The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)
On the Distortion of Voting with Multiple Representative Candidates∗
Yu Cheng Shaddin Dughmi David Kempe Duke University University of Southern California University of Southern California
Abstract voters and the chosen candidate in a suitable metric space (Anshelevich 2016; Anshelevich, Bhardwaj, and Postl 2015; We study positional voting rules when candidates and voters Anshelevich and Postl 2016; Goel, Krishnaswamy, and Mu- are embedded in a common metric space, and cardinal pref- erences are naturally given by distances in the metric space. nagala 2017). The underlying assumption is that the closer In a positional voting rule, each candidate receives a score a candidate is to a voter, the more similar their positions on from each ballot based on the ballot’s rank order; the candi- key questions are. Because proximity implies that the voter date with the highest total score wins the election. The cost would benefit from the candidate’s election, voters will rank of a candidate is his sum of distances to all voters, and the candidates by increasing distance, a model known as single- distortion of an election is the ratio between the cost of the peaked preferences (Black 1948; Downs 1957; Black 1958; elected candidate and the cost of the optimum candidate. We Moulin 1980; Merrill and Grofman 1999; Barbera,` Gul, consider the case when candidates are representative of the and Stacchetti 1993; Richards, Richards, and McKay 1998; population, in the sense that they are drawn i.i.d. from the Barbera` 2001). population of the voters, and analyze the expected distortion Even in the absence of strategic voting, voting systems of positional voting rules. can lead to high distortion in this setting, because they Our main result is a clean and tight characterization of posi- typically allow only for communication of ordinal prefer- tional voting rules that have constant expected distortion (in- 1 dependent of the number of candidates and the metric space). ences , i.e., rankings of candidates (Boutilier and Rosen- Our characterization result immediately implies constant ex- schein 2016). In a beautiful piece of recent work, (An- pected distortion for Borda Count and elections in which each shelevich, Bhardwaj, and Postl 2015) showed that this ap- voter approves a constant fraction of all candidates. On the proach can draw very clear distinctions between voting sys- other hand, we obtain super-constant expected distortion for tems: some voting systems (in particular, Copeland and re- Plurality, Veto, and approving a constant number of candi- lated systems) have distortion bounded by a small constant, dates. These results contrast with previous results on voting while most others (including Plurality, Veto, k-approval, and with metric preferences: When the candidates are chosen ad- Borda Count) have unbounded distortion, growing linearly versarially, all of the preceding voting rules have distortion in the number of voters or candidates. linear in the number of candidates or voters. Thus, the model The examples giving bad distortion typically have the of representative candidates allows us to distinguish voting rules which seem equally bad in the worst case. property that the candidates are not “representative” of the voters. (Anshelevich, Bhardwaj, and Postl 2015) showed more positive results when there are no near-ties for first 1 Introduction place in any voter’s ranking. (Cheng, Dughmi, and Kempe In light of the classic impossibility results for axiomatic ap- 2017) proposed instead a model of representativeness in proaches to social choice (Arrow 1951) and voting (Gib- which the candidates are drawn randomly from the popu- bard 1973; Satterthwaite 1975), a fruitful approach has been lation of voters; under this model, they showed smaller con- to treat voting as an implicit optimization problem of find- stant distortion bounds than the worst-case bounds for ma- ing the “best” candidate for the population in aggregate jority voting with n =2candidates. (Cheng, Dughmi, and (Boutilier et al. 2015; Caragiannis and Procaccia 2011; Kempe 2017) left as an open question the analysis of the Procaccia 2010; Procaccia and Rosenschein 2006). Using distortion of voting systems for n ≥ 3 representative candi- this approach, voting systems can be compared based on dates. how much they distort the outcome, in the sense of leading In the present work, we study the distortion of positional to the election of suboptimal candidates. A particularly nat- voting systems with n ≥ 3 representative candidates. Infor- ural optimization objective is the sum of distances between mally (formal definitions of all concepts are given in Sec- ∗Due to space constraints, this version omits several important 1Of course, it is also highly questionable that voters would be proofs. The full version of this paper can be found on arXiv. able to quantify distances in a metric space sufficiently accurately, Copyright c 2018, Association for the Advancement of Artificial in particular given that the metric space is primarily a modeling Intelligence (www.aaai.org). All rights reserved. tool rather than an actual concrete object.
973 tion 2), a positional voting system is one in which each voter As a by-product of the proof of our main theorem, in writes down an ordering of candidates, and the system as- Lemma 3.3, we show that every voting system (positional signs a score to each candidate based solely on his2 position or otherwise) has distortion O(n) with representative candi- in the voter’s ordering. The map from positions to scores is dates. Combined with the lower bound alluded to above, this known as the scoring rule of the voting system, and for n exactly pins down the distortion of Plurality, Veto, and O(1)- candidates is a function gn : {0,...,n− 1}→R≥0. The approval with representative candidates to Θ(n). For Veto, total score of a candidate is the sum of scores he obtains this result also contrasts with the worst-case bound of (An- from all voters, and the winner is the candidate with maxi- shelevich, Bhardwaj, and Postl 2015), which showed that the mum total score. The most well-known explicitly positional distortion can grow unboundedly even for n =3candidates. voting system is Borda Count (de Borda 1784), in which gn(i)=n − i for all i. Many other systems are naturally 2 Preliminaries cast in this framework, including Plurality (in which voters 2.1 Voters, Metric Space, and Preferences give 1 point to their first choice only) and Veto (in which The voters/candidates are embedded in a closed metric space voters give 1 point to all but their last choice). ∈ In analyzing positional voting systems, we assume that (Ω,d), where dω,ω is the distance between points ω, ω voters are not strategic, i.e., they report their true ranking of Ω. The distance captures the dissimilarity in opinions be- candidates based on proximity in the metric space. This is in tween voters (and candidates) — the closer two voters or keeping with the line of work on analyzing the distortion of candidates are, the more similar they are. The distribution of voters in Ω is denoted by the (measurable) density function social choice functions, and avoids issues of game-theoretic q 3 modeling and equilibrium existence or selection (see, e.g., qω. We allow for to have point masses. Unless there is no (Feldman, Fiat, and Golomb 2015)) which are not our focus. risk of confusion, we will be careful to distinguish between a location ω ∈ Ω and a specific voter j or candidate i at that As our main contribution, we characterize when a posi- location. We apply d equally to locations/voters/candidates. tional voting system is guaranteed to have constant distor- We frequently use the standard notion of a ball tion, regardless of the underlying metric space of voters and B(ω, r):={ω | d ≤ r} in a metric space. For balls candidates, and regardless of the number n of candidates ω,ω (and other sets) B, we write q := q dω. that are drawn from the voter distribution. The characteri- B ω∈B ω An election is run between n ≥ 2 candidates according to zation relies almost entirely on the “limit voting system.” rules defined in Section 2.3. The n candidates are assumed By normalizing both the scores and the candidate index to th to be representative of the population, in the sense that their lie in [0, 1] (we associate the i out of n candidates with his locations are drawn i.i.d. from the distribution q of voters. quantile i ∈ [0, 1]), we can take a suitable limit g of the n−1 Each voter ranks the n candidates i by non-decreasing →∞ scoring functions gn as n . distance from herself in (Ω,d). Ties are broken arbitrarily, Our main result (Corollary 3.2 in Section 3) states the fol- but consistently4, meaning that all voters at the same lo- lowing: (1) If g is not constant on the open interval (0, 1), cation have the same ranking. We denote the ranking of then the voting system has constant distortion. (2) If g is a a voter j or a location ω over candidates i by πj(i) or constant other than 1 on the open interval (0, 1), then the πω(i). The distance-based ranking assumption means that voting system does not have constant distortion. The only πω(i) <πω(i ) implies that dω,i ≤ dω,i and dω,i
974 Based on the votes, a voting system will determine a win- gn(n − 1) = 0 is without loss of generality, because a score- ner w(C) for the set C of candidates, who will often be dif- based rule is invariant under affine transformations. ferent from o(C). The distortion measures how much worse Next, we want to capture the notion that a positional vot- c the winner is than the optimum D(C)= w(C) . We are inter- ing system is “consistent” as we vary the number of can- co(C) ested in the expected distortion of positional voting systems didates n. Intuitively, we want to exclude contrived voting systems such as “If the number of candidates is even, then under i.i.d. random candidates, i.e., E i.i.d. [D(C)]. C ∼ q use Borda Count; otherwise use Plurality voting.” This is Our distortion bounds are achieved by lower-bounding captured by the following definition. co(C) ≥ co. A particularly useful quantity in this context is the fraction of voters outside a ball of radius r around o, Definition 2.3 (Consistency). Let V be a positional voting { | ∈ N} V which we denote by H(r):=1− qB(o,r ). The following system with scoring rules gn n . We say that is lemma captures some useful simple facts that we use: consistent if there exists a function g : Q ∩ [0, 1] → [0, 1] such that for each rational quantile x ∈ [0, 1] and accu- Lemma 2.1. 1. For any candidate or location i, racy parameter >0, there exists a threshold n0 such that − ≥ − − ≤ ci ≤ co + di,o. (1) gn( x(n 1) ) g(x) and gn( x(n 1) ) g(x)+ for all n ≥ n0. We call g the limit scoring rule of V. 2. The cost of any candidate or location i can be written as Intuitively, this definition says that the sequence of scor- ∞ ing rules gn is consistent with a single scoring rule g in ci = (1 − qB(i,r))dr. (2) 0 the limit. Using the fact that gn is monotone non-increasing for each n, it can be shown that g is also monotone non- ≥ 3. For all r 0, the cost of the optimum location o is lower- increasing. We note that gn converges pointwise to g in a bounded by precise and natural sense. Formally, when x ∈ [0, 1] is ra- tional, there exists an infinite sequence of integers n with c ≥ rH(r). (3) o x(n − 1) = x(n − 1) = x(n − 1), and consistency im- Proof. 1. The proof of the first inequality simply ap- plies that g(x) must equal the limit of gn(x(n − 1)) for that plies the triangle inequality under the integral: ci = sequence of values of n. Therefore the limit scoring rule g is ≤ uniquely defined if it exists. ω dω,iqωdω ω(dω,o + di,o)qωdω = co + di,o. All positional voting systems we are aware of are consis- 2. For the second equation, observe that c = E ∼q [d ], i ω i,ω tent according to Definition 2.3. and the expectation of any non-negative random variable E ∞ ≥ X can be rewritten as [X]= 0 Pr[X x]dx. Example 2.1. To illustrate the notion of a consistent posi- 3. For the third inequality, we apply the previous part with tional voting system, consider the following examples, en- ∞ ≥ ≥ i = o, and lower bound 0 Prω∼q[do,ω x]dx compassing most well-known scoring rules. r ∞ Prω∼q[do,ω ≥ r]dx + 0dx = r · H(r). 0 r • In Plurality voting with n candidates, gn(0) = 1 and gn(k)=0for all k>0. The limit scoring rule is 2.3 Positional Voting Systems and Scoring Rules g(0) = 1 and g(x)=0for all x>0. We are interested in positional voting systems. Such systems • In Veto voting with n candidates, g (k)=1for all k< are based on scoring rules: voters give a ranking of candi- n n−1 and gn(n−1) = 0. The limit scoring rule is g(x)= dates, and with each position is associated a score. 1 for all x<1 and g(1) = 0. Definition 2.1 (Scoring Rule). A scoring rule for n candi- • In k-approval voting with constant k, we have gn(k )=1 dates is a non-increasing function g : {0,...,n− 1}→ n for k ≤ min(k − 1,n− 1), and gn(k )=0for all other − [0, 1] with gn(0) = 1 and gn(n 1) = 0. k . The limit scoring rule is g(0) = 1 and g(x)=0for all Definition 2.2 (Positional Voting System). A positional vot- x>0, i.e., the same as for Plurality voting. (This relies ing system is a sequence of scoring rules gn, one for each on k being constant, or more generally, k = o(n).) number of candidates n =1, 2,.... • In k-approval voting with linear k, there exists a constant ∈ ≤ The interpretation of gn is that if voter j puts a can- γ (0, 1) with gn(k)=1for all k γn, and gn(k)=0 didate i in position π (i) on her ballot, then i obtains for all larger k. The limit scoring rule is g(x)=1for j ≤ gn(πj(i)) points from j. The total score of candidate i is x γ and g(x)=0for x>γ. then σ(i)= gn(πω(i))qωdω. The winning candidate is • − k ω The Borda voting rule has gn(k)=1 n−1 (after nor- one with highest total score, i.e., for a set C of n candidates, malization). The limit scoring rule is g(x)=1− x. w(C) ∈ argmax σ(i); again, ties are broken arbitrarily, i∈C • and our results do not depend on tie breaking. The Dowdall method used in Nauru (Fraenkel and Grof- The restriction to monotone non-increasing scoring rules man 2014; Reilly 2002) has gn(k)=1/(k+1). After nor- 1 · n − is standard when studying positional voting systems. One malization, the rule becomes gn(k)= n−1 ( k+1 1). The justification is that in any positional voting system violating limit scoring rule is g(0) = 1 and g(x)=0for all x>0, this restriction, truth-telling is a dominated strategy, render- i.e., the same as for Plurality voting. This is because for ing such a system uninteresting for most practical purposes. every constant quantile x, the score of the candidate at x 1 1 − n→∞→ Given this restriction, the assumption that gn(0) = 1 and is n−1 x 1 0.
975 3 The Main Characterization Result constant) fraction y of all voters, such that y satisfies (4). In this section, we state and prove our main theorem, charac- If the number of candidates n is large enough (a large con- ≥ terizing positional voting systems with constant distortion. stant), standard Chernoff bounds ensure that as r r grows large, most candidates who are running will be from inside V Theorem 3.1. Let be a positional voting system with a B(o, r). In turn, if many candidates inside B(o, r) are run- V sequence gn of scoring rules for n =1, 2,.... Then, has ning, all candidates outside B(o, 3r) are very far down on constant expected distortion if and only if there exist con- almost everyone’s ballot, and therefore cannot win. In par- ∈ ≥ stants n0 and y (0, 1) such that for all n n0, ticular, Inequality (4) implies that the total score of an av- erage candidate in B(o, r) exceeds the maximum possible y(n−1) −1 total score of a candidate outside B(o, 3r). This allows us to y· (g (k) − g ( y(n − 1) )) n n bound the expected distortion in terms of the cost of o. =0 k The case of small n is much easier, since we can treat n −1 (4) n as a constant. In that case, the following lemma is sufficient. > (1 − y) · (1 − gn(k)) . Lemma 3.3. If n candidates are drawn i.i.d. at random from k=n−y(n−1) q, the expected distortion is at most n +1. We prove Theorem 3.1 in Sections 3.1 (sufficiency) and Proof. The proof illustrates some of the key ideas that will 3.2 (necessity). Condition (4) is quite unwieldy. In most be used later in the more technical proof for a large number cases of practical interest, we can use Corollary 3.2. of candidates. We want to bound Corollary 3.2. Let V be a consistent positional voting sys- (1) EC cw(C) ≤ co + EC do,w (C) tem with limit scoring rule g. ∞ = c + Pr [d ≥ r]dr. 1. If g is not constant on the open interval (0, 1), then V has o 0 C o,w (C) constant expected distortion. In order for a candidate at distance at least r from oto win, 2. If g is equal to a constant other than 1 on the open interval it is necessary that at least one such candidate be running. By (0, 1), then V does not have constant expected distortion. a union bound over the n candidates, the probability of this event is at most PrC [do,w (C) ≥ r] ≤ nH(r),so The constant in Theorem 3.1 and Corollary 3.2 depends V ∞ on , but not on the metric space or the number of candi- (2) dates. Corollary 3.2 has the advantage of determining con- EC cw(C) ≤ co + n H(r)dr =(n +1)co. stant expected distortion only based on the limit scoring rule 0 g. The only case when it does not apply is when g(x)=1 Lower-bounding the cost of the optimum candidate from for all x ∈ [0, 1). In that case, the higher complexity of The- C in terms of the overall best location o, the expected dis- orem 3.1 is indeed necessary to determine whether V has tortion is
constant distortion. Fortunately, Veto voting is the only rule cw(C) cw(C) 1 of practical importance for which g(x) ≡ 1 on [0, 1), and it EC ≤ EC = EC cw(C) c c c is easily analyzed. o(C) o o Before presenting the proofs, we apply the characteriza- 1 ≤ · (n +1)co = n +1. tion to the positional voting systems from Example 2.1. Us- co ing the limit scoring rules derived in Example 2.1, Corol- In preparation for the case of large n, we begin with the lary 3.2 implies constant expected distortion for Borda following technical lemma, which shows that whenever (4) Count and k-approval with linear k =Θ(n), and super- holds, it will also hold when the terms on the left-hand side constant expected distortion for Plurality, k-approval with are “shifted,” and the right-hand side can be increased by a k = o(n), and the Dowdall method. factor of 2 (or, for that matter, any constant factor). This leaves Veto voting, for which it is easy to apply The- orem 3.1 directly. Because g (k)=1for all k
976 V 1 Notice that μ, z, n only depend on , but not on the metric Because z>2 and gn is monotone, we get that space or number of voters. n−1 − ≥ 1 n−1 − k=n−Z (1 gn(k)) 2 k=n−s(1 gn(k)). Hence, Let r := inf{r | qB(o,r ) ≥ μ}, so that qB(o, r) ≥ μ, and ≥ − 1 · · 1−z · n−1 − q{ω|do,ω ≥r} 1 μ. (Both inequalities hold with equality ΔS > s μ z k=n−s(1 gn(k)) ≥ unless there is a discrete point mass at distance r from o.) μ z −1 ≥ 1 · (1 − μ) · n (1 − g (k)) = Δ . Consider any r ≥ r and write T := B(o, 3r) and s k=n−s n i S := B(o, r). When n candidates are drawn i.i.d. from q, We now wrap up the sufficiency portion of the proof of the expected fraction of candidates drawn from outside of Theorem 3.1. We distinguish two cases, based on the num- S is exactly H(r) ≤ 1 − μ. Let Er be the event that more ber of candidates n.Ifn
s−1 − from each other. A contains a large α fraction of the popula- k=0 (gn(k + m) gn(s + m)) −1 tion, and thus should in an optimal solution be the one that ≥ Z (g (k + m) − g (Z + m)) k=0 n n the winner is chosen from. We will ensure that with proba- (5) 2(1−z) · n−1 − bility at least 1 , the winner instead comes from B. Because > z k=n−Z (1 gn(k)) . 2
977 −1 B is far from A, most of the population then is far from the in A.Forω ∈ A and x ∈ Iπ, let π (ω) be the position chosen candidate, giving much worse cost than optimal. of ω in π, and define the distance between ω and x to be −1 The metrics underlying A and B are as follows: B will es- x π (ω) dω,x = s + + . sentially provide an “ordering,” meaning that whichever set 4 M! of candidates is drawn from B, all voters in B (and essen- Proposition 3.7. Definition 3.1 defines a metric. tially all in A) agree on their ordering of the candidates. This will ensure that one candidate from B will get a sufficiently Proof. Non-negativity, symmetry, and indiscernibles hold large fraction of first-place votes, and will be ranked highly by definition. Because all distances within clusters are in A A [1, 2], and distances across clusters are more than 2, the tri- enough by voters from , too. will be based on a large ∈ M ω angle inequality holds for all pairs ω, ω A and all pairs number of discrete locations . Their pairwise distances ∈ are chosen i.i.d.: as a result, the rankings of voters are uni- x, x B. ∈ ∈ ∈ formly random, and there is no consensus among voters in Because dω,x [s, s +1]for all ω A and x B, A on which of their candidates they prefer. Because the vote and distances within A or B are at least 1, there can be no ∈ is thus split, the best candidate from B will win instead. shorter path than the direct one between any ω A and ∈ The following parameters (whose values are chosen with x B. Therefore, the triangle inequality is satisfied. foresight) will be used to define the metric space. Now consider a (random) set C of n candidates, drawn • Let c>1 be any constant; we will construct a metric i.i.d. from q. We are interested in the event that the resulting space and number of candidates for which the distortion slate of candidates is highly representative of the voters, in is at least c. the following sense. • ∈ 1 2β+1 · − Let β (0, 2 ) solve the quadratic equation 3β (1 Definition 3.2. Let C be the (random) set of n candidates 1 q E β)=2c − 1. A solution exists because at β = 2 , the drawn from . Let be defined as the conjunction of the 2 following: left-hand side is 3 < 2c − 1; it goes to infinity as β → 0, while the right-hand side is a positive constant. β is the 1. For each location ω ∈ A, the set C contains at most one fraction of voters in the small cluster B. candidate from ω. • − β Let α =1 β denote the fraction of voters in the large 2. At least a 2 fraction of candidates in C is drawn from B cluster A. (and thus at most an α fraction of candidates are from A). • 1+β α Let s = β be the distance between the clusters B and 3. At least an 2 fraction of candidates in C is drawn from A. (Each cluster will have diameter at most 2.) A. 1 1 α 4. No pair x, x ∈ B ∩ C has |x − x | < ( −1)! . • Let α ≥ 2 + 2 >αsatisfy 4α · (1 − α) <α· (1 − α); M such an α exists because the left-hand side goes to 0 as Lemma 3.8 uses standard tail bounds to show that E hap- α → 1. α< 1 is a high-probability upper bound on the 1 pens with probability at least 2 ; then, Lemma 3.9 shows that fraction of candidates that will be drawn from A. whenever E happens, the winner is from B. 4 • Let n0 = 2 > 16; this is a lower bound on the number 1 β Lemma 3.8. E happens with probability at least 2 . of candidates that ensures that the actual fraction of can- E didates drawn from A is at most α with sufficiently high Lemma 3.9. Whenever happens, the winning candidate B probability. is from . • Let n ≥ n0 be the n whose existence is guaranteed by the Proof. Let b be the actual number of candidates drawn from assumption (6) (for y = α and n0). B, and a = n − b the number of candidates drawn from A. 3 E ≥ β · ≤ • Let M = n ; this is the number of discrete locations ω Because we assumed that happened, b 2 n and a · we construct within the larger cluster A. α n. Let CA be the set of candidates drawn from A. Under E, CA contains at most one candidate from each location We now formally define the metric space consisting of ω ∈ A. As a result, because the random distances within A two clusters: are distinct with probability 1, there will be no ties in the Definition 3.1. The metric space consists of two clusters A rankings of any voters. and B. A has M discrete locations, and q has a point mass Let ı be the candidate from B with smallest value x. With of α on each such location. The total probability mass on B probability 1, the x value of ı is unique. Consider some ar- M is qB =1− α, distributed uniformly over the interval [1, 2]. bitrary candidate i ∈ CA from location ω . We calculate the Locations in B are identified by x ∈ [1, 2]. The distances contributions toı and i from voters in B and in A separately, are defined as follows: and show thatı beats i. Because this holds for arbitrary i, the candidate ı or another candidate from B wins. 1. For each distinct pair ω, ω ∈ A, the distance dω,ω is drawn independently uniformly at random from [1, 2]. 1. We begin with points given out by voters in B. By def- 2. For each distinct pair x, x ∈ B of locations, the distance inition of the distances within B, ı is ranked first by all is defined to be dx,x := min(x, x ). voters in B. 3. Partition B =[1, 2] into M! disjoint intervals Iπ of length Voters in Iπ rank the candidates from A according to 1 1/M ! each, one for each permutation of the M locations their order in π. For each ordering of CA, exactly a a!
978 fraction of permutations induces that ordering. In partic- over ı from votes from A is at most ∈ ular, for each k 1,...,a, exactly a 1/a fraction of −2 Δ := α a 2 g (k)+ 2−a g (a − 1) − g (a) voters places i in position k + b. Thus, i obtains a to- A k=0 a n a n n n−1 1 tal of (1 − α) · · g (k) points from voters −2 k=n−a a n = α a 2g (k) − g (a − 1) − g (a) a k=0 n n n in B. Overall, ı obtains an advantage of at least ΔB = − · 1 · n−1 − (1 α) k=n−a(1 gn(k)). a +gn(a − 1) − gn(a) 2. Next, we analyze the number of points given out by voters gn monotone ∈ 2 a−1 in A. The distance from any voter location ω A to ı is at ≤ α · (g (k) − g (a)). x n E a k=0 n n most s+ 4 + M! . Under , no other candidate from B can be at a location x ≤ x + 1 ; therefore, the distance Finally, we can bound (M−1)! from any voter location ω ∈ A to any other candidate · ≤ · a−1 − ΔA a 2α k=0(gn(k) gn(a)) x ∈ B is at least ≤ α α, gn mon. ( −1) −1 ≤ 2α α n (g (k) − g ( α(n − 1) )) x x 1 x n k=0 n n ≥ ≥ ≥ (6), Def. of n dω,x s+ s+ + >s+ + dω,ı, ≤ − · n−1 − 4 4 4(M − 1)! 4 M! 2(1 α) k=n−α(n−1) (1 gn(k)) gn mon. ≤ − · α·(n−1) n−1 − so all voters in A prefer ı over any other candidate from 2(1 α) a k=n−a(1 gn(k)) · a≥α·n/2 B. Hence, ı obtains at least α gn(a) points combined ≤ − · 2α n−1 − 2(1 α) α k=n−a(1 gn(k)) from voters in A. Def. of α − · n−1 − · To analyze the votes from voters in A for candidates < (1 α) k=n−a(1 gn(k)) = ΔB a. from A, we first notice that E and the draw of candi- dates are independent of the distances within A. Hence, Thus, ı beats all candidates drawn from A, and the winner even conditioned on E, the distances dω,ω between lo- will be from B. cations in A are i.i.d. uniform from [1, 2]. In particu- Using the preceding lemmas, the proof of necessity is al- lar, each location ω ∈ A ranks the candidates in C in A most complete. Consider the metric space with all the pa- uniformly random order. Furthermore, for two locations rameters as defined above. By Lemmas 3.8 and 3.9, with ω = ω , the rankings of C are independent; the rea- 1 A probability at least , the winner is from B. The social cost son is that they are based on disjoint vectors of distances 2 of any candidate from B is at least β · 0+(1− β) · (s +1). (dω,i)i∈CA , (dω ,i)i∈CA . We use this independence to ap- On the other hand, the social cost of any candidate from A ply tail bounds. Let ω be the location of i. Voters rank i is at most (1 − β) · 2+β · (s +1)=3. The distortion in this as follows: (1− )·( +1) (2 +1)·(1− ) case is thus at least β s = β β =2c − 1. • Among locations ω without a candidate of their own, 3 3β In the other case (when E does not occur — this happens in expectation, a 1/a fraction of voters will rank i in 1 with probability at most ), the distortion is at least 1, so that position k, for each k =0,...,a− 1. 2 the expected distortion is at least 1 (2c − 1) + 1 · 1=c. • Among the a − 1 locations ω = ω with a candidate of 2 2 their own, in expectation, a 1/(a − 1) fraction of voters will rank i in position k, for each k =1,...,a− 1. 4 Conclusions • Voters at ω will rank i in position 0. When candidates are drawn i.i.d. from the voter distribution, we showed that whether a positional voting system V has ex- For each k, let the random variable Xk be the number pected constant distortion can be almost fully characterized of locations that rank i in position k. By the preceding by its limiting behavior. In particular, if the limiting scoring E M arguments, [Xk]= a , and Xk is a sum of M inde- rule is not constant on (0, 1), then V has constant expected pendent (not i.i.d.) Bernoulli random variables. Hence, distortion; if the limiting scoring rule is a constant other than by the Hoeffding bound, the probability that more than (0, 1) V 2 1on , then has super-constant expected distortion. a a fraction of voters rank i in position k is at most A more subtle condition depending on the “rate of conver- − · 1 · ≤ − 2exp( 2 a2 M) 2exp( n). By a union bound over gence” to the limit rule completes the characterization. all candidates i ∈ CA and all values k =0,...,a− 1, Our Theorem 3.1 currently does not characterize the order with high probability, for all i and k, the fraction of voters of growth of the distortion. With some effort, the proof could · 2 (in A) ranking i in position k is at most α a . Because likely be adapted to the case where the y in the theorem is a the total fraction of voters in A is α, any excess votes for function y(n), which would allow us to characterize the rate some (early) positions k must be compensated by fewer at which the distortion grows with n. votes for other (late) positions k. Relaxing the constraint For specific voting systems, the proof of Theorem 3.1 that the number of votes for each position k must be non- can often be adapted to give tighter bounds. For example, negative, we can upper-bound the total points for i by as- straightforward modifications of the proof can be used to suming that each of the positions k =0,...,a − 2 re- show that the distortion of k-approval or k-veto (where each ceives twice the expected number of votes, while position voter can veto k candidates) for constant k grow as Ω(n). k = a − 1 receives a negative number of votes that com- This matches the O(n) upper bound from Lemma 3.3, giv- pensates for the excess votes. Then, the advantage for i ing a tight analysis of the distortion of these voting systems.
979 Similarly, the sufficiency proof can be adapted to show that Caragiannis, I., and Procaccia, A. D. 2011. Voting almost the distortion of Borda Count is at most 16, for all metric maximizes social welfare despite limited communication. spaces and all n. When the number of candidates grows large Artificial Intelligence 175(9):1655–1671. enough, the expected distortion is in fact bounded by 10. Cheng, Y.; Dughmi, S.; and Kempe, D. 2017. Of the peo- Our results indicate that if one is concerned about sys- ple: Voting is more effective with representative candidates. tematic, and possibly adversarial, bias in which candidates In Proc. 18th ACM Conf. on Economics and Computation, run for office, randomizing the slate of candidates may be 305–322. part of a solution approach. Such an approach can be con- de Borda, J.-C. 1784. Memoire´ sur les elections´ au scrutin. sidered as a step in the direction of lottocracy and sortition Histoire de l’Academie´ Royale des Sciences, Paris 657–665. (Dowlen 2008; Guerrero 2014; Landemore 2013), in which office holders are directly chosen at random from the popu- Dowlen, O. 2008. The political potential of sortition: A lation. Pure lottocracy does well in terms of representative- study of the random selection of citizens for public office. ness of office holders, but one of its main drawbacks is the Imprint Academic. potential lack of competency. As a broader direction for fu- Downs, A. 1957. An economic theory of political action in ture research, our work here suggests devising models that a democracy. The Journal of Political Economy 65(2):135– capture the tension between these two objectives, and would 150. allow for the design of hybrid mechanisms that navigate the Feldman, M.; Fiat, A.; and Golomb, I. 2015. On voting and tradeoff successfully. facility location. arXiv:1512.05868. Acknowledgments Part of this work was done while Yu Fraenkel, J., and Grofman, B. 2014. The Borda count Cheng was a student at University of Southern California. and its real-world alternatives: Comparing scoring rules in Yu Cheng was supported in part by Shang-Hua Teng’s Si- Nauru and Slovenia. Australian Journal of Political Science mons Investigator Award. Shaddin Dughmi was supported 49(2):186–205. in part by NSF CAREER Award CCF-1350900 and NSF Gibbard, A. F. 1973. Manipulation of voting schemes: a grant CCF-1423618. David Kempe was supported in part by general result. Econometrica 41(4):587–601. NSF grants CCF-1423618 and IIS-1619458. We would like Goel, A.; Krishnaswamy, A. K.; and Munagala, K. 2017. to thank anonymous reviewers for useful feedback. Metric distortion of social choice rules: Lower bounds and fairness properties. In Proc. 18th ACM Conf. on Economics References and Computation, 287–304. Anshelevich, E., and Postl, J. 2016. Randomized social Guerrero, A. A. 2014. Against elections: The lottocratic choice functions under metric preferences. In Proc. 25th alternative. Philosophy & Public Affairs 42(2):135–178. Intl. Joint Conf. on Artificial Intelligence, 46–59. Landemore, H. 2013. Deliberation, cognitive diversity, and Anshelevich, E.; Bhardwaj, O.; and Postl, J. 2015. Approx- democratic inclusiveness: An epistemic argument for the imating optimal social choice under metric preferences. In random selection of representatives. Synthese 190(7):1209– Proc. 29th AAAI Conf. on Artificial Intelligence, 777–783. 1231. Anshelevich, E. 2016. Ordinal approximation in matching Merrill, S., and Grofman, B. 1999. A unified theory of vot- and social choice. ACM SIGecom Exchanges 15(1):60–64. ing: Directional and proximity spatial models. Cambridge University Press. Arrow, K. 1951. Social Choice and Individual Values.Wi- ley. Moulin, H. 1980. On strategy-proofness and single peaked- ness. Public Choice 35:437–455. Barbera,` S.; Gul, F.; and Stacchetti, E. 1993. Generalized Procaccia, A. D., and Rosenschein, J. S. 2006. The dis- median voter schemes and committees. Journal of Economic tortion of cardinal preferences in voting. In Proc. 10th Intl. Theory 61:262–289. Workshop on Cooperative Inform. Agents X, 317–331. Barbera,` S. 2001. An introduction to strategy-proof social Procaccia, A. D. 2010. Can approximation circumvent choice functions. Social Choice and Welfare 18:619–653. Gibbard-Satterthwaite? In Proc. 24th AAAI Conf. on Arti- Black, D. 1948. On the rationale of group decision making. ficial Intelligence, 836–841. J. Political Economy 56:23–34. Reilly, B. 2002. Social choice in the South Seas: Electoral Black, D. 1958. The Theory of Committees and Elections. innovation and the Borda count in the Pacific Island coun- Cambridge University Press. tries. International Political Science Review 23(4):355–372. Boutilier, C., and Rosenschein, J. S. 2016. Incomplete Richards, D.; Richards, W. A.; and McKay, B. 1998. Col- information and communication in voting. In Brandt, F.; lective choice and mutual knowledge structures. Advances Conitzer, V.; Endriss, U.; Lang, J.; and Procaccia, A. D., in Complex Systems 1:221–236. eds., Handbook of Computational Social Choice. Cam- Satterthwaite, M. A. 1975. Strategy-proofness and Arrows bridge University Press. chapter 10, 223–257. conditions: Existence and correspondence theorems for vot- Boutilier, C.; Caragiannis, I.; Haber, S.; Lu, T.; Procaccia, ing procedures and social welfare functions. Journal of Eco- A. D.; and Sheffet, O. 2015. Optimal social choice func- nomic Theory 10:187–217. tions: A utilitarian view. Artificial Intelligence 227:190–213.
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