Common Voting Rules s Mximum Likelihoo Estimtors Vinent Conitzer Computer Siene Deprtment Crnegie Mellon University 5000 Fores Avenue Pittsurgh, PA 1513 onitzer@s.mu.eu Astrt Voting is very generl metho of preferene ggregtion. A voting rule tkes s input every voter s vote (typilly, rnking of the lterntives), n proues s output either just the winning lterntive or rnking of the lterntives. One potentil view of voting is the following. There exists orret outome (winner/rnking), n eh voter s vote orrespons to noisy pereption of this orret outome. If we re given the noise moel, then for ny vetor of votes, we n ompute the mximum likelihoo estimte of the orret outome. This mximum likelihoo estimte onstitutes voting rule. In this pper, we sk the following question: For whih ommon voting rules oes there exist noise moel suh tht the rule is the mximum likelihoo estimte for tht noise moel? We require tht the votes re rwn inepenently given the orret outome (we show tht without this restrition, ll voting rules hve the property). We stuy the question oth for the se where outomes re winners n for the se where outomes re rnkings. In either se, only some of the ommon voting rules hve the property. Moreover, the sets of rules tht stisfy the property re inomprle etween the two ses (stisfying the property in the one se oes not imply stisfying it in the other se). 1 Introution Voting is very generl metho for ggregting multiple gents preferenes over set of lterntives, This mteril is se upon work supporte y the Ntionl Siene Fountion uner ITR grnts IIS-011678 n IIS-07858, n Slon Fellowship. Tuoms Snholm Computer Siene Deprtment Crnegie Mellon University 5000 Fores Avenue Pittsurgh, PA 1513 snholm@s.mu.eu suh s potentil presients, joint plns, llotions of goos or resoures, et.. As suh, it is topi of signifint n growing interest in the AI ommunity with pplitions in ollortive filtering [13], plnning mong utomte gents [10, 11], etermining the importne of we pges [1], to nme few. Reent AI reserh hs stuie the omplexity of exeuting voting rules [7], the omplexity of mnipulting eletions [3,, 5], s well s effiient eliittion of the voters preferenes [, 6]. In this pper, we stuy how voting n e interprete s mximum likelihoo estimtion prolem. We elieve this will she new light on, n enhne the ppliility of, oth voting n mximum likelihoo tehniques. To see how voting my e interprete s mximum likelihoo prolem, let us first ontrst the following two views of voting: 1. The voters preferenes over nites re iiosynrti, n there is no sense in trying to moel where they me from. The purpose of voting is solely to fin ompromise nite tht mximizes the omine welfre of the gents, either expliitly or implitly tring off gents utilities ginst eh other.. There is some solute sense in whih some nites re etter thn others, whih is prior to n not epenent on the gents preferenes. Rther, the gents preferenes re merely their noisy estimtes of this solute qulity. The purpose of voting is to infer the nites solute gooness se on the gents noisy signls, i.e., their votes. For the purposes of this pper, we will e onerne with the seon interprettion (whih is espeilly sensile in ontexts suh s oument seletion). It is instrutive to visulize this interprettion s the simple Byesin network given elow. In this network, the oserve vrile is the gents votes, n we seek to estimte the orret outome. The most nturl wy of oing so is to tke the mximum likelihoo estimte of
"orret" outome gents votes the orret outome. 1 Whih outome is hosen s the mximum likelihoo estimte (MLE) epens on the onitionl proility tle for the gents votes noe tht is, it epens on the noise moel. Eh noise moel genertes mximum likelihoo estimtor funtion from gents votes to outomes, n this funtion onstitutes voting rule. The si ie of using n MLE pproh to voting ws introue s erly s the 18th entury, y Conoret [8]. Conoret stuie prtiulr noise moel in whih voter rnke two nites orretly with some given proility p>1/. Conoret solve the ses of n 3 nites. The solution for ritrry numers of nites ws given two enturies lter y Young [15]: he showe tht it oinie with voting rule propose y Kemeny [1]. A slightly extene moel where p is llowe to inrese with the istne etween two lterntives in the orret rnking of lterntives, n the rules tht this proues, hs lso een stuie [9]. However, none of these rules orrespon to ny of the ommonly use voting rules. Shoul the ft tht we o not know how to interpret the ommon voting rules s mximum likelihoo estimtors e viewe s ritiism of the ommon voting rules? This is perhps somewht unfir, euse we my simply not yet hve stuie the right noise moels to generte these rules. Nevertheless, there is no gurntee tht suh noise moels exist. Moreover, it is useful to tully know noise moel for rule: it inreses our unerstning of the rule, it llows us to question the ssumptions in the noise moel (n therefore in the rule), n, where we isgree with these ssumptions, we n moify the noise moel to otin rule tht is more pproprite for our nees. In this pper, we ress these issues y turning the tritionl pproh on its he, y sking: For whih ommon voting rules oes there exist noise moel suh tht the rule is the mximum likelihoo rule for tht istriution? Perhps surprisingly, if we ssume 1 A Byesin (mximum posteriori) interprettion is, of ourse, lso possile: Byes rule gives P (orret outome gents votes) = P (orret outome)p (gents votes orret outome)/p (gents votes). If the istriution over orret outomes is uniform, this expression is mximize y the mximum likelihoo estimte. This pproh les to inonsistent ylil rnkings (e.g. ) with nonzero proility, ut this oes not ffet the mximum likelihoo pproh. tht the voters votes re inepenent given the orret outome, it turns out tht only some rules hve this property. Answering our question hs t lest the following purposes: Rules tht hve the property re in sense more nturl thn ones tht o not, espeilly in settings where informtion ggregtion is the min purpose of voting. Hene, nswering the question will provie some guine in the prolem of hoosing rule. Showing tht rule oes hve the property requires us to onstrut noise moel, for whih the rule is the mximum likelihoo estimtor. Susequently, we n ssess whether the noise moel is resonle, or nees to e moifie to eome resonle. In the ltter se, the moifie noise moel my le to novel n useful voting rule. The rest of this pper is orgnize s follows. In Setion, we efine the ommon voting rules n efine the types of noise moel tht we onsier. In Setion 3, we present our positive results (noise moels tht hve ommon voting rules s their mximum likelihoo estimtor). In Setion, we present tehnique for showing tht no noise moel hs given rule s its mximum likelihoo estimtor, n pply this tehnique to otin our negtive results. Definitions.1 Voting rules We hve set of nites (k. lterntives) C over whih the voters vote. A vote is efine s strit orering (rnking) of the nites in C. A(voting) rule tkes the (vetor of) votes s input, n proues n outome. This outome n either e single nite (the winner), or rnking of the nites (where the top-rnke nite is the winner). Most rules llow for the possiility of ties, n o not speify how ties shoul e roken. In this pper, for our positive results, we o not ttempt to relize ny prtiulr tie-reking rule; our negtive results hol regrless of how ties re roken. We next efine the ommon voting rules tht we stuy. soring rules. Let α = α 1,...,α m e vetor of integers suh tht α 1 α... α m. For eh voter, nite reeives α 1 points if it is rnke first y the voter, α if it is rnke seon et. The sore s α of nite is the totl numer of points the nite reeives. The Bor rule is the soring rule with α = m 1,m,...,1,0. The plurlity rule (k. mjority rule) is the soring rule with α =
1, 0,...,0,0. The veto rule is the soring rule with α = 1, 1,...,1,0. Cnites re rnke y sore. single trnsferle vote (STV). The rule proees through series of m 1 rouns, eh one eliminting one nite. In eh roun, the nite with the lowest plurlity sore (tht is, the lest numer of voters rnking it first mong the remining nites) is rnke t the ottom of the remining nites. Then, tht nite is eliminte from the votes (eh of the votes for tht nite trnsfer to the next remining nite in the orer given in tht vote). The lst remining nite is the winner. Buklin. For ny nite n integer l, let B(, l) e the numer of voters tht rnk nite mong the top l nites. Cnite s Buklin sore is min{l : B(, l) >n/}), n nites re rnke oring to their sores (where lower sores re etter). Tht is, if we sy tht voter pproves her top l nites, then we repetely inrese l y 1, n whenever nite eomes pprove y more thn hlf the voters ( psses the post ), tht nite is ple next in the rnking. When multiple nites pss the post simultneously, ties re roken y the numer of votes y whih the post is psse. mximin (k. Simpson). Let N( 1, ) e the numer of votes tht rnk nite 1 higher thn nite. Cnite s mximin sore is min N(, ) (the nites worst sore in pirwise eletion). Cnites re rnke y sore. Copeln. A nite gins one Copeln point for every pirwise eletion it wins (one point for every suh tht N(, ) > N(,)), n loses one Copeln point for every pirwise eletion it loses (minus one point for every suh tht N(, ) < N(,)). Cnites re rnke y sore. rnke pirs. Sort ll orere pirs of nites (, ) yn(, ), the numer of voters who prefer to. Strting with the pir (, ) with the highest N(, ), we lok in the result of their pirwise eletion ( ). Then, we move to the next pir, n we lok the result of their pirwise eletion. We ontinue to lok every pirwise result tht oes not ontrit the orering estlishe so fr.. Noise moels In this pper, we will ple the following restritions on the noise moel. First, we require tht the noise is inepenent ross votes. Tht is, votes re onitionlly inepenent given the orret outome, s illustrte y the Byesin network elow. Seon, we require tht the onitionl istriution given the orret outome is the sme for eh vote vote 1 "orret" outome vote... vote n (for exmple, the first vote nnot e more likely to gree with the orret outome thn the seon vote). Together, the two restritions mount to the noise eing i.i.. These restritions strengthen our positive results tht show tht ertin rules n e interprete s mximum likelihoo estimtors. As for the negtive results tht ertin rules nnot e so interprete, it turns out tht without ny restritions on the noise moel, the question eomes trivil. Speifilly, if we o not mke the restrition tht votes re rwn inepenently given the orret outome, then ny rule is n MLE, s the following trivil proposition shows. Proposition 1 Any voting rule ρ n e interprete s mximum likelihoo estimtor (if we o not require tht votes re rwn inepenently given the orret outome). Moreover, if the rule is nonymous (it trets ll voters symmetrilly), then the noise moel n lso e nonymous (i.e. the onitionl istriution given the orret outome is the sme for eh vote). Proof: The following noise moel will suffie: given ny orret outome, let the proility of ll vote vetors on whih ρ proues n outome tht is ifferent from the orret outome e 0, n let the proility on ll other vote vetors e positive. We note tht if the rule is nonymous, then this is n nonymous noise moel. Hene, in the reminer of the pper, we mke the ove two restritions. We sy tht rule tht n e viewe s mximum likelihoo estimtor uner these restritions when the outome is winner is n MLEWIV (Mximum Likelihoo Estimtor for Winner uner I.i.. Votes) rule, n one tht n e viewe s mximum likelihoo estimtor uner these restritions when the outome is rnking is n MLERIV (Mximum Likelihoo Estimtor for Rnking uner I.i.. Votes) rule. 3 Voting rules tht n e interprete s MLEs In this setion, we ly out our positive results: we show whih of the ommon voting rules n e interprete s
mximum likelihoo estimtors uner the ssumption of i.i.. votes (oth for the se where the outome is the winner n the se where the outome is rnking). It turns out tht soring rules stisfy oth the MLEWIV n MLERIV properties. Theorem 1 Any soring rule is oth n MLEWIV n n MLERIV rule. n s(r j(w)) Proof: We first show tht every soring rule is n MLEWIV rule. If nite w is the winner in the orret outome, then let the proility (given the orret outome) of vote j tht rnks nite w in position r j (w) e proportionl to s(rj(w)) (where s(r) is the numer of points nite erives from eing rnke rth in vote). Thus, the proility of votes 1 through n given the( orret outome is pro- ) portionl to s(rj(w)) =. Of ourse, s(r j (w)) is extly nite w s sore. Now, the prolem of fining the mximum likelihoo estimte of the orret outome is the prolem of hoosing nite w to mximize the former expression. This is one y hoosing the nite with the highest sore n s(r j ()). Hene every soring rule is n MLEWIV rule. We next show tht every soring rule is lso n MLERIV rule. If the nites re rnke 1... m in the orret outome, then let the proility (given the orret outome) of vote j tht rnks nite i in position r j ( i ) e proportionl to m (m +1 i) s(rj(i)). Thus, the proility of votes 1 i=1 through n given the orret outome is proportionl ( to ) n m (m+1 i) s(rj(i)) = m s(r j( i)) (m+1 i). i=1 i=1 Of ourse, s(r j ( i )) is extly nite i s sore. Now, the prolem of fining the mximum likelihoo estimte of the orret outome is the prolem of leling the nites s i so s to mximize the former expression. Beuse m + 1 i is positive n eresing in i, this is one y leling the nite with the highest sore s(r j ()) s 1 (therey mximizing the numer of ftors m + 1 1 = m in the expression), the nite with the seon highest sore s, et. Hene every soring rule is n MLERIV rule. STV, on the other hn, stisfies only the MLERIV property (we will see lter tht it violtes the MLEWIV property). Theorem The STV rule is n MLERIV rule. Proof: Let the nites e rnke 1... m in the orret outome. Let δ j ( i ) = 1 if ll the nites tht re rnke higher thn i in vote j re ontine in { i+1, i+,..., m }, n let δ j ( i )=0 otherwise. Then, let the proility of vote j e proportionl to m k δj(i) i, where, for every i, 0<k i <1, i=1 n k i+1 is muh smller thn k i. Thus, the proility of votes 1 through n given the orret ( outome ) δ j( i) is proportionl to n m k δj(i) i = m ki. i=1 i=1 Agin, the prolem of fining the mximum likelihoo estimte of the orret outome is the prolem of leling the nites s i so s to mximize this expression. To o so, we must first minimize the numer of ftors k m, euse these ftors ominte. The numer of suh ftors is δ j ( m ), whih is the numer of votes tht rnk the nite tht we lel m first. Hene, we must rnk the nite tht is rnke first the fewest times lst (s m ). Next, we must minimize the numer of ftors k m 1. The numer of suh ftors is δ j ( m 1 ), whih is the numer of votes tht rnk the nite tht we lel m 1 either first, or seon fter m. Hene, we must rnk the nite tht is rnke first the fewest times fter m is remove seon-to-lst (s m 1 ) et. Thus the mximum likelihoo estimte of the rnking is extly the STV rnking. Hene, STV is n MLERIV rule. Voting rules tht nnot e interprete s MLEs In this setion, we will show tht some rules nnot e interprete s mximum likelihoo estimtors uner the restrition tht the votes re i.i.. (given the orret outome). To o so, we rely on the following lemm. Lemm 1 For given type of outome (e.g. winner or rnking), if there exist vetors of votes V 1, V suh tht rule ρ proues the sme outome on V 1 n V, ut ifferent outome on V 1 +V (the votes in V 1 n
V tken together), then ρ is not mximum likelihoo estimtor for tht type of outome uner i.i.. votes. Speifilly, if there exist vetors of votes V 1, V suh tht rule ρ proues the sme winner on V 1 n V, ut ifferent winner on V 1 + V, then ρ is not n MLEWIV rule; if there exist vetors of votes V 1, V suh tht rule ρ proues the sme rnking on V 1 n V, ut ifferent rnking on V 1 + V, then ρ is not n MLERIV rule. 3 Proof: Let s e the outome tht ρ proues on V 1 n on V. Given istriution suh tht s rg mx s P (V 1 S = s ) n s rg mx s P (V S = s ) (where S is the orret outome), we hve s rg mx s P (V 1 S = s )P (V S = s ) = rg mx s P (V 1 + V S = s ). But s is not the outome tht ρ proues on V 1 + V, so ρ is not mximum likelihoo estimtor for the istriution. vote 1... vote k vote k+1... vote n V1 "orret" outome Figure 1: Grphil illustrtion of Lemm 1. If the mximum likelihoo estimte for the orret outome given only the votes V 1 oinies with the the mximum likelihoo estimte given only the votes V, then it must lso oinie with the mximum likelihoo estimte given ll the votes V 1 + V. Below, when we use Lemm 1 to show tht rule is neither n MLEWIV nor n MLERIV rule, we will exhiit vetors of votes V 1 n V suh tht the rule proues the sme rnking on them (n hene lso the sme winner), ut the rule proues ifferent winner on V 1 +V (n hene lso ifferent rnking). Theorem 3 The Buklin rule is neither n MLEWIV nor n MLERIV rule. Proof: We will pply Lemm 1 to oth ses. Let V 1 ontin two votes e, n one 3 Suh proxil outomes re perhps reminisent of known prox in whih sttistil tests in two ifferent supopultions oth suggest tht tretment is helpful, ut when the t of the two tests re ggregte, this suggests tht the tretment is tully hrmful [1]! We note tht proof using Lemm 1 tht rule is not MLEWIV is not suffiient to show tht the rule is not MLERIV, euse in the proof, V 1 n V my proue the sme winner ut ifferent rnkings. V vote e. The following esries t whih points the nites pss the n/ votes mrk. is rnke the top nite y two votes; is rnke mong the top two nites y ll votes; is rnke mong the top three nites y ll votes; n is rnke mong the top four nites y ll votes. Hene the rnking proue y the Buklin rule on V 1 is e. Let V ontin two votes e, one vote e, n one vote e. The following esries t whih points the nites pss the n/ votes mrk. is rnke mong the top three nites y ll votes; is rnke mong the top three nites y three votes; is rnke mong the top four nites y ll votes; n is rnke mong the top four nites y three votes. Hene the rnking proue y the Buklin rule on V is e, the sme s on V 1. Now, onsier the rnking tht the Buklin rule proues on V 1 + V. The following esries t whih points the nites pss the n/ votes mrk. is rnke mong the top two nites y five votes; is rnke mong the top two nites y four votes; is rnke mong the top three nites y five votes; n is rnke mong the top four nites y six votes. Hene the rnking proue y the Buklin rule on V 1 + V is e. We hve lrey shown tht STV is n MLERIV rule, whih implies tht the onition of Lemm 1 for n MLERIV rule oes not hol. Still, it is interesting to see iretly why this onition oes not hol. Suppose tht we hve votes V 1 n votes V, on whih STV proues the sme rnking. The nite m rnke lst y STV in oth of V 1 n V reeives the lowest numer of votes in oth ses, n therefore must lso reeive the lowest numer of votes in V 1 + V, n e rnke lst in this se s well. Then, the nite m 1 rnke seon lst in oth of V 1 n V reeives the lowest numer of votes in oth ses fter the removl of m, n therefore must lso reeive the lowest numer of votes in V 1 +V fter the removl of m, n e rnke seon lst in this se s well et. Hene the rnking proue y the STV rule on V 1 +V must gree with tht proue on V 1 n V. However, this oes not yet imply tht the Lemm fils on STV for the MLEWIV property, n in ft it oes not: Theorem The STV rule is not n MLEWIV rule. Proof: We will pply Lemm 1. Let V 1 ontin three votes, four votes, n six votes
. Given these votes, rops out first; its three votes trnsfer to, who then hs seven votes, one more thn. Hene, wins the eletion on V 1 uner the STV rule. Let V ontin three votes, four votes, n six votes. Thus, V hs the sme votes s V 1, exept the roles of n re swithe. Hene wins the eletion on V uner the STV rule. Now, onsier the set of votes V 1 + V. n eh reeive nine votes, wheres reeives only eight votes. Hene, rops out first n nnot win. The remining rules re ll se on pirwise eletions. For these, it is useful to onsier the pirwise eletion grph etween the nites, in whih there is irete ege from nite to nite with weight w if efets y w votes in their pirwise eletion. For exmple, if the votes re n, then the pirwise eletion grph is: Proof: We will pply Lemm 1 to oth ses. Let V 1 relize the following pirwise eletion grph (y Lemm ): A nite s Copeln sore is the numer of outgoing eges minus the numer of inoming eges. Therefore, the sores in this eletion re s follows: gets points, gets 1 point, gets 0 points, gets 1 point, n e gets points. Hene the rnking proue y the Copeln rule on V 1 is e. Let V relize the following pirwise eletion grph (y Lemm ): e e In the remining proofs, when we pply Lemm 1, it will e esier not to give the votes in V 1 n V, ut rther only the pirwise eletion grphs for V 1 n V. (The pirwise eletion grph of V 1 + V n e inferre from these grphs y summing their eges.) Of ourse, this pproh is legitimte only if we n show tht there o inee exist votes for V 1 n V tht relize these grphs. This is the purpose of the following lemm. Lemm For ny pirwise eletion grph G whose weights re even-vlue integers, votes n e onstrute tht relize G. Proof: To inrese the weight on the ege from nite to y without ffeting ny other weights, we n the following two votes (where 1,,..., m re the remining nites): 1... m, m m 3... 1. Hene we n relize ny pirwise eletion grph G with even-vlue integer weights. The Copeln sores in this eletion re s follows: gets points, gets 1 point, gets 0 points, gets 1 point, n e gets points. Hene the rnking proue y the Copeln rule on V is e, the sme s on V 1. The pirwise eletion grph of V 1 +V is the following: The Copeln sores in this eletion re s follows: gets points, gets 1 point, gets 0 points, gets 1 point, n e gets points. Hene the rnking proue y the Copeln rule on V 1 + V is e. e Theorem 5 The Copeln rule is neither n MLEWIV nor n MLERIV rule. Theorem 6 The mximin rule is neither n MLEWIV nor n MLERIV rule.
Proof: We will pply Lemm 1 to oth ses. Let V 1 relize the following pirwise eletion grph (y Lemm ): 8 6 10 In this eletion, s worst pirwise efet is y 6 votes; s worst pirwise efet is y 8 votes; s worst pirwise efet is y 10 votes; n s worst pirwise efet is y 1 votes. Hene the rnking proue y the mximin rule on V 1 is. Let V relize the following pirwise eletion grph (y Lemm ): 1 8 10 1 In this eletion, the pirwise rnkings re loke in s follows. First, is loke in; then, is loke in; then is inonsistent with the rnkings loke in lrey, so the next pirwise rnking loke in is ; n finlly, is loke in. Hene the rnking proue y the rnke pirs rule on V 1 is. Let V relize the following pirwise eletion grph (y Lemm ): 6 1 1 16 1 6 10 8 In this eletion, gin, s worst pirwise efet is y 6 votes; s worst pirwise efet is y 8 votes; s worst pirwise efet is y 10 votes; n s worst pirwise efet is y 1 votes. Hene the rnking proue y the mximin rule on V is, the sme s on V 1. The pirwise eletion grph of V 1 +V is the following: 6 8 In this eletion, s worst pirwise efet is y votes; s worst pirwise efet is y votes; s worst pirwise efet is y 6 votes; n s worst pirwise efet is y 8 votes. Hene the rnking proue y the mximin rule on V 1 + V is. Theorem 7 The rnke pirs rule is neither n MLEWIV nor n MLERIV rule. Proof: We will pply Lemm 1 to oth ses. Let V 1 relize the following pirwise eletion grph (y Lemm ): In this eletion, the pirwise rnkings re loke in s follows. First, is loke in; then, is loke in; then is inonsistent with the rnkings loke in lrey, so the next pirwise rnking loke in is ; n finlly, is loke in. Hene the rnking proue y the rnke pirs rule on V is, the sme s on V 1. The pirwise eletion grph of V 1 +V is the following: 1 16 1 0 10 In this eletion, the pirwise rnkings re loke in s follows. First, is loke in; then, is loke in; then, is inonsistent with the rnkings loke in lrey, so the next pirwise rnking loke in is ; then, is inonsistent with the rnkings loke in lrey, so the next (n lst) pirwise rnking loke in is. Hene the rnking proue y the rnke pirs rule on V 1 + V is. Among the ommon voting rules tht we hve stuie, none is n MLEWIV rule ut not n MLERIV rule. So, one my woner if perhps the MLEWIV property implies the MLERIV property. To see tht this is not the se, onsier hyri rule A tht first hooses winner oring to some MLEWIV rule B, 18
n proues the rnking of the remining nites oring to rule C whih is not n MLERIV rule. A is n MLEWIV rule euse B is, ut it is not n MLERIV rule euse C is not. 5 Conlusions n future reserh Voting is very generl metho of preferene ggregtion. We onsiere the following view of voting: there exists orret outome (winner/rnking), n eh voter s vote orrespons to noisy pereption of this orret outome. Given the noise moel, for ny vetor of votes, we n ompute the mximum likelihoo estimte of the orret outome. This mximum likelihoo estimte onstitutes voting rule. In this pper, we ske the following question: For whih ommon voting rules oes there exist noise moel suh tht the rule is the mximum likelihoo estimte for tht noise moel? The following tle summrizes our results for the rules isusse in this pper. MLERIV MLERIV Soring rules Hyris of MLEWIV (inl. plurlity, MLEWIV n Bor, veto) MLERIV Buklin MLEWIV STV Copeln mximin rnke pirs Clssifition of voting rules isusse in this pper. We elieve the tehniques tht we use to prove these results shoul e esy to pply to other rules s well. There re mny questions left to e nswere y future reserh, inluing t lest the following. How resonle re the noise moels tht we gve to show tht ertin rules re MLEWIV or MLERIV rules? To the extent tht they re not resonle, n we improve them? Do these improve noise moels le to the sme rules, or ifferent (n possily ltogether new) ones? For the rules tht we showe re not MLEWIV or MLERIV rules, n we still interpret them s mximum likelihoo estimtors if we relx somewht the ssumption of votes eing rwn inepenently? (If we hve no restrition t ll, then Proposition 1 shows tht this n lwys e one.) Alterntively, re there other rules tht n e interprete s mximum likelihoo estimtors uner our inepenene ssumption, while prouing outomes tht re lose to those proue y rules tht nnot e so interprete? Finlly, re there rules tht re not MLEWIV/MLERIV ut for whih Lemm 1 nnot e use to show this, or oes Lemm 1 in ft lso provie us with suffiient onition for rule eing MLEWIV/MLERIV? Referenes [1] Alon Altmn n Moshe Tennennholtz. Rnking systems: The PgeRnk xioms. In Proeeings of the ACM Conferene on Eletroni Commere (ACM- EC), Vnouver, Cn, 005. [] Vinent Conitzer, Jerome Lng, n Tuoms Snholm. How mny nites re neee to mke eletions hr to mnipulte? In Theoretil Aspets of Rtionlity n Knowlege (TARK), pges 01 1, Bloomington, Inin, USA, 003. [3] Vinent Conitzer n Tuoms Snholm. Complexity of mnipulting eletions with few nites. In Proeeings of the Ntionl Conferene on Artifiil Intelligene (AAAI), pges 31 319, Emonton, Cn, 00. [] Vinent Conitzer n Tuoms Snholm. Vote eliittion: Complexity n strtegy-proofness. In AAAI, pges 39 397, Emonton, Cn, 00. [5] Vinent Conitzer n Tuoms Snholm. Universl voting protool tweks to mke mnipultion hr. In Proeeings of the Eighteenth Interntionl Joint Conferene on Artifiil Intelligene (IJCAI), pges 781 788, Apulo, Mexio, 003. [6] Vinent Conitzer n Tuoms Snholm. Communition omplexity of ommon voting rules. In ACM- EC, Vnouver, Cn, 005. [7] Anrew Dvenport n Jynt Klgnnm. A omputtionl stuy of the Kemeny rule for preferene ggregtion. In AAAI, pges 697 70, Sn Jose, CA, USA, 00. [8] Mrie Jen Antoine Niols e Critt (Mrquis e Conoret). Essi sur l pplition e l nlyse l proilite es eisions renues l plurlite es voix. Pris: L Imprimerie Royle, 1785. [9] Mohme Drissi n Mihel Truhon. Mximum likelihoo pproh to vote ggregtion with vrile proilities. Tehnil Report 011, Deprtement eonomique, Universite Lvl, 00. [10] Eithn Ephrti n Jeffrey S Rosenshein. The Clrke tx s onsensus mehnism mong utomte gents. In AAAI, pges 173 178, Anheim, CA, 1991. [11] Eithn Ephrti n Jeffrey S Rosenshein. Multigent plnning s ynmi serh for soil onsensus. In IJCAI, pges 3 9, Chmery, Frne, 1993. [1] John Kemeny. Mthemtis without numers. In Delus, volume 88, pges 571 591. 1959. [13] Dvi M Pennok, Eri Horvitz, n C. Lee Giles. Soil hoie theory n reommener systems: Anlysis of the xiomti fountions of ollortive filtering. In AAAI, pges 79 73, Austin, TX, 000. [1] Donl G. Sri. A hoti explortion of ggregtion proxes. SIAM Review, 37:37 5, 1995. [15] Peyton Young. Optiml voting rules. Journl of Eonomi Perspetives, 9(1):51 6, 1995.