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CONSTRAINED QUADRATING VOTING: AN ALTERNATIVE VOTING MODEL
Daniel S. Perez
Economic Game Theory
Cleveland State University - Ahuja School of Business
11/8/2016
“It has been said that democracy is the worst form of Government except for all those other forms that have been
tried from time to time.”
-Winston Churchill
Democracy is a systemof government inwhich all eligible membersof astate electand
designate control of the groupby majority rule.Inthe west,democracyispracticedthrough voluntary
participationinaone-person-per-vote system. Thoughimplementedforitspropensityto produce
desirable electionoutcomes withoutcompromising social equality,the traditional votingsystemis
flawedandbecome highlyinefficientovertime.GlenWeyl,aneconomistatthe Universityof Chicago
suggestssuchinefficiencyisthe resultof simple-majorityvotingrules whichpreventvotersfrom
expressingthe intensityof theirpreference forone alternative overanother-- thatinabilitytoexpress
intensityof preferencecanleadtodisinterest,diminishingparticipationandultimately inefficientsocial
outcomes.He suggeststhatif a minorityof votersstrongly preferone candidate andamajorityof voters
weaklypreferanother, general welfare couldbe improved by allowingthe minority topurchase extra
votes.Voters withstrongerpreferencescan thenobtainvictoryfortheirpreferred Candidate whiletheir
paymentscanbe usedtosupport social issues.LalleyandWeyl (2016) purpose a methodcalled
quadraticvotingthat isdesignedtocapture these efficiencygains.
In thismethod,votersare encouragedtopurchase asmany votesastheydesire,whilethe cost
of eachadditional vote increasesinthe square of the numberof votespurchased. Lalley andWeyl show
that thismethod- undertechnical conditions, hasvery desirable efficiencyproperties inlarge

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CONSTRAINED QUADRATING VOTING: AN ALTERNATIVE VOTING MODEL
populationswithprivatevalues andpreferences.Despite the academicappeal of LalleyandWeyl’s
mechanism,there are manysituationswhere allowingindividualstopurchase anunlimitednumberof
voteswouldbe politicallyunpalatable. Inlarge scale elections,suchamethodwouldlikelybe
interpretedasfavoringthose withextreme levelsof wealthandinsmallervotes,committee members
mightbe uncomfortable explicitlyassigningamonetaryvalue totheirchoices. Ineithercase,any
scheme torebate or otherwise utilizethe votingpaymentswouldundoubtedly generate some
controversy.
Thispaperconsidersa modifiedversionof quadraticvotingwhere the costfunctionforvotesis
still quadratic,buteachvoterisrequiredtovote at leastonce and limitedtocastingnomore than most
three votes(initial vote is free,anduptotwo at quadraticcost).For simplicity,attentionisrestrictedto
an environmentwiththree voters,twocandidatesandpublicvalues.Twovotershave extreme,
opposingpreferences,while the finalvoterismoderate.
The resultinggame is showntohave a unique Nashequilibriumwhere the voterswithstrong
preferencesrandomizebetweenbuyingzeroandtwoadditional votes.Inabaseline case where the
opposingvotershave equal intensityof preferences,the equilibriumgeneratesanoutcome thathas a
75% probabilityof agreeingwiththe majorityrule,while allowingthe minoritytowinthe remaining25%
of the time. Althoughthisdoesnotimproveefficiency,itensurescontinuedparticipationfromminority
voters.
 Two Players:A and B
 Three Voters: Voter 1, Voter 2, and Voter 3
 Expected Utility Function:Voter i receives utility ui (A) if CandidateA wins and ui (B) if CandidateB
wins.In this caseVoter 1 strongly prefers CandidateA, while Voter 2 strongly prefers CandidateB:
 u1 (A) = u2 (B) = 10 and u1 (B) = u2 (A) = 0
 Voter 3 is moderate and is assumed to have a slightpreference for CandidateA:
 u3 (A) = 5.1 whileu3 (B) = 4.9
 If this slightpreference were reverse, the remaininganalysis would proceed
unchanged except that the roles of Voter 1 and Voter 2 would be reversed.
 Strategies for each voter: vote once, twice or three times for their preferred candidate.The firstvote
is free; the second vote is $1 and the third vote is $4 (total costof $5). The total costs enter linearly

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CONSTRAINED QUADRATING VOTING: AN ALTERNATIVE VOTING MODEL
into each voter’s final payoff.For instance,if Voter 2 buys one vote and CandidateB wins,Voter 2’s
final payoff is u2 (B) -1 =9
 Note 1: The firstvote is free, but no one is permitted to abstain.
 Note 2: Payoffs are common knowledge
 Note 3: It is assumed that players only vote for their preferred candidates.Again,
sincepayoffs arecommon knowledge there will never be any reason to vote against
your preferred candidate.
 First,note that the moderate has a dominant strategy: vote one for A (becausethe difference in
payoffs for the moderate between A and B is < 1, the moderate has no incentive to ever ‘purchase’a
singlevote)
 Therefore, the focus of the game is the strategic interaction between Voter 1 and Voter 2. The
followingtableillustrates theelection outcome for each possiblestrategy,given that the moderate
votes once for A.
Voter 1 | Voter 2 1 votes for B 2 votes for B 3 votes for B
1 vote for A A wins Tie B wins
2 Votes for A A wins A wins Tie
3 Votes for A A wins A wins A wins
The nexttable illustratesthe expectedpayoffstoeachplayergiventhe costof purchasingvotes
and the electionoutcomesabove,withthe assumptionthattiesare brokenatrandom.In eachcell,
Voter1’s payoff appearsfirstandthe bestresponsesare bolded.
Voter 1 | Voter 2 1 votes for B 2 votes for B 3 votes for B
1 vote for A 10,0 5,4 0,5
2 Votes for A 9,0 9,-1 4,0
3 Votes for A 5,0 5,-1 5,-5
For example, ifVoter 1 votes once and Voter 2 twice, a tie results. Voter 1 pays nocosts so his expected utilityis
(1/2) * 10+ (1/2) * 0 = 5. Voter 2 pays a cost of 1 for sure; her expected utilityis therefore (1/2) * 9 + (1/2) * (-1) = 4.

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CONSTRAINED QUADRATING VOTING: AN ALTERNATIVE VOTING MODEL
In thiscontext,votingtwice isneverthe bestresponse for Voter2.Moreover,giventhat Voter2
will nevervote twice, Voter1will alsonevervote twice. Therefore, tofindthe Nashequilibriumof this
game,it sufficestoconsiderthe followingsimplifiedpayoff matrix:
Voter 1 | Voter 2 1 votes for B
(Q)
3 votes for B
(1-Q)
1 vote for A
(P)
10,0 0,5
3 Votes for A
(1-P)
5,0 5,-5
Since there isnocell where eachvoterischoosinga bestresponse tothe other’saction,there is
no pure-strategyNashequilibriumforthisgame.Tofinda mixedstrategyequilibrium, assumethat
Voter1 votesonce for A withprobability p andvotesthree time forA withprobabilityof 1-p,while
Voter2 votesonce for B withprobabilityof q andthree timesforB withprobabilityof 1-q.GivenVoter
2’s randomization,the expectedpayoff to Voter1from votingonce is10q+0 (1-q) =10q, while the
expectedpayoff fromvotingthree timesis5q+5(1-q)=5. Voter1 is thusindifferentbetweenthesetwo
actionsif 10q=5 or q =1/2. GivenVoter1’srandomization,the expectedpayoff to Voter2from voting
once is 0p + 0(1-p) = 0, while the expectedpayoff fromvotingthree timesis 5p+(1-p)(-5)=10p-5. Voter2
isthus indifferentbetweenthese twoactionsif 10p-5 =0 or p=1/2. Therefore,there isaunique mixed-
strategyNashequilibriumof thisgame inwhichVoters1and 2 bothrandomize withequal probability
betweenvotingonce andvotingthree timesfortheirpreferredcandidate. Voter3,of course simply
votesonce for A.
Give these preferences,intraditionalmajority-rule votingCandidateA wouldwinthe election
twovotesto one.However,inthisgame, Candidate Bwill winif Voter3 buysthree voteswhile Voter1

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CONSTRAINED QUADRATING VOTING: AN ALTERNATIVE VOTING MODEL
onlybuysone.Perequilibrium,the oddsof thishappeningare (1/2) * (1/2) = (1/4). Therebyallowing
Candidate Bto winthe election25%of the time.Froman efficiencystandpoint,thisisundesirable.
Voters1 and 2 have opposingpreferencesthatare equal inintensity,while Voter3prefers Candidate A
– soany reasonable notionof social efficiencywoulddemandthat Candidate A winthe election.In
addition, Voter1couldguarantee that Candidate A winsbypurchasingthree votes,butbuyingthree
votesforsure is not part of an equilibriumstrategy.Insteadbothplayersmix andrandomizationof
Voter2 is such thatVoter1 isindifferentbetweenbuyingthree votestowinforsure andvotingonly
once,whichoffersthe possibilityof winningforfree butalsothe riskof losing.
Considerchangingthe model byincreasingthe intensitywithwhich Voter2(the minorityvoter)
prefersCandidate B.Specifically,letu2(B) =10 + ⍺,where ⍺>0. The new payoff matrix is:
Voter 1 | Voter 2 1 votes for B 2 votes for B 3 votes for B
1 vote for A 10,0 5,4+⍺ 0,5+2⍺
2 Votes for A 9,0 9,-1 4,0
3 Votes for A 5,0 5,-1 5,-5
Again, Voter2 will nevercast2 votesand giventhis, Voter1will nevercasttwovotes. So, it
sufficestoconsiderthe reducedmatrix.
Voter 1 | Voter 2 1 votes for B
(Q)
3 votes for B
(1-Q)
1 vote for A
(P)
10,0 0,5+2⍺
3 Votes for A
(1-P)
5,0 5,-5

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CONSTRAINED QUADRATING VOTING: AN ALTERNATIVE VOTING MODEL
As before,thisgame hasnopure-strategyNashequilibrium.Tofindthe mixedstrategy
equilibrium,againassume that Voter1 votesonce forA withprobabilityof 1-p,whileVoter2votesonce
for B withprobabilityof q andthree timesforB with probabilityof 1-q.Voter1’spayoffshave not
changedso he isagain indifferentbetweenhistwoactionswhen q=1/2.Voter2,on the otherhand, still
receivesanexpectedpayoff of 0fromcasting one vote,butnow receivesandexpectedpayoff of:
p(5+2⍺) + (1-p)(-5) =10p + 2⍺p-5from castingthree votes. Voter2is indifferentbetweenthese choices
if 0 = 10p + 2⍺p-5 or p = 5/(2(5+⍺)).Therefore,inthe unique mixedstrategyequilibrium, Voter2
continuestorandomize withequal probabilitybetweenvotingonce andvotingthree times. Voter1,
howevernowvotesonce withprobability5/(10+2⍺) andthree timeswiththe remainingprobability.The
followingtable illustrates Voter1’srandomizationsas⍺ changes:
The firstrow correspondstothe original case.Then,as ⍺ increases,the probabilitythat Voter1
votesonlyonce declines.As ⍺ growsarbitrarilylarge,this probability growsarbitrarilysmall.The
equilibriummayfeelcounterintuitive,as ⍺ increases,the strengthof Voter2’spreferencesincrease—
but Voter2’s behaviordoesnotchange.Instead Voter1’sstrategychanges.Byvotingthree timeswith
higherprobability, Voter1 decreasesthe probabilitythat CandidateBwill win,evenif Voter2votes
three times.Inthe equilibrium,thiseffectexactlyoffsetsthe greaterutilitythat Voter2will receive if B
doeswin,leavingVoter2still indifferentbetweenvotingonce andvotingthree times.Indeed,the
⍺ P
0 ½
5 ¼
15 1/8

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CONSTRAINED QUADRATING VOTING: AN ALTERNATIVE VOTING MODEL
probabilitythatCandidate Bwinsisnow(1/2) * (5/(10+2⍺)),whichisdecreasingin ⍺ (forinstance,itis¼
when⍺=0, 1/8 when ⍺=5,and 1/16 when ⍺=15). Unfortunately,thisshowsthatthe restrictiontothree
voteshasseverelyaffectedthe efficientpropertiesof quadraticvoting:asthe preferencesof the
minorityvotergetstronger,the likelihoodof the minority Candidatewinningactuallyfalls.
In conclusion,amodificationof quadraticvotingwhere individualsare limitedtothree votesina
simple environmentwiththree voters,twocandidates,andpublicvalues,the votinggame hasaunique
Nashequilibriumwherethe voterswithopposingviewseachrandomizebetweenvotingonlyonce and
votingas manytimesas possible.Althoughthe restrictiontothree votesmaybe politicallyappealing,in
the environmentstudiedhere,itcausesquadraticvotingtolose itsefficiencybenefitsascomparedto
the traditional majorityrule.
References
Weyl, E. Glen, and Steven P. Lalley. "Quadratic Voting." Quadratic Voting by Steven P. Lalley, E.
Glen Weyl :: SSRN. Department of Statistics, University of Chicago, 13 Feb. 2012. Web. 04 Dec.
2016. <https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2003531>.
(Glen Weyl) - Department of Statistics, University of Chicago
(Steven Lalley) - Microsoft Research New York City; Yale University