfully. Acknowledgments
Part of this work was done while Yu Cheng was a student at the University of Southern California.Yu Cheng was supported in part by Shang-Hua Teng’s Simons Investigator Award. Shaddin Dughmiwas supported in part by NSF CAREER Award CCF-1350900 and NSF grant CCF-1423618. DavidKempe was supported in part by NSF grants CCF-1423618 and IIS-1619458. We would like to thankanonymous reviewers for useful feedback.
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