Misrepresentation in District Voting

Transcription

1 Proceedigs of the Twety-Fifth Iteratioal Joit Coferece o Artificial Itelligece (IJCAI-6) Misrepresetatio i District Votig Yoram Bachrach Microsoft Research UK yobach@microsoft.com Omer Lev Uiv. of Toroto Caada omerl@cs.toroto.edu Yoad Leweberg Hebrew Uiv. of Jerusalem Israel yoadlew@cs.huji.ac.il Yair Zick Caregie Mello Uiv. USA yairzick@cs.cmu.edu Abstract Votig systems i which voters are partitioed to districts ecourage accoutability by providig voters a easily idetifiable district represetative, but ca result i a selectio of represetatives ot represetative of the electorate s prefereces. I some cases, a party may have a majority of the popular vote, but lose the electios due to districtig effects. We defie the Misrepresetatio Ratio which quatifies the deviatio from proportioal represetatio i a district-based electio, ad provide bouds for this ratio uder various votig rules. We also examie probabilistic models for electio outcomes, ad provide a algorithm for approximatig the expected Misrepresetatio Ratio uder a give probabilistic electio model. Fially, we provide simulatio results for several such probabilistic electio models, showig the effects of the umber of voters ad cadidates o the misrepresetatio ratio. Itroductio A votig system is a method by which voters choose betwee several alteratives ad their opiios are aggregated, ultimately choosig a wier (or wiers). Democratic coutries, i priciple, aim to have a represetative outcome, by havig a legislature roughly represetative of the public s beliefs, ad i some coutries, by havig the chief executive elected directly by the public. However, may democracies use a district based system for the selectio of their legislature (most promietly, the Westmister system ad the US system). I district based schemes, voters are divided ito geographically based districts, ad each oe selects a represetative to the legislature. The selectio mechaisms differ: Westmister ad US systems use plurality, Frace uses plurality with ruoff, while Australia uses STV. Typically, cadidates i each costituecy are members of political parties, ad some systems have the majority party form the executive (others, such as the US, have a similar process for selectig the chief executive). While we use the term district, other terms iclude electoral district (US), ridig (Caada), ad costituecy (UK). Sigle-member district electios provide voters a sigle ad easily idetifiable district represetative, ecouragig service ad accoutability. However, the proportio of seats i the legislature belogig to a party may be very differet from the proportio of voters supportig that party i the overall populatio; this is kow as the referedum paradox [Nurmi, 999]. The disparity betwee the popular vote ad the district vote has bee a source of cotetio i US electios; by redistrictig costituecies ( gerrymaderig ), political parties have maipulated the electios [Issacharoff, 00; Erikso, 97]; ideed, the US Voter Rights Act of 965 icludes several provisios that require chage i cogressioal districts i several states to be approved by federal authorities [Schuck, 987]. Moreover, such a discrepacy is caused ot oly by gerrymaderig, but is built ito district-based mechaisms. I the US presidetial electio of 876, the losig cadidate, Samuel Tilde, got 6% more votes tha the wier, Rutherford Hayes, a occurrece that happeed twice sice. The electoral college, through which the presidet is elected, displays the problem eve more acutely; i the US 99 presidetial electio a cadidate that garered 8.9% of the vote, Ross Perot, was ot represeted at all. I the UK system, i 95 the Coservative party lost the popular vote to the Labour party while still wiig a strict parliametary majority; i geeral, while o party has received a majority of the popular vote i a British electio sice 93, all but two electios resulted i a parliametary majority for oe of the parties. Similar scearios have ufolded i Caada, Australia ad elsewhere. Such problems may also arise i ay multi-level decisio makig process. So, if a orgaizatio or a sesor aalysis system (e.g., a automated car), attempts to decide o its ext step based o iputs from sub-systems employig their ow decisio makig processes (e.g., each sesor family is a district, ad sesors are votig betwee themselves), they may also ecouter such a problem, as a small amout of sigals may cause the system to reach a wrog outcome. Cosider the followig example. Two political parties, A ad B, compete i seve districts of equal size, D,D,...,D 7. Both parties ru a cadidate i every district, ad the plurality votig rule is used to determie the wier. Now, suppose that i districts D,...,D 4, 60% of the vote goes to the represetative of party A; o the other 8

2 had, i districts D 5,D 6 ad D 7 the represetatives of party B take 00% of the vote. The fial vote tally shows party A wis the electio, eve though party B has early twice as may voters! As the example above shows, a miority (those i favor of party A) may rule over the majority (those i favor of party B). We refer to the outcome i this case as a misrepresetatio of the voters prefereces. District based electoral systems ted to be more stable : they ted to result i a smaller umber of cadidates, ad thus less fragmetatio of the parliametary body. However, misrepresetatio is a iheret byproduct of electoral stability; ideed, stability comes at the heavy price of potetially overridig the prefereces of most voters. Oe way to quatify the degree to which a system skews the true desires of the voters as captured by the total support for each party is to examie the ratio betwee the umber of those who voted for the most popular party i geeral, ad the umber of voters who voted for the wier. Whe more people voted for the losig party tha for the wiig party this ratio is larger tha oe. The higher this ratio is, the more proouced the misrepresetatio effect. Our Cotributio We examie the issue of misrepresetatio due to district-based systems uder several promiet votig rules. We first provide a metric for quatifyig the misrepresetatio effect, both for electios with two parties ad for electios with more tha two parties, which we call the Misrepresetatio Ratio. We provide bouds o the Misrepresetatio Ratio for several votig rules, depedig o the umbers of voters, cadidates ad districts. Fially, we provide some simulatio results regardig the misrepresetatio ratio i certai scearios, to examie its values uder various settigs. Related Work Votig district represetability has bee the topic of much public debate ad research for two ceturies, sice the advet of redistrictig i the US ( gerrymaderig ), ad i the UK, redistrictig due to a attempt to follow populatio chages i successive reform bills. Most of the research o these issues focused o historical [Egstrom, 006; Butler, 99], sociological [Lubli, 997] ad political sciece [Erikso, 97] issues. Particularly sice the Votig Rights Act of 965, much legal research has also dealt with district based system, though usually focusig o particular coutries (maily though ot solely the US) [Schuck, 987; Issacharoff, 00]. Political sciece research o misrepresetatio mostly focused o its possible occurrece, history or statistical aalysis [Feix et al., 004; Miller, 0; 04; Lahrach ad Merli, 00; Grofma et al., ; Tagia, 00; Felsethal ad Miller, 05]. Some research defies voter misrepresetatio as distace of parliametary seat allocatio compared to pure represetatioal settigs [Gallagher, 99; Felsethal ad Machover, 999; Gelma et al., 00; Feix et al., 008], usig cocepts such as the Bazhaf idex. Maipulatio by voters is well studied [Xia et al., 009; Zuckerma et al., 008; Procaccia et al., 007], as opposed to istitutioal maipulatio. Our work is similar to votig cotrol problems (see [Hemaspaadra et al., 007] ad the survey by [Faliszewski et al., 00]), icludig prelimiary work o dividig voters ito groups [Erdélyi et al., 05]. Optimal gerrymaderig strategies have bee studied i [Owe ad Grofma, 988; Friedma ad Holde, 005]; however, these have mostly focused o party scearios, as i the US. I establishig our bouds, we ecouter the problem of fidig the miimal score a cadidate ca wi with, give a scorig rule; Xia [0] studies a similar problem, but his results do ot directly apply to our settig. Our problem is related to the bi packig problem with a costat umber of bis. This problem was believed to be NP-hard, but recet advaces [Stille, 008; Jase et al., 03] established that it is i P [Goemas ad Rothvoß, 04]. Prelimiaries We have a set of voters, V, ad each voter i V has a preferece order (without ties) over the cadidate set C, deoted i. For every c, c 0 C, we say that i prefers c to c 0 if c 0 i c. We deote the set of liear prefereces over C as L(C). A votig rule is a fuctio f : L(C)! C, whose iput is a fiite list of liear prefereces over C (a profile) of size (most votig rules are well defied for ay >0) ad whose output is a cadidate c C. Votig rules are ofte assumed to output a set of cadidates, but sice i our settig we are oly iterested i the wiers of the electio, the output of f is a sigle cadidate (f may icorporate some fixed tie-breakig rule). Sice the set of voters ad their prefereces is costat throughout the paper, we overload otatio ad defie f over subsets of V : give a set Q V, we let f(q) be the output of f over the prefereces of the voters i Q. A votig rule is eutral if the outcome is ivariat uder cadidate ame chages. More formally, a votig rule f is eutral if for ay two cadidates c ad c 0, if every voter i V ow raks c i the positio of c 0 ad vice versa, the outcome of f is uchaged if the wier was either c or c 0, is c 0 if the wier was c, ad is c if the wier was c 0. We say a votig rule f is score-mootoe if it iduces a score for every cadidate c C (i.e., each cadidate eds up with some real umber, ad the oe with the maximal oe is chose), ad the followig holds: give two preferece profiles R =( i) iv ad R =( i ) iv, if for all 0 0 a, b C \{c} we have a i b () a i b, ad uder R, c is i a positio that is o lower tha its positio i R, the c s score uder R is at least as high as its score i R. These properties hold for the commo votig rules which will be used i this paper. Scorig Rules: A scorig rule for m cadidates is defied by a vector of scores, (,..., m ), where m =0, > 0. Give a voter i V, let rak i (c) be the rak of cadidate c i the preferece order of i. Give a set of voters Q V, let score c (Q) = P iq rak i(c). Give a set of voters Q V, the output of Scorig (Q) is a member of arg max cc score c (Q) (if there are ties, we break them accordig to some tie-breakig rule). May scorig rules are widely used. For example, the plurality rule, where =ad j =0for all j, the veto rule, where j =for all j<m, ad m =0, both of which are specific Assumig m =0is o loss of geerality: if m > 0 score vector ca be ormalized by settig j = j m. 8

3 examples of the family of k-approval scorig rules, i which =... = k =ad k+ =... = m =0. Aother commo scorig rule is the Borda rule, where j = m j for all j {,...,m}. Copelad Rule: Give a set of voters Q V, we say that a cadidate c beats c 0 i a pairwise electio uder Q if a majority of the members of Q prefer c to c 0. For each cadidate c C, we let score Cp c (Q) to be the umber of cadidates that c beats i a pairwise electio, mius the umber of cadidates that beat c. The Copelad votig rule outputs Copelad(Q) = arg max cc score Cp c (Q). We ote that both scorig rules ad Copelad have a atural otio of a cadidate score, ad were coceptualized as such i [Bartholdi et al., 989]. This will be istrumetal i defiig votig misrepresetatio, as we do i the followig sectio. Defiitio.. Give a votig rule f which iduces a cadidate score, we let score f (c, Q) to be the score of c C uder f, whe the voter set is Q V. 3 The Misrepresetatio Ratio We are iterested i settigs where the set of voters is partitioed ito districts: these are z disjoit sets D,...,D z whose uio is V. The electio wier uder this model is determied by applyig the votig rule f to D,...,D z ; the cadidate who wis the greatest umber of districts is the wier (ties are broke usig some tie-breakig rule). Defiitio 3.. Give a partitio of voters ito z districts D = {D,...,D z }, let w be the wier of the electio whe voters are partitioed ito districts as per D. The misrepresetatio ratio is the ratio betwee the maximum score of ay cadidate uder f, ad the score of w. Formally: MR(V,D,f)= max cc score f (c, V ). score f (w, V ) Note that MR(V,D,f) ; if MR(V,D,f) = the the wier of the district electios completely captures the popular vote (as measured by f). The higher MR(V,D,f), the less popular the wiig cadidate is i the eyes of the people; thus, district electios with a high misrepresetatio ratio are oes where voters prefereces are ot appropriately aggregated, due to the effects of district electios. Remark 3.. I this work we assume all districts have the same umber of voters. Without ay assumptios o the umber of voters i each district, the worst case misrepresetatio ratio (Defiitio 3.) ca be arbitrarily high: cosider ` + districts. Suppose ` +of the districts have oly oe voter, ad ` of them have M voters, where M is a very large umber. There are two cadidates, A ad B; cadidate A wis all votes i the ` districts where there are M voters, ad cadidate B wis all votes i the ` +districts holdig oe voter. Thus, the total umber of votes for B is O(`), ad the total umber of votes for A is O(M`), resultig i a arbitrarily high misrepresetatio. Our results ca be exteded by icorporatig a additioal parameter: max D,D0 D D D 0 ; however, i the iterest of space ad clarity, we assume that districts are of equal size. 4 Bouds o the Misrepresetatio Ratio I what follows, we establish upper bouds o the worst-case misrepresetatio ratio; we show that our bouds are tight, i.e. that there exist district electios where the boud holds. Furthermore, we always refer to the district votig istace hv,c,( i) iv, D,fi as oe that maximizes MR, where C = m, D = z, for all D D: D =, ad f is some eutral score-mootoe rule. We assume that w C is the wier of the district electios; that is, w wo a plurality of the districts. We mark d(c) for c C as the umber of districts wo by cadidate c. We assume that i the case of a tie for first place i a district, ties are broke agaist the district electio wier; moreover, the district electio wier must wi a strict plurality of the districts. Ituitively, i order to establish our boud, we wat to create the worst possible electio. Such a electio would have the cadidate w wi by as small a margi as possible, with some other cadidate p 6= w beig as popular as possible while losig a plurality of the districts. Give a score iducig votig rule f, we let L (f) be the maximal score that a cadidate ca get while still losig a voter electio, ad M (f) be the maximal score that a cadidate ca get ad wi a voter electio. For example, L (Plurality) =, ad M (Plurality) =. The followig lemma (whose proof is i the appedix) offers some isights regardig the umber of districts wo by each cadidate i a istace maximizig MR. Lemma 4.. If f is a score-mootoe, eutral rule the there is a district votig istace maximizig MR such that:. if w does ot wi the electio i a district D, the w is raked last by all voters i D.. If the cadidate p wis i D, the it is raked first by all voters i D; if ot ad the wier is ot w, the the score of p from D is L (f). 3. d(p) =d(w) if m> or z is odd, otherwise d(p) = d(w). 4. Scorig Rules We begi our ivestigatio by boudig MR whe f is a scorig rule. We begi with a simple techical lemma. (proof i the appedix). Lemma 4.. For every scorig rule f = Scorig, where =(,..., m ), if >, L (f) = ( ) + ( + ). If =, deote 0 as the maximal i such that i <, the L (f) =( ) + 0. Corollary 4.3. If f = Scorig the (z+)l (f) for ay z>. M (f) Proof. Give a vector = (,..., m ) we have that M (f) =. Now, by Lemma 4., we kow that L (f), from which the claim trivially follows. The followig lemma (full proof i the appedix) tells us the umber of districts that must be wo by the wiig cadidate i a district electio that maximizes MR. 83

4 Lemma 4.4. Let hv,c,( i) iv, D,fi be a district votig istace which maximizes MR, where f is a scorig rule. z (umber of districts) ca be writte as `m+r (`, r N[{0} ad r<m). There is a MR maximizig istace where the umber of districts that w wis is at most ` +, ad every other cadidate wis at least ` districts. Proof Sketch. Markig as x w the score w receives i districts wo by w, ad x p the score p receives i those same districts, we observe that MR ca be writte as x p + M (f) L (f) + x w (z +)L(f) M(f) ; () x wd(w) Takig a derivative of () o d(w) shows that () is maximized whe d(w) is miimized, from which the lemma is derived. The followig lemmata (proofs i the appedix) discuss the score that w receives i districts that it wis. Lemma 4.5. For every scorig rule f, if z, z>3 (i.e., umber of voters i each district is far larger tha the umber of districts), the score give to w i districts i which it wis must be the miimal possible score eeded to wi a district i a istace maximizig MR. 3 Lemma 4.6. For k-approval votig rules, the miimal score a wier ca get i a electio is k m +if k mod m. Otherwise, it is k m +. Armed with Lemmata , we ow proceed to aalyze specific votig rules. Theorem 4.7. Suppose that = qm + s ad z = `m + r, where q, s, `, r N [{0} ad s, r < m; the MR(V,D, Plurality) is at least (z +)( ) q + (` +)(q +) ad at most + + q + + (z +)( ). (` +)(q +) I particular, MR(V,D, Plurality) is i (m ). Proof. By Lemma 4.4 ad Corollary 4.3, we kow that w wis either ` +or ` +districts. Pluggig i the values L (Plurality) = ad M (Plurality) = ito (), we have that MR(V,D, Plurality) equals x p + + (z + )( ) + x w x w d(w) (x w ad x p deote the score of w ad p, respectively, whe w wis a district). We are left just with determiig the value of x p ad x w. For reasos similar to the oes detailed i Lemma 4., it holds that x p = x w. Due to Lemma 4.5, MR(V,D, Plurality) is maximized whe x w is miimal; We have that w receives q +votes if s, ad q +otherwise. Pluggig this ito () we obtai the desired result. 3 This is ot trivial, as possible to icrease w s score to allow p to receive a higher score as a d place cadidate. The secod expressio i the upper boud of Theorem 4.7, d e+ q+ ca be upper bouded by q+, which is at most m. Thus, if the umber of voters domiates the umber of cadidates, this expressio has little effect o MR. The secod expressio ca be upper bouded as follows (z +)( ) z apple (` +)(q +) (` +)(q +) apple m. A similar lower boud of m which cocludes the proof. ca be show as well, Note that tightess is achieved as our costructed expressios were depedet o particular votig profiles (as described i Lemma 4.), ad hece carry o to these expressios. As some parliametary systems require ot a plurality of districts to become a wier, but a majority, we also ote the MR i these cases (the proof ca be foud i the appedix). Corollary 4.8. If the umber of districts eeded for a victory is above 50%, MR for plurality is (m). The followig theorem (proof i the appedix) discusses the MR for k-approval scorig rule. Theorem 4.9. Suppose that the umber of districts is expressed as z = `m + r, where `, r N [{0} ad r<m; the MR(V,D, k-approval) for k> is at least (z + ` +) (z ` +) + (` +)( k m +) ad at most (z + ` +) (z `) + (` +)(. k m +) m I particular, MR(V,D, k-approval) is i k. Corollary 4.0. MR(V,D, Veto) is i (m). Oe of the mai challeges i computig a closed form formula for MR for geeral scorig rules is that oe must first decide what is the miimal score that w ca obtai while wiig a district for a give score vector. This problem ca be thought of as a bi packig problem: cadidates ca be thought of as bis, ad the scores must be packed ito them. It is oly recetly that a polyomial time algorithm has bee proposed for bi packig problems with a costat umber of types (also commoly referred to as the oe-dimesioal cuttig stock problem) [Goemas ad Rothvoß, 04]. Thus, for geeral scorig rule we offer looser bouds o the umber of votes eeded to wi: Let S = P m j= j; the the miimal umber of votes eeded to wi a district is at most S +, ad at least m S m : we allocate the scores as evely as possible amog the cadidates, ad break the tie i favor of the wier usig at most poits. Of course, i some cases this ca be improved, but it depeds o, ad o the divisibility of S ad m. The followig theorem (proof i the appedix) uses these loose bouds to boud MR for the Borda scorig rule. Theorem 4.. MR(V,D, Borda) is i (m ). 84

5 Algorithm Mote-Carlo MR Approximatio : procedure EXPECTED-MR ( M, B, ", ) : s = l B l " m 3: T =0 4: for i =to s do 5: Sample a electio outcome E from M 6: w = arg max cc score(c, V ) // The wier 7: s m = max cc score(c, V ) // Maximal score = sm s w // sampled MR 9: T = T + R samp i 0: retur ˆr = T s // average of sampled MRs 8: R samp i 4. Copelad Whe usig the Copelad votig rule, oe ca get a udefied value for MR, as a score of 0 is possible for the wier. Example 4.. Let us have two idetical districts, each cotaiig voters with the preferece w p a ad 0 voters who have p w a. A third district cotais 4 voters with the preferece p w a. w wis the first districts, becomig the ultimate wier. But, lookig globally, p s Copelad score is, while w s Copelad score is 0, makig MR(V,D, Copelad) udefied. The Copelad score ca be additively adjusted by addig to each cadidate s score a fixed amout that is larger tha m. However, Copelad s performace remais bad, as is captured by Theorem 4.3 (see appedix for proof). Theorem 4.3. Uder Copelad, the wier w may have the worst possible Copelad score, while aother cadidate has the best possible Copelad score. 5 The Misrepresetatio Ratio Uder Ucertai Votes I Sectio 4 we established bouds o the misrepresetatio ratio by costructig pathological examples: settigs where districtig effects were so proouced as to cause a extremely upopular cadidate to wi the electios, despite the existece of a clearly better alterative. We ow take the average-case, rather tha the worst-case, approach, ad ask how commo are istaces where misrepresetatio is high. We do this i the form of a probabilistic geerative model, utilizig partial iformatio to iform our assumptios o the geeral populatio. Ay istatiatio of the model is a votig domai for which we ca compute the misrepresetatio ratio. Thus, MR is a radom variable ad we evaluate the expected MR. A aive solutio is to exhaustively search over the space of possible electio outcomes; for each such outcome we ca compute its probability of occurrig uder the geerative model, ad the MR value for that outcome; we ca the sum the product of the two across all outcomes to get the expected MR. However, such a exhaustive search is itractable, as the space of outcomes ca be prohibitively large, especially whe there are may cadidates, voters ad districts. We propose a alterative approximate solutio, based o a Mote-Carlo algorithm. Our algorithm requires a boud.4..8 Chart Title Ɛ Figure : The effects of the oise o the MR o the MR for the examied domai, ad such ca be foud for may cases (see previous sectio). We assume the geerative model is give i the form of a black-box, which outputs a sampled electio outcome, cosistig of the votes of every voter i every district. We further assume that the wier of the electio ca be computed i polyomial time. 4 Deote the geerative model as M, ad by r = E M (MR) the expectatio of MR uder the model M. Our proposed algorithm is probably approximately correct : give two parameters, " ad, the algorithm returs a approximatio ˆr to r, such that with high probability the retured value ˆr is very close to r, so that ˆr r apple". The ruig time of the algorithm depeds o " ad ; it is quadratic i " ad logarithmic i. Our proposed algorithm is a Mote-Carlo algorithm, but it is oly appropriate to votig rules where there is a kow boud o the possible MR values. 5 The miimal MR value is (as this is the ratio betwee the maximal score of ay cadidate ad the score of a specific cadidate, amely the wier). Give a boud H o the maximal MR i a domai, we refer to the MR value rage as B = H. The rutime of our algorithm is quadratic i B. The method is give i Algorithm, ad we provide a proof for its correctess. Theorem 5.. The value retured by Expected-MR is a ", approximatio for the expected MR uder M: with probability at least the retured value ˆr is withi a distace " of r = E M (MR), i.e: ˆr r apple". Proof. We ote that the R samp i computed iside the loop is the MR i a specific istatiatio of a electio outcome E sampled from the geerative model M (see Defiitio 3.), so each R samp i is a radom variable, whose expectatio is r = E M (MR) (i.e. E[R samp i ]=r). Our algorithm computes T = P s i= Rsamp i, the sum of s i.i.d draws, each of which has a value of r i expectatio, so E[ˆr] =E[ T s ]=r. We use Hoeffdig s iequality [Hoeffdig, 963] to show that the umber of samples s that we use achieves the desired accuracy " ad cofidece. Hoeffdig s iequality states that if R,...,R are idepedet radom variables, where each 4 Not all votig rules admit a polyomial wier determiatio algorithm. As our algorithm samples electio outcomes, its rutime i this case would ot be polyomial. 5 Our method is aki to Mote-Carlo methods used for aalyzig votig uder various forms of ucertaity [Fatima et al., 008; Bachrach et al., 00; 0; Bachrach ad Shah, 03]. 85

6 Voters i every district Voters i every district Chart Title Series3000 Series Series34000 Series4 Series55000 Voters i every district (a) MR where the prefereces are uiformly draw, uder Borda scorig rule. (b) MR where the prefereces are draw (c) MR where the prefereces are draw by a Mallows model, uder plurality. by a Mallows model, uder Copelad votig rule. Figure : Simulatios results R i is bouded so that R i [a i,b i ], ad if T = P s i= R i, s the Pr( T E[T ] s") apple exp P ". i= (bi ai) is the MR i a electio outcome obtaied uder the geerative model M, so R samp i is bouded i the rage [,H] (i.e., by our assumptio, the MR value rage is B = H ). Applyig Hoeffdig s boud ad substitutig B l " for s, we get Pr apple as required. R samp i T s 6 Simulatios E[T ] r " We ow use our Algorithm Expected-MR to aalyze the MR i several votig domais. We begi with a oisy versio of the example domai described i the itroductio. We fix C = {w, p}, ad the umber of districts to : 6 districts of type A ad 5 of type B, modelig heterogeeous ad homogeous districts respectively. I type A districts, every voter v votes radomly with Pr[v votes for w] = + ", ad Pr[v votes for p] = "; i type B districts, Pr[v votes for w] =", Pr[v votes for p] = ", for " 0,. Figure shows the averaged MR as a fuctio of the oise " (x-axis). Each poit is average MR obtaied for may electios sampled usig this probabilistic model; MR first icreases as the oise " grows, util it reaches a sweetspot from which it drops. This idicates that for some models the model oise may ot have a mootoe effect o the MR. I our secod experimet, we fix the umber of districts to 5, ad rage the umber of voters i every district from 00 to We examied MR of electios with m {3, 4,...,7} cadidates. Figure a shows the averaged MR where the preferece of every voter i every district is uiformly draw from the set of all m! possible orders of cadidates, uder Borda scorig. The figure shows that icreasig the umber of voters teds to lower MR. This is ot surprisig, as all cadidates will likely have early the same score. Next, we cosider the Mallows model [Mallows, 957] for geeratig voter prefereces, where we assume that there is a groud truth for every district, (represetig the commo rakig of cadidates i that district) ad dispersio parameter,. Uder the Mallows model every voter compares every pair of cadidates idepedetly ad raks them correctly (accordig to ) with probability. For every district, was draw uiformly at radom ad U(, ). We used the plurality scorig rule (Figure b) ad Copelad votig rule (Figure c). As predicted by our theoretical results, MR grows whe there are more cadidates. Uder Copelad, a fixed amout of m was added each cadidate s score so that MR would be positive. Our simulatio results idicate that votig misrepresetatio may occur i several atural domais. Our secod experimet is fair i the sese that there is o preferred cadidate, ad yet the MR values are quite high. Also, our theoretical results agree with experimets i some atural domais. 7 Coclusios This work aalyzes district-based electios. We demostrate the represetability issues that arise i such electios, ad show tight bouds o misrepresetatio. We further show that misrepresetatio is a commo occurrece uder various atural voter distributios, ad that its effect may ot dimiish eve whe the umber of voters is large. District based electios ted to uder-represet smaller parties; this is a log observed pheomeo (ad, i some coutries, a welcome stabilizig feature). However, we do ot focus o smaller parties, but rather show that the prefereces of large majorities may be completely urepreseted (the UK electios of 95, where the Labour party with 48.8% support lost the electios, ad the Coservative party with 44.3% ot oly beat it, but had a strog parliametary majority, is just oe example of these occurreces). Research ito the istitutioal bias i votig procedures, beyod cotrol issues, is oe which we thik deserves more attetio by the computatioal social choice commuity. Very few electio systems i the world are proportioal, ad the effect this has o the expressio of voters views has mostly focused (i political sciece research) o how small miorities are hurt. As our aalysis shows, large majorities may also be affected. Further research is eeded with regards to other votig methods. Moreover, further ad complemetary cocepts may be developed, idicatig ufairess, lack of represetatio ad other problems with various votig procedures (parliametary etrace bouds, commo i some coutries, are a obvious cadidate for such directios). I additio, while we have focused o a outcome of a sigle wier, a coalitioal aalysis of district settigs may also be of iterest. 86

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