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SocialChoiceandWelfaremanuscriptNo. (willbeinsertedbytheeditor) 2 Social Choice Under Incomplete, Cyclic Preferences 0 0 2 Majority/Minority-BasedRules, andComposition-Consistency n JobstHeitzig a J Institut fu¨r Mathematik, Universita¨t Hannover, Welfengarten 1, D-30167 Hannover, Ger- 9 many,e-mail:[email protected],fax:+495117625803 2 ] Thedateofreceiptandacceptancewillbeinsertedbytheeditor O C . Abstract Actual individual preferences are neither complete (=total) nor anti- h t symmetricingeneral,sothatatleasteveryquasi-ordermustbeanadmissibleinput a toasatisfactorychoicerule.Itisarguedthatthetraditionalnotionof“indifference” m inindividualpreferencesismisleadingandshouldbereplacedbyequivalenceand [ undecidedness. 1 In this context, ten types of majority and minority arguments of different v strength are studied which lead to social choice rules C : (S;R) 7→ C(S;R) ∈ 5 P(S)\{∅}thatacceptprofilesRofarbitraryreflexiverelations.Theserulesare 8 discussed by means of many familiar, and some new conditions, including im- 2 munityfrom binaryarguments.Moreover,itis provedthateverychoicefunction 1 0 satisfying two weak Condorcet-type conditions can be made both composition- 2 consistent and idempotent, and that all the proposed rules have polynomialtime 0 complexity. / h t Keywords algorithm,majority,quasi-order,timecomplexity,tournament. a m : v 1 Introduction i X Inthispaper,weshallreconsiderthequestionofhowagroupofindividualsshould r a proceedtomaketheirdecisioniftheywanttorealizeexactlyoneoutofacertain set X of alternatives. We shall do this under the usual assumptions that (i) all informationrelevantforthechoiceiswhichindividualspreferwhichalternatives towhichotheralternatives,andthat(ii)theresultoftheprocedureshouldnotonly be some alternative(s) that are “best” in X, but rather some “best” alternative(s) foreachnon-emptysubsetSofX.Forthesakeofsimplicity,wewillnotconsider “degrees” of preference here, although there are some good arguments that, for Iwouldliketothanktheeditorsforcallingmyattentiontoarticles[?]and[?]. 2 JobstHeitzig example,fuzzyrelationsareamoreprecisemodelofactualindividualpreferences. Still, the ideas presentedhere can probablybe translated into a fuzzy preference setting,particularly,theclassesofgeneralizedmajority/minorityrelationswewill studybelowmayperhapsbeinterpretedasfuzzysocialpreferences.Butalthough the introducedaxioms of immunity from binary argumentscan be interpreted as axioms for one individualwith fuzzy preferences(as in [?], for example) rather than a society with exactpreferences,theyare notintendedfor andhardlymake senseinthefuzzycase. Ifwe understand(i)in a way thatdenieseitherthe existenceof “degrees”of individual preference, or their measurability, or at least their relevance, we can assume that the individual preferences are given as binary relations on X, and since their significance lies in the comparison of different alternatives, we may adopt the convention to use reflexive relations only. We will see, however, that it isneitherjustified nor(formostpurposes)necessaryto assume thatindividual preferencerelationshaveanyspecialpropertiesotherthanreflexivity. Asusual,wedealwithafinitesetN ofindividuals,andwithoutlossofgener- alityletN ={1,...,n}andn>1.Iwillusei,j,...asvariablesoverindividuals. ThepossiblealternativesbuildasetX,anditisimportantthatX isfinite,too,but thisisnotanactualrestrictionsince,inallpracticalsituations,therewillalwaysbe onlyfinitelymanyfeasiblealternatives.Wemayaswellassumethat|X|=m>2. Alternativeswillbedenotedbyvariablesx,y,.... The motivation for (ii) is this: Because X is meant to contain also those al- ternativesthatmightturnouttobeactuallyimpossibleafterthedecisionismade, and because, on the other hand, there are situations in which two or more alternatives appear in the individual preferences in completely equivalent ways, we shall consider social choice rules C, i.e. algorithms that provide the group with a(multivalued)choicefunctionratherthanasinglealternative.Thisisafunction C :P(X)\{∅}→P(X)\{∅}thatassignstoeachnonemptyS ⊆X a(hopefully small)subsetC(S) ⊆ S ofalternativesthatIwill callacceptablehere.Then,as soon asthe setS of feasible alternativesis knownto the group,theycan choose somex ∈ C(S)eitherrandomly,orbydelegatingthatdecisiontosomeindivid- ual, orin whateveralternativeway.In fact, some authorsdiscuss algorithmsthat directlyassignaprobabilitydistributiontoSratherthanasubset.Buteveninsuch amodel,whichismoregeneralthanthepresentone,adiscussionofchoicerules remains important, since it is rather natural to define an induced choice rule by lettingC(S)bethesetofalternativesreceivinganonzeroprobability. In order to have a fixed interpretation of the input to the algorithm C, let us assumethat,foreachpair(x,y)∈X×X eachindividuali∈N hasbeen askedwhetherornotshethinksthat alternativexisatleastasdesirabletoherasalternativey. Theinputtothealgorithmisthentheresultingprofile(ofindividualpreferences) onX,i.e.thetupleR = (R ,...,R )ofreflexivebinaryrelationsonX,where 1 n xR y holdsif and only if i has answered “yes” to the abovequestion.To make i clearfromwhichprofileRtheoutputchoicefunctionisderived,thesetC(S)will insomeplacesmoreaccuratelybedenotedbyC(S;R). SocialChoiceUnderIncomplete,CyclicPreferences 3 An isomorphism between profiles R on X and R′ on X′ is here a bijection ϕ : X → X′ between the two sets of alternatives for which there is another bijectionψ : N → N′ betweenthetwosetsofindividualssuchthatxR y ⇐⇒ i ϕ(x)R′ ϕ(y)forallx,y ∈ X andi ∈ N.Inthiscase,RandR′ willbecalled ψ(i) isomorphic.Forexample,theidentitymapid isan isomorphismbetweenR = X (R ,...,R )andRψ := (R ,...,R )foreverypermutationψ : N → N 1 n ψ(1) ψ(n) ofindividuals.ThroughoutthispaperIwilladoptaverybroadideaofequalityand independence:We will only consider rules that are anonymousw.r.t. individuals andneutralw.r.t.alternatives,whichcanbesummarizedinthefollowingcondition ofisomorphisminvariance: (Iso) C(ϕ[S];R′)=ϕ[C(S;R)] wheneverϕisanisomorphismbetweenRandR′. WemaythenthinkofRandRψ asbeingessentiallythesameprofile.Moreover, ourruleswillbeindependentofirrelevantalternatives,i.e.fulfill (I) C(S;R)=C(S;R| ), S where R| = (R | ,...,R | ) and R | = R ∩(S ×S) are the restrictions S 1 S n S i S i ofRandR toS,respectively.Theconditionexpressestheideathat,forachoice i of feasible alternatives, only (preferences about) feasible alternatives should be relevant. This has the nice consequence that C is already determined as soon as C(X;R)isknownforallX andallprofilesRonX. 2 Atleastallquasi-ordersmustbeadmissibleindividualpreferences Tobeginwith,agivenprofileRprovidesuswiththefollowingadditionalrelations: P :=R \RopistheasymmetricpartofR ,whereRop ={(x,y):(y,x)∈R }, i i i i i i andasymmetricmeansthatxP yexcludesyP x.P containstheexpressedstrict i i i preferences of i. The rest of R is its symmetric part E := R ∩Rop, the ex- i i i i pressed equivalences of i. Although, in case of complete preferences, this relation isusually called“indifference”andis thenconsequentlydenotedbythe let- ter I, in case of incomplete preferences the above terminology is better for two reasons: It resembles standard order-theoretical terminology (cf. [?]), and, what is especially important in this context, E must not be confused with the rela- i tion xU y :⇐⇒ ¬(xR y ∨ yR x) which, according to the interpretation we i i i startedwith,encodestheexpressedundecidednessofiwhethertopreferx, ory, orwhethertovaluethemasequallydesirable. Being undecided does not automatically imply being also unconcerned, be- causeitmaysimplybethecasethatapersonmeasuresthealternativeswithmore than just one criterion and that x is better with respect to one aspect but y with respecttoanother.Mypointisthat,infact,inthefewestsituationseachindivid- ual uses only one criterion to evaluate alternatives.Let us thereforeconsiderthe situationofanindividualiwithasetC ofdifferentcriteriasothateachcriterion i c∈C provideshimwithadifferentpreferencerelationR .Thereisnolegitima- i ic tion(andnonecessity)toforbidhimtoconsiderxatleastasdesirableasy ifand onlyifxisatleastasgoodasy withrespecttoallhiscriteria.Moreprecisely:It 4 JobstHeitzig well makessense and shouldthereforesurely be legitimate for i to representhis preferencesbytherelationR := R .Itshouldalsobehisrighttouseas i c∈Ci ic manyandas independentcriteria ashe likes. But thenthe resulting R mightbe i T anypartialorder(i.e.reflexive,transitive,antisymmetric,butnotnecessarilycom- plete),evenifallusedcriterialeadtolinearlyordered (i.e.complete,1 transitive, and antisymmetric) preferences R . This follows from a basic order-theoretical ic fact(cf.[?]): Lemma1Any partial order on a finite set X is an intersection of at most |X| many linear orders on X, any quasi-order on X an intersection of at most |X| manytotalquasi-ordersoftheform(X ×X)\ (X \S)×S forsomeS ⊆X. Inthelatterkindoftotalquasi-orderstherearetw(cid:0)osubsetsS,X(cid:1) \Sofequallyde- sirablealternativessuchthatthosefromS arepreferredtotherest.Theynaturally arisefrombinarycriteria,thereforeLemma1impliesthatevenifonlybinarycrite- riaareused,R mightstillbeevenanyquasi-order(i.e.reflexiveandtransitive,but i notnecessarilycompleteorantisymmetric).Theimmediateandinmyeyesindis- putableconsequenceofthisisthat,contrarytothecompletenessassumptionthat occurs throughoutthe literature, every quasi-order on X must be an admissible individualpreferencerelation. A quite differentargumentagainstthe completenessassumptioncomesfrom thefactthatimighthavedifferentintentionswhenexpressingxE y ratherthan i xU y: On the one hand, saying “x and y are equally desirable to me” can be i interpreted as a vote not to distinguish between x and y, i.e., a vote for having eitherbothornoneoftheminC(S).Ontheotherhand,xU ymaybeinterpreted i asthestatement“Idonotwanttodecideaboutxandy”inthesensethatiwantsto delegatethedecisionaboutxandytothoseindividualsthathavemoreinformation about,ormoreinterestinthedistinctionbetweenxandy.Suchadelegationisan oftensensibleand,inpractice,verycommonbehaviour,especiallywhen(i)there are manyalternatives,or (ii) some pairsof alternativesdiffer only in their effect onfewindividuals,or(iii)someindividualshaverestrictedinformation. Before we turn to the conceptof majority and minority, let us finally remember thatithasalsobeennoticedthat,insomecases,individualpreferencesmaycon- taincycles.But,despitethefactthatwehavetobeabitcarefulwithPareto-type principles then (see below), there seems to be no problem at all in dealing with cyclicpreferences,too,atleastnotwhenthepreferenceinformationthatistaken intoaccountbytheruleCinanycaseonlyconsistsofsomecardinalitieslikethe following: r :=|{i∈N :xR y}|, xy i p :=|{i∈N :xP y}|, xy i e :=|{i∈N :xE y}|=e =r −p , xy i yx xy xy u :=|{i∈N :xU y}|=u =n−d , xy i yx xy 1 A note on terminology: in order theory, complete relations are usually called “total” instead,whilea“complete”quasi-orderisoneinwhichinfimaandsupremaexist. SocialChoiceUnderIncomplete,CyclicPreferences 5 d :=|{i∈N :xR y∨yR x}|=d =p +p +e , and xy i i yx xy yx xy d :=|{i∈N :xP yforsomex,y ∈S}|. S i Insomeplacesbelow,Iwillidentifyirreflexiverelationswiththeirreflexivecoun- terpartsanduseasomewhatsloppyarrownotation;forexample,A = a ⇄ c → b←a,d→bshouldbereadas A={(a,a),(b,b),(c,c),(d,d),(a,c),(c,a),(c,b),(a,b),(d,b)}. Onlyincaseofquasi-orders,themoreconvenientHassediagramscan,andwillbe used;theaboveexamplewouldthereforeratherlooklikethis: ac d A= (cid:31) (cid:30) . b Analogouslytotheorder-theoreticnotionsof“minimal”and“smallest”elements, let us call an alternative x A-optimal if, for all y, yAx implies xAy. As there are not always optimal elements of A| , we will often use its transitive hull S tr (A) = ∞ (A| )k, thatis, thesmallestquasi-orderonS containingA| . If S k=0 S S Aisatournament,thatis,completeandantisymmetric,thesetoftr (A)-optimal S S elementsisjustthetop-cycleofA| . S 3 Binaryargumentssupportedbymajorities Despite some irritating phenomenathat are related to the conceptof “majority”, thisnotionissurelythemostimportantideainthetheoryofsocialchoice.Itiswell knownthatwecan’texpectanyalternativetohavemajoritysupport,butwhatever exactdefinitionofmajoritywemayadopt,analternativexshouldonlybeaccept- able tothe groupif itis atleast in somesense “defendable”againstargumentsa majorityofthe groupmightgiveagainstx. Themostimportanttypeof suchar- gumentsseemstobewhatmaybecalled“binary”arguments:Apartofthegroup might argue that another alternative y is more desirable to them than x and that thereforexshouldnotbeacceptable.Since,foreveryalternativetheremightbea majoritythatfavoursadifferentalternative,weshouldthinkaboutpossibilitiesto counterandrefusesomeofthesebinaryargumentssoastomakeachoicepossi- ble.Assumethatwehaveinsomewaydecidedwhichkindsofbinaryargumentsto considerimportant,andletyAxdenotethefactthatthegivenprofileRcontains suchanargumentforyagainstx.Nowconsiderthefollowingpossibleconditions onC: (wIm ) Ifx∈C(S),y ∈S\{x},andyAx, A theremustbez ∈S\{y}withzAy. (Im ) Ifx∈C(S),y ∈S,andyAx,thenxtr (A)y A S (i.e.theremustbez ,...,z ∈S withxAz A···Az Ay). 1 m 1 m (sIm ) Eachx∈C(S)mustbeoptimalintr (A) A S (i.e.ifx∈C(S),y ∈S,andytr (A)x,thenalsoxtr (A)y). S S 6 JobstHeitzig Thecondition(wIm )ofweakA-immunityclaimsthatwhenyisusedinanargu- A mentagainstx,xcanonlybeacceptableiftheproposed“better”alternativey is subjecttoasimilarargument. PerhapsoneshouldatleastrequirethestrongerpropertyofA-immunity(Im ) A which claims that the argument yAx must even be answered with a sequence of similar argumentsthatlead back to x. Thisis because the existenceof such a sequenceeffectivelydemonstratesthattheargumentyAxisdestructiveintwore- spects:(i)Itcannot“consistently”betakenintoaccountwithouttheriskofmaking allalternativesinacceptable(ratherthanjustx):iftheargumentyAxissuccess- ful,thensoshouldbeallothersinthesequence,whichwouldresultinexcluding x,y,z ,...,z atonce.(ii)Theargumentisalsosomehowuselessforitssupport- 1 m ers,becauseitdoesnotplaceyinanessentiallybetterpositionthanx. The appeal of the even more restrictive strong A-immunity (sIm ) is that it A treats the situation more symmetrically: Even if there is only a sequence of arguments leading from y to x instead of a direct one, there shall also be such a sequenceleadingback. Let us look at some possible concretizations of the notion of majority, that is, somepossibledefinitionsofAasafunctionofR.Thefollowingbinaryrelations onX encode(proper)majoritiesofdifferenttypesandstrengthsαintermsofthe individualpreferences:ForallS ⊆X,x,y ∈S,and 1 <α61,let 2 xM y :⇐⇒ p >αn, (=⇒p >0) α xy xy xN y :⇐⇒ r >αn, (=⇒r >0) α xy xy xMSy :⇐⇒ p >αd >0, α xy S xNSy :⇐⇒ r >αd >0, α xy S xB y :⇐⇒ p >αd >0, α xy xy xD y :⇐⇒ r >αd >0, α xy xy xP y :⇐⇒ p >α(p +p )>0, α xy xy yx xR y :⇐⇒ r >α(r +r ), p >0 α xy xy yx xy xU y :⇐⇒ p >p , p +u >αn α xy yx xy xy (⇐⇒r >r , r 6(1−α)n), and xy yx yx xE y :⇐⇒ p >p , p +e >αd α xy yx xy xy xy (⇐⇒r >r , r >αd ). xy yx xy xy (For α = 1, P , R , U and E should, of course, not be confused with the α α α α preferencesofindividual1.) Mostofthesedefinitionshaveincommonthattheindividualsconstitutingthe majoritysharesomeopinionaboutxandyandbuildafractionofatleastαofall insomewayrelevantindividuals.Moreover,theremustatleastbeoneindividual within a majority. That unifying opinion can be either strictly preferring x to y (which correspondsto using the numberp in M , B , and P , for example), xy α α α orjustconsideringxatleastasdesirableasy(whichcorrespondstousingr in xy SocialChoiceUnderIncomplete,CyclicPreferences 7 N ,D ,andR ).Therelevantindividualsareeitherall(wherenisused),orthose α α α that are not undecided about x and y (which correspondsto using d ), or only xy thosethatexpressastrictpreferenceaboutxandy(asinP ).Althoughr +r α xy yx doesnot,ingeneral,enumeratesomesubsetofindividuals,R hastheappealthat, α similarly to P , it considersthose majorityarguments“equallystrong”in which α theproportionr /r isconstant.ThisisbecausexR yismainlyequivalentto xy yx α r /r > α (theconditionp 6=0isaddedonlytoensureantisymmetryand xy yx 1−α xy theinclusionsthatarestatedbelow).Insomeofthedefinitions,d occursinstead S ofn,whichmodelskindsof“semi-relative”majorities.Thiscouldprovideacom- promisebetweenusingabsolutemajoritiesandthepossiblerequirementthatC(S) shouldbeatleastindependentofindividualsthatarecompletelyundecidedabout allfeasiblealternatives.Ontheotherhand,itmightbeproblematicifwhetheror notindividuali is countedwould dependon perhapsjust one alternative’sbeing feasibleornot. Thelasttwodefinitionsdeserveamoredetailedexplanation:Theyexpressthe ideathatinthefirstplaceonlyindividualswithstrictpreferencesforxoverywill raise the corresponding argument against y, but that then additional individuals mayjointhemtobuildamajority.Forexample,ifonlyslightlymorepeoplestrictly preferxtoythanytox,theformermightpersuadeallwhoequallydesirexandy tosupporttheirargument,sothatintheendaverystrongmajorityevolves(relative toalldecidedindividuals);thisisassumedinthedefinitionofE .Onecouldalso α assumethat,onthecontrary,theundecidedindividualsarepersuadedtoconstitute an(absolute)majority,whichleadstoU . α Since α > 1, it can easily be seen that, despite N , NS, and D , all of the 2 α α α aboverelationsareantisymmetric.Moreover,wehavesome,perhapsunexpected, inclusionsthatareshowninthelowerpartofthediagramonpage10:(i)Whenα grows,therelationsobviouslyshrink(whichisrepresentedbythedottedlines). (ii)M ⊆MS ⊆B ⊆R ⊆P ⊆E ⊆D , α α α α α α α becausen>d >d ,r =p +e ,and S xy xy xy xy αd +e >α(r +r )>α(p +p )+e >αd . xy xy xy yx xy yx xy xy Similarly, M ⊆ N ⊆ NS ⊆ D , MS ⊆ NS, and B ⊆ U . (iii) For α α α α α α α α α 6 n , xR y is equivalentto p 6= 0 and r /r > n , the latter be- 2n−1 α xy xy yx n−1 ing equivalentto r > r . Then xR y is already implied by xE y, so that xy yx α α R =P =U .(iv)xU yimplies¬yN x. α α α α 1−α+1/n Summarizingthissection,wehavethefollowingclassesofmajorityrelations: M and N encode absolute majorities, MS and NS semi-relative majorities, α α α α B andD relativeones,P andR proportionalones,andE andU encode α α α α α α persuadedmajorities.Moreover,M,MS,B,P,E andU willbecalledthestrict types,andtherestnon-strict. 8 JobstHeitzig 4 Rulesbasedonimmunityfromclassesofbinaryarguments ForafixedprofileR,eachofthesets{tr (A ) : 1 <α6 1},whereAisoneof S α 2 thetentypesM,N,MS,...,isachainofquasi-orders.Nowitisimportantthat X isfinite:Thenthechainisalsofinite,andthefollowingLemmaapplies: Lemma2Every chain2 Q ⊆ ··· ⊆ Q of finite quasi-orders has a common 1 m optimalelement,i.e.somexthatisQ -optimalforallk 6m. k Proof. Since Q is finite, the set S of its optimal elements is not empty. Let m m S ⊆ S bethe(alsonon-empty)setofoptimalelementsofQ | .Then m−1 m m−1 Sm eachx∈S isalsoQ -optimal,because m−1 m−1 yQ x=⇒yQ x=⇒y ∈S =⇒xQ y. m−1 m m m−1 Thus,denotingthesetofoptimalelementsofQ | byS ,weinductivelyget k Sk+1 k S ,thestillnotemptysetofallcommonoptimalelements. (cid:3) 1 Note that, consequently, the set of common optimal elements can be found in polynomialtime. The above observation enables us to fulfill (sIm ) not only A for one specific majority relation, say M , but at once for a whole class 1/2+ε of majority relations of the same type but of different strength: We may simply define C(S) as the set of common optimal elements of, for example, the chain {tr (M ): 1 <α61}.Thenanyx∈C(S)canbedefendedagainstasequence S α 2 ofargumentsyM z M ···M z M xwithasequenceofequallystrongM- α 1 α α m α typearguments.ExchangingM byanyoftheothertypesofmajorityrelationsand varyingthelowerandupperboundsforα,wegetalargenumberofmajority-based ruleswithverygoodimmunityproperties.Letusintroducethefollowingnotation: For 1 6 β < γ 6 1,therulebasedonthechain{tr (A ) : β < α 6 γ},where 2 S α A is one of M, N, MS, ..., will be denoted by A . For example, we just (β,γ] introducedtheruleM . (.5,1] It seems quite natural to me to take minorities in the same way into account as majorities:SupposeweapplythedefinitionsofM ,...,U evenfor0<α6 1, α α 2 exceptthat we dropall of the “ > 0”-requirementshere. ThenD , P , and R α α α becomecompleterelationsforα6 1 (butonlyU andE remainantisymmetric 2 α α in general),and in the inclusions,R and P changeplaces: Still M ⊆ N ⊆ α α α α NS ⊆D ,butnow α α M ⊆MS ⊆B ⊆P ⊆R ⊆D α α α α α α andU ⊆E =R .Moreover,R =P ,andnolongerB ⊆U . α α n/(2n−1) 1/2 1/2 α α Now,requiring(Im ) as well for 0 < α 6 1 would ensure thatalso a mi- Aα 2 nority argumentyA x will be successful if only if it can be taken into account α “consistently”.Insuchacaseitisnotonlyharmlesstogivetheminoritythis“direct” power to exclude y, but it may even be indicated in certain situations: (i) 2 Chains(towers/nestedfamilies)ofotherkindsofrelationsarealsoanimportanttoolin modellinguncertainorstochasticallyvaryingpreferencesofoneindividual,cf.[?,?] SocialChoiceUnderIncomplete,CyclicPreferences 9 Whenanabsoluteorsemi-relativemajoritytypeisusedandmanyindividualsare undecided(becauseoftoofewinformation,forexample),or(ii)whenastricttype other than P is used and many individualsequally desire x and y (because they are notaffectedby theirdistinction,forexample),minorityargumentsshouldbe takenintoaccount. Moreover,if we even require (sIm ) for 0 < α 6 1, a minority argument Aα 2 yA x which is refused as a single argument, may still have “indirect” power: α Other such arguments can help constituting a sequence of arguments leading to the exclusion of x, if at least one of them is non-refusable.While indirectinflu- enceworkswithallofthetypes,directinfluenceofaminorityisonlypossibleifA isnoneofthecompletetypesD,P,orR:If,forexample,yD xforsomeα6 1 α 2 butfornoα′ > 1,thenr 6 1d ,thusr > 1d >αd ,hencealsoxD y. 2 yx 2 yx xy 2 xy xy α However,inthecrucialsituations(i)and(ii)describedabove,asubgroupofindi- vidualsthatwouldbeaminorityofoneoftheothertypesmightwellconstitutea majorityoftypeD,P,orR. The rules A with 0 6 β 6 γ 6 1 will be called minority-based rules, (β,γ] 2 thosewith06β 6 1 <γ 61mixedrules. 2 Asforthequestionwhichofthepossibletypesofmajority/minoritytouse,itmay turnoutthatthiscannotbecompletelydecidedintheusualaxiomaticway.Thus,in additiontotheshortaxiomaticdiscussioninthenextsection,therulesshouldalso becomparedfromamorepracticalperspective.Surely,sometypeslikeM,N,B, P, and D look more “simple” or “natural” than others, and E and U are based onquestionablebehaviouralimputations.Thereisagoodreasontoprefersomeof the“larger”types:WhatevertypeAweuse,anx∈C(S)maybecomesubjectto anargumentsupportedbyamajorityofa differenttype,sayyA′ x. Itmaythen α stillpossibletorefuseyA xonthebasisofasequenceofA -argumentsleading α α backtoy,ifonlyA′ ⊆ A ,i.e.,theruleusingAisinsomesense alsoimmune α α fromargumentscorrespondingtotypessmallerthanA. Following an argument from Section 2, another criterion is that xE y and i xU y shouldnotalwayshavethe sameeffect,whichis apointagainstusingM i andP. Inall,thenon-persuaded,relativetypesD(beingthelargestsuch)andB (being the largestthat allows“direct” influenceof minorities)seem to give the best compromisesso far.Asforthe rangeofα, its lowerboundβ shouldbe takenas smallaspossibleinordertokeeptheresultingsetC(S)assmallaspossible.On the other hand, taking the upper bound γ not to large could be a sort of protec- tionofminorities,sincethenitwouldbepossibletocounterargumentsofstrength > γ byasequenceofargumentsofstrengthγ.However,eveninsuchacaseone wouldprobablyaddtr (A )tothechain,thecorrespondingruleswillbedenoted s 1 by A Depending on how much power and protection we want to give mi- (β,γ],1 norities,onecouldforexampleusetherulesD ,D ,B ,B , or (.5,1] (0,1] (0,1] (0,2/3],1 evenB . (0,.5],1 10 JobstHeitzig Fig.1 Majorityandminorityrelations (cid:8)(cid:8)(cid:8)D...η (cid:8) . (cid:8) .. (cid:8)(cid:8)R..η .... (cid:8)(cid:8)P..η .... .... B.η ... ... ... . . . . . . .... .... .... (cid:8)(cid:8)(cid:8)D....5 . . . (cid:8) . .. . . (cid:8) .. B.....5(cid:8)(cid:8)P.5 = R.5HHEη ....... η6.5 . . minorities . . . . .. . majorities .... (cid:8)(cid:8)D...5+ε α>.5+ε (cid:8)..... (cid:8)(cid:8)R....5+ε=P....5+ε=E....5+ε ..... 0<ε6 4n1−2 (cid:8). . . . . (cid:8) .. .. .. .. .. Uη ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . U...5+HεH ... .... .... .... ... .... B...5+ε .... .... ... (cid:8)(cid:8)D..α .... .... .... ... (cid:8)(cid:8)E..α ...(cid:1)(cid:1). ... .... ... (cid:8)(cid:8)P..α (cid:8)(cid:8). N..αS UαMHH...S(cid:1)(cid:1)(cid:8)B......α(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)R(cid:8)...(cid:8)α(cid:8)(cid:8)(cid:8)(cid:8)(cid:8).(cid:8)(cid:8)(cid:8)N...........α(cid:1)(cid:1). .α (cid:8) . ...(cid:1)(cid:1)(cid:8)..(cid:8)(cid:8) .... M.α (cid:8)N1 . (cid:8) .. (cid:8) . (cid:8) ... (cid:8)(cid:8) ... (cid:8)(cid:8) . (cid:8) . (cid:8) M1 5 Someconditionsdiscussed Pareto-principles.Theideathatyisunacceptableifallindividuals(strictly)prefer xtoitcanonlyworkwhenthereisaminimalamountofrationalityinatleastone individual’spreferences.Thus,inasettingwherecyclicindividualpreferencesare explicitly admitted, adequate formulationsof this idea must take possible cycles intoaccount: