Lec20

ChapterThirty-ThreeWelfareSocialChoiceDifferenteconomicstateswillbepreferredbydifferentindividuals.Howcanindividualpreferencesbe“aggregated〞intoasocialpreferenceoverallpossibleeconomicstates?AggregatingPreferencesx,y,zdenotedifferenteconomicstates.3agents;Anna,Bill,andCindy.Usesimplemajorityvotingtodecideastate?AggregatingPreferencesMorepreferredLesspreferredAggregatingPreferencesMajorityVoteResultsxbeatsyAggregatingPreferencesMajorityVoteResultsxbeatsyybeatszAggregatingPreferencesMajorityVoteResultsxbeatsyybeatszzbeatsxAggregatingPreferencesMajorityVoteResultsxbeatsyybeatszzbeatsxNosociallybestalternative!AggregatingPreferencesMajorityVoteResultsxbeatsyybeatszzbeatsxMajorityvotingdoesnotalwaysaggregatetransitiveindividualpreferencesintoatransitivesocialpreference.Nosociallybestalternative!AggregatingPreferencesAggregatingPreferencesRank-ordervoteresults(lowscorewins).AggregatingPreferencesRank-ordervoteresults(lowscorewins).x-score=6AggregatingPreferencesx-score=6y-score=6Rank-ordervoteresults(lowscorewins).AggregatingPreferencesx-score=6y-score=6z-score=6Rank-ordervoteresults(lowscorewins).AggregatingPreferencesx-score=6y-score=6z-score=6Nostateisselected!Rank-ordervoteresults(lowscorewins).AggregatingPreferencesx-score=6y-score=6z-score=6Nostateisselected!Rank-ordervotingisindecisiveinthiscase.Rank-ordervoteresults(lowscorewins).ManipulatingPreferencesAswell,mostvotingschemesaremanipulable.I.e.oneindividualcancastan“untruthful〞votetoimprovethesocialoutcomeforhimself.Againconsiderrank-ordervoting.ManipulatingPreferencesThesearetruthfulpreferences.ManipulatingPreferencesThesearetruthfulpreferences.CindyintroducesanewalternativeManipulatingPreferencesThesearetruthfulpreferences.CindyintroducesanewalternativeManipulatingPreferencesThesearetruthfulpreferences.Cindyintroducesanewalternativeandthenlies.ManipulatingPreferencesThesearetruthfulpreferences.Cindyintroducesanewalternativeandthenlies.Rank-ordervoteresults.x-score=8ManipulatingPreferencesThesearetruthfulpreferences.Cindyintroducesanewalternativeandthenlies.Rank-ordervoteresults.x-score=8y-score=7ManipulatingPreferencesThesearetruthfulpreferences.Cindyintroducesanewalternativeandthenlies.Rank-ordervoteresults.x-score=8y-score=7z-score=6ManipulatingPreferencesThesearetruthfulpreferences.Cindyintroducesanewalternativeandthenlies.Rank-ordervoteresults.x-score=8y-score=7z-score=6-score=9zwins!!Bonusinfo:ForMoreinfoaboutdiffernetvotingrules,pleasereadDonaldSaari’sbooks.DesirableVotingRuleProperties1.Ifallindividuals’preferencesarecomplete,reflexiveandtransitive,thensoshouldbethesocialpreferencecreatedbythevotingrule.2.Ifallindividualsrankxbeforeythensoshouldthevotingrule.3.Socialpreferencebetweenxandyshoulddependonindividuals’preferencesbetweenxandyonly.DesirableVotingRulePropertiesKennethArrow’sImpossibilityTheorem:Theonlyvotingrulewithallofproperties1,2and3isdictatorial.DesirableVotingRulePropertiesKennethArrow’sImpossibilityTheorem:Theonlyvotingrulewithallofproperties1,2and3isdictatorial.Implicationisthatanondictatorialvotingrulerequiresgivingupatleastoneofproperties1,2or3.SocialWelfareFunctions1.Ifallindividuals’preferencesarecomplete,reflexiveandtransitive,thensoshouldbethesocialpreferencecreatedbythevotingrule.2.Ifallindividualsrankxbeforeythensoshouldthevotingrule.3.Socialpreferencebetweenxandyshoulddependonindividuals’preferencesbetweenxandyonly.SocialWelfareFunctions1.Ifallindividuals’preferencesarecomplete,reflexiveandtransitive,thensoshouldbethesocialpreferencecreatedbythevotingrule.2.Ifallindividualsrankxbeforeythensoshouldthevotingrule.3.Socialpreferencebetweenxandyshoulddependonindividuals’preferencesbetweenxandyonly.Giveupwhichoneofthese?SocialWelfareFunctions1.Ifallindividuals’preferencesarecomplete,reflexiveandtransitive,thensoshouldbethesocialpreferencecreatedbythevotingrule.2.Ifallindividualsrankxbeforeythensoshouldthevotingrule.3.Socialpreferencebetweenxandyshoulddependonindividuals’preferencesbetweenxandyonly.Giveupwhichoneofthese?SocialWelfareFunctions1.Ifallindividuals’preferencesarecomplete,reflexiveandtransitive,thensoshouldbethesocialpreferencecreatedbythevotingrule.2.Ifallindividualsrankxbeforeythensoshouldthevotingrule.Thereisavarietyofvotingprocedureswithbothproperties1and2.SocialWelfareFunctionsui(x)isindividuali’sutilityfromoverallallocationx.SocialWelfareFunctionsui(x)isindividuali’sutilityfromoverallallocationx.Utilitarian:SocialWelfareFunctionsui(x)isindividuali’sutilityfromoverallallocationx.Utilitarian:Weighted-sum:SocialWelfareFunctionsui(x)isindividuali’sutilityfromoverallallocationx.Utilitarian:Weighted-sum:Minimax:SocialWelfareFunctionsSupposesocialwelfaredependsonlyonindividuals’ownallocations,insteadofoverallallocations.I.e.individualutilityisui(xi),ratherthanui(x).Thensocialwelfareis

whereisanincreasingfunction.SocialOptima&EfficiencyAnysocialoptimalallocationmustbeParetooptimal.Why?SocialOptima&EfficiencyAnysocialoptimalallocationmustbeParetooptimal.Why?Ifnot,thensomebody’sutilitycanbeincreasedwithoutreducinganyoneelse’sutility;i.e.

socialsuboptimalityinefficiency.UtilityPossibilitiesOBOA00UtilityPossibilitiesOBOA00UtilityPossibilitiesOBOA00UtilityPossibilitiesOBOA00UtilityPossibilitiesOBOA00UtilityPossibilitiesOBOA00UtilityPossibilitiesOBOA00Utilitypossibilityfrontier(upf)UtilityPossibilitiesOBOA00Utilitypossibilityfrontier(upf)UtilitypossibilitysetSocialOptima&EfficiencyUpfisthesetofefficientutilitypairs.SocialOptima&EfficiencyUpfisthesetofefficientutilitypairs.SocialindifferencecurvesSocialOptima&EfficiencyUpfisthesetofefficientutilitypairs.SocialindifferencecurvesHighersocialwelfareSocialOptima&EfficiencyUpfisthesetofefficientutilitypairs.SocialindifferencecurvesHighersocialwelfareSocialOptima&EfficiencyUpfisthesetofefficientutilitypairs.SocialindifferencecurvesSocialoptimumSocialOptima&EfficiencyUpfisthesetofefficientutilitypairs.SocialindifferencecurvesSocialoptimumisefficient.FairAllocationsSomeParetoefficientallocationsare“unfair〞.E.g.oneconsumereatseverythingisefficient,but“unfair〞.Cancompetitivemarketsguaranteethata“fair〞allocationcanbeachieved?FairAllocationsIfagentAprefersagentB’sallocationtohisown,thenagentAenviesagentB.AnallocationisfairifitisParetoefficientenvyfree(equitable).FairAllocationsMustequalendowmentscreatefairallocations?FairAllocationsMustequalendowmentscreatefairallocations?No.Whynot?FairAllocations3agents,sameendowments.AgentsAandBhavethesamepreferences.AgentCdoesnot.AgentsBandCtradeagentBachievesamorepreferredbundle.ThereforeagentAmustenvyagentBunfairallocation.FairAllocations2agents,sameendowments.Nowtradeisconductedincompetitivemarkets.Mustthepost-tradeallocationbefair?FairAllocations2agents,sameendowments.Nowtradeisconductedincompetitivemarkets.Mustthepost-tradeallocationbefair?Yes.Why?FairAllocationsEndowmentofeachisPost-tradebundlesare

andFairAllocationsEndowmentofeachisPost-tradebundlesare

andThen

andFairAllocationsSupposeagentAenviesagentB.I.e.FairAllocationsSupposeagentAenviesagentB.I.e.

Then,foragentA,FairAllocationsSupposeagentAenviesagentB.I.e.

Then,foragentA,

Contradiction.