Arrow axiom and full rationality for fuzzy choice

Arrowaxiomandfullrationalityforfuzzychoice

functions

IrinaGeorgescu∗

TurkuCentreforComputerScience,˙AboAkademiUniversity,Institutefor

AdvancedManagementSystemsResearch,Lemmink¨aisenkatu14,FIN-20520

Turku,Finland

Abstract

AclassicalresultforcrispchoicefunctionsshowstheequivalencebetweenArrowax-iomandthepropertyoffullrationality.InthispaperwestudyafuzzyformofArrowaxiomformulatedintermsofthesubsethooddegreeandofthedegreeofequality(offuzzysets).WeprovethatafuzzychoicefunctionverifiesFuzzyArrowAxiomifandonlyifitis(fuzzy)fullrational.WealsoshowthattheseconditionsarealsoequivalentwithweakandstrongfuzzycongruenceaxiomsWFCAandSFCA.ItisstudiedtheArrowindex,anewconceptthatindicatesthedegreetowhichafuzzychoicefunctionverifiestheFuzzyArrowAxiom.

Keywords:Fuzzychoicefunction;Fullrationality;ArrowAxiom

1Introduction

Therationalityofaconsumerisafrequentresearchtopicinclassicalconsumer

theory.ByUzawa[18]”therationalityofaconsumermaybedescribedby

postulatingthattheconsumerhasadefinitepreferenceoverallconceivable

bundlesandthathechoosesthosecommoditybundlesthatareoptimumwith

respecttohispreferencesubjecttobudgetaryconstraints”.

Samuelson’stheoryofrevealedpreference[11]expressestherationalityof

aconsumerintermsofsomepreferencerelationsassociatedwithademand

function.Uzawa[17]andArrow[1]havedevelopedarevealedpreferencetheory

inanabstractframework.TheworkofUzawaandArrowwascontinuedby

Richter[10],Sen[12,13,14],Suzumura[15,16]andmanyothers.

Following[16],achoicespaceisapair󰀕X,B󰀖whereXisanon-emptyuni-

verseofalternativesandBafamilyofnon-emptysubsetsofX.Inthetermi-

nologyofconsumersthepair󰀕X,B󰀖iscalledabudgetspace;theelementsofX

∗Correspondingauthor.Tel.:+358-2-2153339;fax:+358-2-215-4809.E-mailaddress:irina.georgescu@abo.fi(I.Georgescu)

1arecalledbundlesandthesetsinBarecalledbudgets.ThenasetS∈Bcan

betakenasanavailablesetofalternatives.

Achoicefunction(=consumer)onachoicespace󰀕X,B󰀖isafunctionC:B→P(X)whichtoanyS∈Bassignsanon-emptysubsetC(S)ofS.

C(S)iscalledthechoicesetofS.BytherationalityofCwemeantofinda

preferencerelationQonXsuchthatforanyavailablesetSthechoicesetC(S)

coincideswiththesetofQ-greatestelementsofS.ThusCisrationalandQis

arationalizationofC.IfQisreflexive,transitiveandtotalthenwesaythatC

isfullrational.

Theresultsin[1,12,13]areobtainedassumingthatBcontainsthenon-

emptyfinitesubsetsofX.Inthisframework,by[16],p.28,thefullrationality

ofachoicefunctionCisequivalenttothefollowingaxiomintroducedbyArrow

in[1]:

(AA)ForanyS1,S2∈B,ifS1⊆S2thenS1∩C(S2)=∅orS1∩C(S2)=C(S1).

Theaimofthispaperistoobtainanextensionofthisresultinthecontext

offuzzychoicefunctions.

In[2]Banerjeeintroducedaclassoffuzzychoicefunctionsandstudiedtheir

fuzzyrevealedpreferencetheory.Weworkwithamoregeneralgeneraldefinition

offuzzychoicefunctions.Banerjeefuzzifiesonlytherangeofachoicefunction;

inourapproachboththedomainandtherangeofachoicefunctionaremade

offuzzysubsetsofauniverseofalternativesX.Papers[6,7]developatheory

ofrevealedpreferenceforthesefuzzychoicefunctions.

Section2containssomepreliminaryresultsontheresiduatedstructureof

therealinterval[0,1]andsomebasicthingsonfuzzyrelations.

Somebasicdefinitionsandresultsonfuzzychoicefunctionsareincludedin

Section3.WeformulatetwohypothesesH1andH2asanaturalfuzzyextension

ofthesituationin[1,12,13,14,17].Therevealedpreferencetheoryoffuzzy

choicefunctionsdevelopedin[6,7]isbasedonthesehypotheses.

Section4concentratesthemaincontributionsofthispaper.Weformulate

theFuzzyArrowAxiom(FAA)intermsofthesubsethooddegreeI(.,.)and

thedegreeofequalityE(.,.)[4].WeprovethatifthehypothesesH1andH2arefulfiledthenFAAisequivalenttothe(fuzzy)fullrationality.Then

weshowthattheseconditionsarealsoequivalentwithweakandstrongfuzzy

congruenceaxiomsWFCAandSFCA[6].Themaintoolinprovingthisresult

isthemanipulationoftheresiduumproperties.

InSection5wedefinetheArrowindexofafuzzychoicefunctionandthe

similarityoffuzzychoicefunctions.Westudythebehaviourofthissimilarity

relationwithrespecttotheArrowindex.

2Preliminaries

Thissectioncontainssomepreliminarymatterwithrespecttotheresiduated

structureof[0,1]andsomebasicnotionsonfuzzyrelations.Thebackgroundis

givenby[4,8,21].

2Foranya,b∈[0,1]wedenotea∨b=max(a,b)anda∧b=min(a,b).

Moregenerally,forany{ai}i∈I⊆[0,1]wedenote󰀄

i∈Iai=sup{ai|i∈I}and

󰀃

i∈Iai=inf{ai|i∈I}.Then([0,1],∨,∧,0,1)becomesaboundeddistributive

lattice.Furthermore,[0,1]isacompletedistributivelattice.Thenonecan

defineanewbinaryoperation→on[0,1],calledimplicationorresiduation:a→b=󰀂{c∈[0,1]|a∧c≤b}.

Asimplecalculationyields

a→b=󰀁1ifa≤bbifa>b

Thebiresiduum↔isabinaryoperationon[0,1]definedbya↔b=(a→b)∧(b→a).

Thefollowingtwolemmascollectsomebasicpropertiesoftheresiduum.

Lemma2.1[4,8]Foranya,b,c∈[0,1]thefollowingpropertieshold:

(1)a∧b≤ciffa≤b→c;

(2)a∧(a→b)=a∧b;