Paradoxes of Preferential Voting

7/22/2019 Paradoxes of Preferential Voting

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Paradoxes of Preferential VotingAuthor(s): Peter C. Fishburn and Steven J. BramsReviewed work(s):Source: Mathematics Magazine, Vol. 56, No. 4 (Sep., 1983), pp. 207-214Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2689808 .

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Paradoxes of Preferential otingWhat an go wrongwithophisticated oting ystemsdesigned to remedy problems of simpler systems.PETER C. FISHBURNBell LaboratoriesMurrayHill, NJ 07974STEVEN J.BRAMSNew YorkUniversityNew York,NY 10003

Preferentialoting, evelopedby Thomas Hare [12] in the 1860's, s still used for majorelectionsnAustralia,reland ndSouthAfrica,s well as for ocalelectionsnmany ountries.From ts nception,t has beentouted s a wayto promote ull xpressionf electors' referencesand to ensuremaximum nd equitable onsiderationf each elector's ote.Whenused to fillseveral eats n a legislature,referentialoting rovides epresentationor iableminoritiesndtends odistributeeats nproportiono thenumbers f voterswhofavor hedifferentarties.It seeks o do all this n thebasis of a single referentialranked) allotby transferringotes,inpartor nwhole, romhemost ndleastpopular andidatesocandidateswithntermediatesupport. he mostpopular, lected irst,on'tneedtheir surpluses,"nd the eastpopular annever vercome heir deficits,"o the ransfersfboth urplusesnd deficitsothe ntermediatecandidates etermineshich fthesewin.Whenthere re n voters nd c seatsaretobe filled,transfersremadesequentiallyntil candidates ttain hevotequotaneededfor lection. hequota s usually efined s[C ]+

where racketsignifyhe nteger art fthe rgument. e shalluse this onceptater.Despite ts tendencies o promotendividual nterestsnd fairrepresentation,referentialvoting as several urprisingndpotentiallyamning efects.We shallbeginby llustratingourof these hroughn apocryphaltoryfan electionmong hree ontendersormayor f a smalltown. n thisdeliberatelyimple ase,a candidate anked irstn more han50 percent f theballots s elected; f there s no suchcandidate, he one withthe fewest irst-placeotes sscratched,hen heoneoftheremainingwowho rankshighern moreballots s elected. incethis rocedures tantamountopluralityvote-for-one)oting ollowedya two-candidateunoffelection,hedefectsrparadoxes evelopednour tory pply lso to the ommon lurality-runoffscheme.Thestory'sour aradoxes re summarizedere or eferencendfor eaderswhomaywish otest hem n theirwn.

NO -SHOW PARADOX: The addition f dentical allotswith andidate ranked astmaychange hewinnerromnotherandidate o x.THWARTED-MAJORITIESPARADOX: A candidate ho andefeat very ther andidateindirect-comparisonajorityotesmay ose the lection.MULTIPLE-DISTRICTS PARADOX: A candidatean winn eachdistricteparately,et osethegenerallectionn the ombinedistricts.

VOL. 56, NO. 4, SEPTEMBER 1983 207

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MORE-IS-LESS PARADOX: If the winnerwere rankedhigher y some voters, ll elseunchanged,hen notherandidatemight avewon.Following he tory, e shalldiscuss eneral roblems onfrontingoting chemes nd mentioninteresting athematical ork n the ubject.We then eturn o preferentialoting o illustratetwo other aradoxes hat riseonly n morecomplex ituations.We concludewith note onparadoxprobabilities.

A funny hing appened n the wayto the pollsMr. and Mrs. Smith's ar brokedown on theway to thepolls ust before losing ime.TheSmiths erentenselynterestedn a tightaceformayor ftheirown mongMrs.Bitt,Mr.Huffand Dr. Wogg.The ballotformayor skedeach voter o rank hethree andidates rom irst hoice o thirdchoice. hetownspeoplenew hat he lection ouldbe decided y the imple referentialotingmethod, hich adbeen nstitutedy ocalreferendumomeyears arlier. veryonentownwaspleasedthat hey sed such sensible rocedure or lecting hehead of theirocal government.The Smithswereof one mind bout thecandidates. heyfavored itt o Huff o Wogg, nd

thereforeothwouldhavevotedBHW.Although heyikedMrs. Bittbest, heywere lmost sfondofMr. Huff ut disliked ndmistrustedr. Wogg.Much to their egret,he Smith's arproblem reventedhem rommakingt tothepollsbefore losing ime.Manyof their ellow ownspeopleid.When Mrs. Smith penedhernewspaper henextmorning,er yewas caught ya headline roclaimingHuffElected s 1,608Go toPolls."Sheand herhusbandweredelightedhatDr.Wogghadnotwon.Theydid feel twingefregrethattheir riend, rs.Bitt,wasbeaten. erhaps heir oteswouldhavemadea difference.As Mrs.Smith eadon,shenoted hatno candidate adgotten nough irst-placeotes owinoutright. rs.Bitthadbeenscratched ecause he had thefewestirst-placeotes, nd Mr. Huffwent n to beat Dr. Woggbya pluralityf 917 to 691.Toward heend of the rticle,n an insidepage,Mrs.Smith ead thetabulationf howthe1,608 oters ast their allots hownnFIGURE 1.

Totals Rankings H overW WoverH417 BHW 417 082 BWH 0 82143 HBW 143 0357 HWB 357 0285 WBH 0 285324 WHB 0 3241608 917 691

FIGURE IIt madeher feelgood that he and her husbandwould have votedwith he argest f thesixgroups.As the rticle ad noted arlier,Mrs.Bittbarelyostouton the nitial ount inceBitt,Huff ndWogghad first-placeallies f 499 (417+ 82), 500 (143+ 357),and 609 (285+ 324)respectively.

208 MATHEMATICS AGAZINE



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Mrs.Smith ealizedwhen he readthis hatMr. Huff atherhanMrs. Bittwouldhavebeenscratchedfsheand herhusband advoted.At easttheir riend, rs.Bitt,wouldhavemadethe"runoff"ftheirar hadnotbroken own.Beforeeaving orher ob as an actuary ith n insuranceompany eadquarterednthenexttown,Mrs.Smith ecided o seewhatwouldhave happened fshe and herhusbandhadvoted.Hertabulations shownnFIGuRE 2.Totals Rankings B overW W overB

419 BHW 419 082 BWH 82 0143 HBW 143 0357 HWB 0 357285 WBH 0 285324 WHB 0 3241610 644 966FIGURE 2

To her hagrin,he awthat heir oteswouldhavemadeDr.Wogg hewinnerven hough ewas ranked ast on their allots!This so shocked er that hechecked erfigureshree imes.When hey efused ochange,t hither:thewhole hing ependedn whowasscratchedfterheinitial ount.WithBitt ut,Huffwins;withHuff ut,Woggwins.Even f300 morepeoplehadvotedBHW,Dr.Woggwould tillhavewon.Mrs. Smithwas beginningowonderfthe town's rocedure or lecting mayorwas thatsensible fter ll.Thatevening, hile eviewingerfiguresgain,Mrs.Smith ecame wareofanotheruriousfact. he realized hat hewinner,Mr.Huff,wouldhave beaten itherMrs. Bitt 824 to 784) orDr. Wogg 917 to691) in a direct otebetween hetwo.The "majorityandidate"-that s,thecandidatewho could have beaten achoftheothersn direct airwise otes-had indeedwon.However,ftheSmiths advoted, hen otonlywould heirast choicehavewonbutthewinner,Dr.Wogg,wouldnothavebeenthe andidate avorednseparate airwiseontests o eachoftheother andidates.At thispoint,Mrs.Smith uspectedhat heir lection roceduremight e more han littleflawed ndwonderedffurtherrobingmight ncover ther nusual ossibilities.hevowed omaketime or his ver heweekend.The Smith's ownhad twovoting istricts,alled East andWest.When theweekend ameround,Mrs. Smithdecided to compare heoutcomewithwhat might avehappened n theseparate istricts.he suspected hat hewinner,Mr.Huff,might ave ost n one if not bothdistricts. hepaperhadreportedhat 88 people voted n theEast and 1,020hadvoted n theWest.Moreover,tgave theEast-Westplits hown nFIGURE 3.Totals Rankings East West

417 BHW 160 25782 BWH 0 82143 HBW 143 0357 HWB 0 357285 WBH 0 285324 WHB 285 391608 588 1020FIGURE 3

Applyingreciselyhe ame lection uleusedfor hegeneral lectiono theEast andtheWestseparately, rs.Smith ound hatMrs. Bittwouldhave won nbothdistricts!he felt hiswastrulymazing incebothHuff ndWogghadsizablemajoritiesverBitt n the verall lectorate,so therewasnowaythatMrs.Bitt ouldhavewon n the ombined istricts.VOL. 56, NO. 4, SEPTEMBER 1983 209



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Moreover,s Mrs. Smith oted, multiple-districtinnerikeMrs.Bitt ouldbe a "minoritycandidate" n the sense thatthiscandidatewould be defeated y every ther andidate ndirect-comparisonotes. She also realized hat thisanomaly ould arise only when differentcandidateswere cratchedn thefirst oundsnthe everal districtlections."n fact,Huffwasscratchedn theEast,whereasWoggwasscratchedntheWest.Mrs.Smithwas nowconvinced hat hehad a very trongaseagainst he upposedlyensiblesystem sed to elect hemayor ftheir own.Atherrequest, he hairman f the ocal ElectionBoard calleda specialmeetingf theboard toreview erfindings.On the nightbefore heboard meeting,s she was goingoverher figures,Mrs. Smithdiscovered nother rregularity.hileponderingwhat would have occurredf she and herhusband ad voted seeFIGURE2), she realized hatftwoor more fthe82 voterswith ankingBWH had movedWogg nto first lace (WBH), thenBittrather hanHuffwould have beenscratchednd Huff ather hanWoggwouldhavewon. n otherwords,n increasensupport orDr. Woggwouldhave changed imfrom winner o a loser!Extraordinary,hought rs. Smith,as sheprepared erflip harts orherpresentationo theElection oard.The next day Mrs. Smith o impressed he board that theydecided to appoint a selectpanel-chaired by Mrs. Smith,fcourse-to recommend better lection rocedure.n particu-lar, the boardcharged hepanelwithdevising system hatwould avoid all the paradoxesuncoveredyMrs.Smith.At thepanel'sfirstmeeting,ne memberuggestedhat hey etain anked oting ut simplyuse theballots to determine hich f the several andidateswas themajorityandidate.Heexplainedhat hiswoulddirectlyesolve heThwarted-Majoritiesaradox nd,moreover,ouldalso take are of Mrs.Smith's ther hree aradoxes.Mrs.Smith espondedhat hiswas a very ood deaup toa point, ut thatt wouldnot olveall their roblems.hehadbeenreading p on the ubject ndproceededo tell hepanelaboutthemostfamous aradoxof them ll, variously nown s "Condorcet's henomenon"4],the"paradoxofvoting,"nd the paradoxofcyclicalmajorities."Condorcet's henomenonccurswhenevery andidate s beatenby some other andidateunderdirect-comparisonoting.Mrs. Smithpointedout thatthiswas not thecase in theirelection,ut twascertainlyossible. orexample,f1,600 otal allotshadbeencast,with

400 for BHW500 for WBH700 for HWB,thenBitt eatsHuff 00 to700,Huff eatsWogg1,100 o500, ndWoggbeatsBitt1,200 o 400.At thispoint, nother anelmemberuggestedhatperhaps heir roblemswouldvanish fthey sedthemethod hat is odgeusedto choose tspresident.hismethod wards points o afirst-placeote,1point oa second-place ote, nd 0 points oa third-placeote.Thewinnersthecandidatewith hemostpoints.He noted hat t couldbe extendedn a straightforwardaywhen here re more han hree andidates.

210 MATHEMATICS MAGAZINE



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The panel determinedhat this point-scoringystem-sometimeseferred o as Borda's"method f marks" 2], [5]-would resolve ll of Mrs. Smith's aradoxes,with the possibleexceptionf theThwarted-Majoritiesaradox.A quickreview f theelection ata showed hatthemajorityandidate,Mr.Huff,wouldhavewonunder hepoint-scoringystem. owever,hepanel lso noticed hatf50 or so BHW voters ad preferred ogg o Huff nd votedBWH, thendespite hefact hatMr.Huffwouldremain hemajorityandidate, r. Woggwouldwinunderthepoint-scoringystemseeFIGURE4).Totals Rankings B Points H Points W Points

367 BHW 734 367 0132 BWH 264 0 132143 HBW 143 286 0357 HWB 0 714 357285 WBH 285 0 570324 WHB 0 324 6481608 1426 1691 1707FIGuRE

Confusedndtired,hepanel agreed hat hey addone enough or ne meeting. heirnextmeeting asset for hefollowing ednesday.Problems f voting ystems

Weend ourstorytthis oint ecause,na sense,t has no end.Thepanelcouldmeet oreverwithout eing ble to fulfilltscharge rom heElection oard to avoidall four aradoxes. his sbecausetheres a metaparadoxurkingnthebackground hich,nsimplifiedorm,aysthatnoelection rocedureansimultaneouslyesolveMrs. Smith's econd nd third aradoxes.Letus elaborate.Weassume, s before,hat oters ank he andidatesrommost referredoleast preferred. ith fixednumber fcandidates, ut any potential umber fvoters, oung[21] seealso [22]) showednoneof themostmathematicallylegant apers n the ubject hatnorder o avoid theMultiple-Districtsaradox s well s to satisfyundamentalquity onditionsfor otersndcandidates,ne must se a type fpoint-scoringystem.n an attempto avoidtheThwarted-Majoritiesaradox,t s necessaryoassignmorepoints o a first-placeotethan o asecond-placeote, nd so forth, hich f course akes areofthe No-Show nd More-Is-LessParadoxes.However, iven ny etofdecreasingoint aluesfor hevarious laces, t s always ossible oconstructnexamplewith majorityandidatewho s notelected ythepoint-scoringystem.nfact, earlywohundred ears goCondorcetecognizedhatt spossible o constructxampleswith majorityandidatewho s notelected y anypoint-scoringystem ith ecreasing ointvalues 4], 9].Forexample,f there re seven oters uch hat3 have BHW2 have HWB1 has HBWI has WBH,thenB has a 4-to-3majorityvereachofH andW,butH beatsB under very oint-scoringsystemhat ssignsmore oints o a second-placeotethan o a third-placeote.Manyother roblemsndparadoxes hat laguepreferentialotingndother lectionystemsseem o have surfacednly ecently.he More-Is-Lessaradox, etter nown n the iteraturesthemonotonicityaradox,was shown ySmith20]to affect irtuallyll successive-eliminationproceduresased on point coring. urther esults n themonotonicityaradox ppear n [10].Within he pecificontextfpreferentialoting,heMore-Is-Lessaradox nd theMultiple-Dis-trictsaradox re discussedn[6], 7].As far as we know,the No-ShowParadox,which s closelyrelated o the More-Is-LessParadox, s not discussed lsewhere. owever, nother o-show aradoxseemsto have beenVOL. 56, NO. 4, SEPTEMBER 1983 211



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discovered any ears go 14], 17].Thisother aradox ays hat neofthe andidates lected ypreferentialoting ouldhaveendedup a loser fadditional eoplewhoranked im nfirst lacehad actually oted.Anexample f this aradox ppearsn thenext ection.The paradoxes iscussed ere ndelsewhere9], 15]reveal nly he urface ffectsf deeperaspects f aggregationtructures,uch as thosedeveloped y Young 21]. Recentwork n thesestructuresas stimulatedn largemeasure yKenneth rrow's lassic mpossibilityheorem1].This theoremhows hat few imple nd appealing onditions or ggregatingiverse ankingsinto consensus ankingre ncompatible. umerous ariants f Arrow's heorem ow exist 8],[13], 19], and thesehave been oined by related esults 11], 13], 16], 18] which how thatvirtuallyvery ensible lection rocedure ormulticandidatelectionss vulnerable o strategicmanipulationy voters.n otherwords, herewillbe situationsn which omevoters an benefitby voting ontraryotheirrue, rsincere, references.Anexample fthe atter henomenonccursn our toryftheSmiths.fthey ad voted heirtrue referencerder, HW,then r.Woggwouldhavewonunder referentialoting. owever,if they ad votedHBW, or any other rder hatdid not have Mrs. Bitt n first lace, thenMr.Huffwouldhavewon.Hence,by voting trategicallyi.e.,falsely),heSmithswouldhavehelpedtoelect heirecond hoiceH) ratherhan heirast choiceW).More paradoxes fpreferentialoting

Additionalflaws n preferentialoting an arise only when thereare more than threecontenders. e shall llustratewoof these fter escribing general,ndwidely sed,procedureforpreferentialoting.In thegeneral ase,voters ank he andidates rommostpreferredo eastpreferredn theirballots.To be elected, candidatemust eceive quota q ofweightedotes.Each voter eginswith otingweight .First-place otes re tallied or ach candidate; hosewith or more re elected.fc' areelected n this irstound nd0 < c' < c, then heweightfeach voterwhose irst hoicewas electeds decreased rom to a nonnegativeumberO fthereis no "surplus" ver uota) so that he umofall weights ecomes qc'. Elected andidates reremoved rom he ballots, nd newrounds ollow ntil candidates re elected, s describedbelow.After emoval f the lected andidates, nelected andidatesmoveup n theballot ankingsofill n top places, nd theprocesss repeatedwith new,weightedally funelectedandidatesnow n first lace.Again, is usedas the uotafor lection. heprocess ontinuesntil ither llc seatshavebeenfilled,r nounelectedandidate ets t eastq weightedotes n the atest ally.In the atter ase,thecandidatewith he mallestweighted irst-placeally s scratched,allotrankingsbut not voterweights)rerevised ccordingly,nd theprocess ontinues ntil seatsare filled.

Insteadof our earlier tory, upposenow thatBitt, oxx,Huff nd Woggarevying or woseatson the own ouncil,ndthat100peoplevote s follows:34 BHFW25 FBHW26 HWBF9 WBFH6 WHFB

The quota s 34, so Bitt s elected irst. inceexactly 4 people votedforBitt n first lace, theweightsfthese oters re reduced o0, leaving25 FHW26 HWF9 WFH6 WHF




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Sincenoneoftheothers eaches hequota,Wogg s scratched. hen Foxx,who has 34 (25 + 9)votes o 32 (26+ 6) forHuff,wins he econd eat.Now suppose ivemore oxx supportersFBHW) had voted, iving34 BHFW30 FBHW26 HWBF9 WBFH6 WHFB

The newquota s (105/3)+ I = 36. Sinceno candidate eaches hequota,Wogg s scratched:34 BHF9 BFH30 FBH26 HBF6 HFB

At thispointBittpasses the quota with 3 (34 + 9) votes and is elected s before. ince Bittexceeded he uota by7 first-placeotes, ach ofher43 supportersetains /43ofa vote, ivingaggregatesf1.5 H from itt's urplus

30 FH26 HF6 HFSinceHuffnowsurpasses hequotawith 7.5 (26+ 6 + 5.5),hebecomes he secondcandidateelected. husFoxx, winnernthefirstase,becomes loserwhen ivemore oters howupwithhim nfirst lace.Our final aradoxwassuggestedya statementn a recent allot f a professionalocietyhatlisted ight andidates or our eatson the ociety's ominatingommittee3].The electionwasconducted y preferentialoting. ocietymembers ere dvised o mark andidatesn order fpreferencentil heyweregnorantr ndifferentoncerningandidates hom hey idnotrank.Thepreferentialoting ystem escribedarliers easilymodifiedoaccount or artial ankings:if a voter'smarked andidates re removedr scratched efore ll seats arefilled,hatvotersthen reated s ifhe never otedn thefirst lace.The ballot tatementlluded oin thepreceding aragraphlaimed hat there s no tacticaladvantageo be gained y markingew andidates." IGuRE2,suitablymodified,hows hat hisis false. uppose gainthatFoxx is in therace for wocouncil eats longwithBitt,Huff, ndWogg, nd that otes repreciselyhe ame as shownn FIGuRE2, except hatFoxx s thefirstchoice f all 1,610 oters:

Totals Rankings419 FBHW82 FBWH143 FHBW357 FHWB285 FWBH324 FWHB1610

ThenFoxxwins seat, ndmattersroceed s beforewhenhe sremoved romheballots, ivingDr.Wogg heother eat.

VOL. 56, NO. 4, SEPTEMBER1983 213



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But suppose hatMr. and Mrs. Smith ad voted ustF instead f FBHW, .e.,had voted nlyfor heir irsthoice. hen, fter oxx sremoved, erevertoFIGURE ,whereMr.Huffwins heother eat.Byvoting nly or heir irsthoice, heSmiths revent heirast choice romwinningthe econd eat.Thisexample rovides second nstance f how somevotersmightnduce referredutcomesby misrepresentingheir rue references.n thepresentase,misrepresentationakes heform fa deliberateruncationf one's ranking ather han false ut complete anking.Paradox probabilities

Although irtuallyll voting ystems or lections ith hree r more andidates an producecounterintuitivend disturbingutcomes, referentialoting s especially ulnerable ecauseofits sequential liminationnd vote-transferrovisions. evertheless,his ystems stillwidelyused nseveral ountries.Defenders f preferentialoting-and there ave been many ver the past century-mightargue that the paradoxes of preferentialoting re not a problembecause theyoccur soinfrequentlynpractice. heywould,wepresume,laim hat few ontrivedxamples houldnotdeter s from sing carefullyefinedystemhathasprovedtsworthncountless lections.Although robabilitiesf paradoxes avebeen estimatedn other ettings9],we know f noattempts o assessthe ikelihoodsf theparadoxes fpreferentialoting iscussed bove, andwould propose his s an interestingossibilityor nvestigation.s it indeedtrue hat eriousflaws n preferentialoting uch as the No-ShowParadox and the More-Is-Less aradox aresufficientlyare s to causenopracticaloncern?

References[1] K. J. Arrow, ocial Choice and IndividualValues, 2nd ed., Wiley,New York, 1963.[2] Jean-Charles e Borda, Memoire ur es elections u scrutin, istoirede l'Academie Royale des Sciences,Paris, 1781.[3] S. J.Brams,The AMS nominating ystems vulnerable o truncation f preferences, otices Amer. Math.Soc.,29 (1982) 136-138.[ 4] Marquis de Condorcet, ssai sur 'application e l'analyse la probabilite es decisions endues la pluralitedes voix,Paris, 1785.[ 5] A. de Grazia, Mathematical erivation f an election ystem,sis, 44 (1953) 42-51.[ 6] G. Doron,The Hare voting ystems inconsistent,oliticalStudies, 7 (1979) 283-286.[ 7] G. Doron and R. Kronick, Single transferable ote: an example of a perverse ocial choice function,AmericanJournal fPoliticalScience, 1 (1977) 303-311.[ 8] P. C. Fishbum,The Theory fSocial Choice,Princeton niversity ress,1973.[9] , Paradoxesofvoting, merican oliticalScience Review, 8 (1974) 537-546.[10] ' Monotonicity aradoxes nthetheoryfelections, iscreteApplied Mathematics, (1982) 119-134.[11] A. Gibbard,Manipulation fvoting chemes: general esult, conometrica,1 (1973) 587-601.[12] T. Hare, The Election of Representatives, arliamentary nd Municipal: A Treatise, Longman, Green,London, 1861.[13] J. S. Kelly,Arrow mpossibility heorems,Academic Press,New York, 1978.[14] J. C. Meredith, roportional epresentationn Ireland,Dublin, 1913.[15] R. G. Niemi and W. H. Riker,The choice ofvoting ystems,cientific merican, 34 (1976) 21-27.[16] P. K. Pattanaik, trategy nd Group Choice, North-Holland, ew York,1978.[17] Reportof the Royal CommissionAppointed o Enquire nto ElectoralSystems, MSO, London,1910.[18] M. A. Satterthwaite,trategy-proofnessnd Arrow's onditions: xistence nd correspondenceheorems orvoting rocedures nd social welfare unctions,ournal fEconomicTheory, 0 1975) 187-217.[19] A. K. Sen,CollectiveChoice and Social Welfare, olden-Day,San Francisco,1970.[20] J.H. Smith,Aggregationfpreferences ithvariable lectorate, conometrica,1 (1973) 1027-1041.[21] H. P. Young,Social choicescoring unctions,IAM J.Appl. Math.,28 (1975) 824-838.[22] H. P. Young and A. Levenglick, consistent xtension f Condorcet's lectionprinciple, IAM J.Appl.Math.,35 (1978) 285-300.


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Paradoxes of Preferential Voting