MathematicsofVotingSystems

TanyaLeiseMathematics&Statistics

AmherstCollege

Arrow’sImpossibilityTheorem

1) Nospecialtreatmentofparticularvotersorcandidates

2) Transitivity• A>BandB>CimpliesA>C• Nocycles

3) Monotonicity• Avoterchangingtheirballotinawayfavoringcannotcausethatcandidate’soverallrankingtogodown.

4) Independenceofirrelevantalternatives• Overallrelativerankingoftwocandidatesdependsononlytheirrelativerankingonvoterballots

A

BC

Whyindependenceofirrelevantalternativesmatters:1995FigureSkatingWorldChampionship

• RankingspriortoMichelleKwanskating:

• 1st place:ChenLu(China)

• 2nd place:NicoleBobek (USA)

• 3rd place:SuryaBonaly (France)

• RankingsafterjudgingofMichelleKwan:

• 1st place:ChenLu(China)

• 2nd place:SuryaBonaly (France)

• 3rd place:NicoleBobek (USA)

• 4th place:MichelleKwan(USA)

Plurality:whoevergetsthemostvoteswinsStrengths• Simpleballottofillout• Transparentresults• Easytounderstand• Monotonic

Weaknesses• Votesplitting• Spoilers• Tacticalvoting• Negativecampaigning

• 1860USPresidentialElection• AbrahamLincoln• StephenDouglas• JohnBreckinridge• JohnBell

Borda count

PointsystemforfieldofNcandidates,e.g.,• N-1pointsfor1st place• N-2pointsfor2nd place• …• 0pointsforlastplace

(orotherpointscheme,forinstance,weighting1stplacemoreheavily)

Strengths• Takesintoaccountfullsetofpreferences• Canpromotecompromisecandidates• Monotonic

Weaknesses• Vulnerabletostrategicvoting,suchasburyingfavorite’smainrivals

Borda count:1999baseballMVPelections

28voters14pointsfor1st place9pointsfor2nd place8pointsfor3rd place7pointsfor4th place…

http://www.baseball-reference.com/awards/awards_1999.shtml

Approvalvoting

• Voteforallcandidatesyoufindacceptable• Mayreducevotesplittingandsupportthirdparties• Notasexpressiveasrankedmethods

Saari’s example:• 9,999votersstronglysupportA,findBmarginallyacceptable,andstronglyopposeC• 1voterstronglysupportsC,findsBmarginallyacceptable,andstronglyopposesA

Pairwisecomparison/Condorcetmethod

• Winnerbasedonhead-to-headmatchesofallpossiblepairingsofcandidates

• Beatpath/CSSDtakesintoaccountmarginsofvictoryusingaweighteddirectedgraphcalculation

• Condorcetwinner:candidatewhowinsallhead-to-headmatches

• Condorcetwinnercriterion:whenaCondorcetwinnerexists,thatcandidateshouldwintheelection.

Instantrunoffvoting(IRV)/rankedchoice

• Eliminatecandidatewithleast1st placevotes• Moveupcandidatesandrepeatuntilsinglewinnerleft

• Burlington,VT2009mayoralraceusedIRV• IRVwinnerwasKiss,followedbyWrightthenMontroll• Montroll wasCondorcetwinner• IfKisshadwonmore1st placevotes,hewouldhavelost

IRVisnotmonotonicIRVdoesn’tsatisfyCondorcetwinnercriterion

Gibbard-SatterthwaiteTheorem

• Tacticalvoting:dishonestvotingtoimproverankingofyourpreferredcandidate.• Allrankedvotingsystemswithnospecialtreatmentofparticularvotersorcandidatesaresusceptibletotacticalvoting.

Gerrymandering

• Incumbent(sweetheart)• RuledOKbycourt

• Racial• VotingRightsActof1965

• Partisan• Noclearmeasure

• Packingandcracking

http://www.redistrictingthenation.com

BakervsCarr,1962SupremeCourtcase• “Oneperson,onevote”• Eachindividualisweightedequallyinapportionment(doesn’tmatterwhetherlegallyabletovoteornot)

• Establishedrightoffederalcourtstoreviewredistrictingmaps(redrawnevery10yearsaftercensus)• FoundTennesseedistrictmapunconstitutional

• Districtsdidnotreflectmovementofpopulationtocities• 2/3ofrepresentativeselectedby1/3ofthestatepopulation

github.com/JeffreyBLewis/congressional-district-boundaries

CoopervsHarris:NorthCarolinadistrictmap• SupremeCourtruled5-3earlierthisweekthatDistricts1and12exhibitunconstitutionalracialgerrymandering• District12electedAfrican-American-favoredcandidateswith64-72%ofvote• NewmapincreasedpackingofAfrican-Americanvoters

2003-13map:7Demto6Repseatsin2011 2013-16map:10Repto3Demseatsin2015

QuantifyingpartisangerrymanderingEfficiencygap

• StephanopoulosandMcGhee• Assesses“wastedvotes”in2-partyelection• Ifapartylosestheelection,allofthatparty’svotesarewasted.

• Ifapartywinstheelection,thevotespast50%arewasted.

• Sumwastedvotesforeachpartyacrossthedistrictsinthatstate• Finddifferenceintotalwastedvotesbetweenthe2parties,dividedbytotal#ofvotes

6Red:4Blue1vs4wasted

1 Red:9Blue1vs4wasted

Efficiencygapof30%

QuantifyingpartisangerrymanderingEfficiencygapofzerodoesn’timplyproportionalrepresentationDistrict Red Blue Winner Wastedvotes

1 6 4 Red 1vs4

2 6 4 Red 1vs4

3 6 4 Red 1vs4

4 4 6 Blue 4vs1

5 3 7 Blue 3vs2

6 3 7 Blue 3vs2

7 3 7 Blue 3vs2

8 3 7 Blue 3vs2

9 3 7 Blue 3vs2

10 3 7 Blue 3vs2

Total 40 60 25vs25

• 100votersin10districts• 40totalRedvoters• 60totalBluevoters

• Redwins3districts• Bluewins7districts

• Efficiencygap=0• BiasedtowardBlue

FelonydisenfranchisementintheUS• Dependsonstatelaws• OverallinUS,7.7%ofblackadultsdisenfranchised,comparedto1.8%ofnon-blackadults.• Largeprisonpopulationsalsousedasformofgerrymandering(countaspopulationbutcan’tvote)• Stateswithmostseverelaws:

• Florida(21%ofAfrican-Americansdisenfranchised)

• Kentucky(26%)• Virginia(22%)• Upto40%ofblackmendisenfranchisedinthesestates http://politicalmaps.org/6-million-lost-voters-state-

level-estimates-of-felony-disenfranchisement-2016/

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