Endogenous Coalition Formation With Identical Agents

Fiaschi, Davide Working Paper Endogenous Coalition Formation With Identical Agents Quaderni - Working Paper DSE, No. 309 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Fiaschi, Davide (1998) : Endogenous Coalition Formation With Identical Agents, Quaderni - Working Paper DSE, No. 309, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4999 This Version is available at: https://hdl.handle.net/10419/159151 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ EndogenousCoalitionFormationwith Identical Agents1 DavideFiaschi2-Pier MarioPacini3 UniversitàdiPisa DipartimentodiScienzeEconomiche September 29,1998 1Inspiteof thispaper derivesfromajoint workof thetwo authors,thesections 1,3and4canbeattributedtoPierMarioPacini,whilethesections2and5to DavideFiaschi. 2d…[email protected] [email protected] 1Introduction Cooperativebehaviour oftenemergesat agroup,rather thansocial level.In manyinstancesweobservetheformationof independent andsometimecompeting groups,teams,clubs,cooperatives(coalitionsfor short)eachof them persecutingthesamegoal(inturnprovisionofcommodities,maximization ofpro…ts,raising ofpublicfunds,standardsofbehaviouretc.).Examples of thisbehaviour arenumerousbothat microandmacrolevel:scienti…cresearchgroups,universitydepartments,consumers’associations,…rmsasorganizations,consumptionandproductioncooperatives,industrialdistricts, internationalcommercialtreatisesamongcountriesareall instancesofvolunteer agreementsamong independent partiesthat coalesceto obtainasame goal.Oncecoalitionsform,societyispartitionedinacoalitionalstructure. Thisfact hasalreadybeenobservedandstudiedinanumber of worksfrom di¤erent pointsof view:amongthem[7], [1], [11] dealt withtheproblemof coalitionformationfortheprovisionofpublicgoods, [8] withtheproblem oftheformationofcooperativesofworkerswithdi¤erent workingcapacity, [6], [2], [3], [4] dealt moredirectlywiththeproblemof coalitional structures formationinmoregeneral settings(see [10] for anexcellent review).Inthese worksithasbeenstressedthattheformationofcooperatinggroupsisa¤ected byamoralhazardproblemwheneverprivateactionscannotbemonitored; thisfactmaycontributetodeterminethe extent towhichcooperationcan besustainedasaself-enforcing equilibrium.Thequotedworks show that the cooperativeagreement is self-sustainedif thecoalitionsizeisunder acertain threshold,whereasit wouldnot beincentivecompatibleinlarger coalitions; theseconclusionsaremuchinthespirit of [13] (see also[12]) Thisworktakesasimilarapproachbutwithadi¤erentstrategy:…rst we…ndasetofconditionsunderwhichcooperationcanemergeingroups that aresubsetsof thewholepopulation; thenwedeterminethelimitsto the sustainabilityof cooperativebehaviour withincoalitionsand…nallywestudy howasocietywill formacoalitionalstructure.Weapproachtheproblem intwosteps;…rstwe examineagents’behaviourwithineachofthepossiblecoalitionscanformandweinvestigatethekindofequilibriathatwe canexpect toemerge,coalitionstakenasgiven:weshowthat the equilibriaoftheinfra-coalitionalinteractiondependonindividualcharacteristics (outsideopportunities)of agents,returnsto scaleof theavailabletechnology forthetransformationofcooperative e¤ortsinoutputandthedistributive rulegoverningtheallocationoftheobtainedoutput.Onceweknowhow peoplebehaveinacoalition,weproceedbackwardandstudywhatkindof coalitionalstructurescanforminsuchawaythatno onehasanincentive todeviatefrom.Inthis stageweshall see that the equilibriumcoalitional 1 structuresdependontheinstitutionalsettingsrulingthedegree ofsocial mobilitywithinpopulation.We examinetwoextremeinstitutional arrangements:inthe…rstweassumetheretobeperfectmobilityandcapacityto changecoalitions,while,intheother,weassumeanextremelylimitedmobilityduetothefact thatevenasinglevetocanforbidtheformationofa coalitional structure.Asweshall see,thesetwoextremecaseswill giverise totwodi¤erent setsof equilibriumcoalitional structures. Inour settingagentscanengageinpre-playnon-bindingcommunication amongthemselves(see [8]).Bythistheycanagree onpareto-dominating, self-enforcingequilibriainanyoccasionaselectionproblemisconcernedwith. However,just becausesuchpre-playtalksareunbinding,noonecanbeforced toperformactionsnot compatiblewithindividual incentivesand,for example,cooperatewhenonlygrouprationalitywouldcall forit.Thereforewe adopt amixedapproachinwhichwesupposeagentscandeal eachother and constraint themselvesinabinding manner,but infull respect of theindividualfreedomnontoparticipateinanyagreementwhichprovidesindividual (andcoalitional aswell)incentivesto deviatefrom.Thisapproachisnot new intheliteratureandwerefer tothe extensiveworkof [9] and[10] for further reference. Theworkisclearlylimitedbysomestrongassumptionsthatwemake; amongthem ² thefact that all individualsareidentical ² thefact that thedistributionruleis…xed ² thefact that wedonot examineintermediateinstitutional settings,in whichsocial mobilitymaybejust partiallyinhibited. Athroughout workshouldget ridof theseassumptions; however,inthis work,wekeepthemas simplifyingdevicesinorder tocarryout ananalysis capableofmanageableresults.Section2describesthebasic characteristics ofthe economy;Section3…ndswhatwill bethe equilibriumself-enforcing outcomeswithinanypossiblecoalition;Section4analyzesthe equilibrium outcomesofthecoalitionalstructureformationprocess underthetwodifferent institutional settingsmentionedabove; …nallySection5examinesand comparesthepropertiesof thetwodi¤erent kindsof equilibriumcoalitional structures.Conclusionsclosethepaper withsomeinsight intofurther work. 2 2Descriptionofthemodel Thereisapopulation= of Iagentsindexedbyi; theyaresupposedidentical inall characteristicsandendowedwithoneunitoftimeandoneunitof aphysicalcapital.Capitalisusedwithlabourtoproduceaconsumption commodityYbymeansof apubliclyandfreelyavailabletechnology.There arenoalternativeusesfor capital. AgentscanproducethecommodityYeitherinisolationorwithina group.It takesplaceinagroupwhenagentsinasubsetSµ=pool their capitalunits;thenagentsindependentlydecidetheamountoftheirown timetospendonproduction.Thecontributionof thecapital endowment is anecessaryandsu¢cientconditiontohavetheright toreceiveashareof theproducedoutput. GivenacoalitionSfor theproductionand distributionof thecommodity Y,let XSdenotethecardinalityof S.AcoalitionSisproperif 2·XS·I; ifXS=1,Sisasingleton.Acoalitionalstructureisapartitionof=into subsetsandisdenotedby¾=fS1;S2;:::;Sj;:::;SJg,whereJ2N+isthe number of non-emptygroupsinthecoalitional structure¾.Anycoalitional structuremust satisfythefollowingproperties: SJ j=1Sj== 8(h;j),h6=j,Sh\Sj=;,Sh;Sj2¾. Let §betheset of all possiblecoalitional structuresin=. 2.1Individual characteristics Let lidenotetheamount of timethat agent ispendsonproduction,li2[0;1]. If anagent choosestoparticipateintoacoalitionShemust …rst contribute hisowncapitalendowment;then hehastodecidetheamountoftimeli todedicatetoproduction;hence(1¡li)istheleisuretime.Themorehe spendshistimeonproduction,themorethetotal output increases; but one unit of leisuregiveshiman utilityof !1.Anagent canalwaysdecidenot to participateinacoalition;insuchacasehekeepshiscapital endowment to produceinautharchy. Agents’utilitydependsonconsumptionofcommodityandonleisure. For simplicitypurposes,theutilityfunctionisassumedtobelinear inall its argumentsandtakestheform Ui=yi+(1¡li)¢! whereyiisthequantityagent i receivesof commodityY. 1Sinceagentsaresupposedidentical,theoutsideopportunitywill not beindexedbyi. 3 Anagentspendinghistimeonproductionwill becalledacooperator,a defectorotherwise.Agentscanchoosetocontribute evenafractionof their workingcapacity,i.e.0·li·1,but theyarenotallowedtoplaymixed strategies. Finallyweassumethat,whereasindividual capital contributionscanbe publiclyobserved,liisavariableknownonlytoagent iandcannot bedirectly monitoredbyanyone else. 2.1.1Technological characteristics ThetechnologyavailablefortheproductionofYcanberepresentedbya productionfunctionF(K;L),whereKisthecapitalstockandListhe amountoflabour;clearly,inanycoalition,0·L·Ksinceno onewill contributehisownworkinge¤ortinacoalitioninwhich hehasnoright toreceivetheproducedoutput.WeassumeF(:;:)tosatisfythefollowing properties. Axiom1The productionfunctionF(K;L)is suchthat 1.F(K;L)=A¢minfK®;L®g 2.2¢!>A>! 3.®>0 NoticethatAxiom1togetherwith0·L·KyieldtoF(K;L)= A¢L®.Theparticularformof theproductionfunctionisasimplifying device, allowing for returnsto scaleto labour invariant to theamount of theworking e¤ort put intotheproductionprocess;themagnitudeof thereturnstoscale dependsonthevalueof®.Thefact that thescaleparameterAisgreater thantheoutsideopportunityservesonlytoinducetheperforming ofthe workinge¤ort whenanagent actsalone,whilethefact that 2¢!>Aserves toexcludethat cooperationmayariseinpropercoalitionsforreasonsthat are not just strategic.It isworth noticing that theproductionwithinacoalition ofsizeKhasnoexternale¤ectontheproductionofothercoalitions(see [10]). 2.1.2Distributivecharacteristics OncethetotaloutputYisproducedinagivencoalitionbytheworkinge¤orts ofsomeofitsmembers,itisthenredistributedwithinthesamecoalition. Thedistributionof Yisgovernedbyanequal sharingruleRthat weassume 4 tobethesameinall coalitions;inotherwordsall membersofacoalition that contributedtheir capital endowmentshavetheright toreceiveansame shareof thetotal output Y,independentlyof therespectivecontributionin termsofworkinge¤ort.Thisassumptionismotivatedbythefact that,li beingunobservableandthecapitalassetshavingequalvalues,thereareno meanstodiscriminateamongtheparticipantstoacoalition. Tobemorepreciseontheformof R,takeageneric coalitionSof cardinalityXSandsaythat YSisthetotal output producedwithinSbymeans of theworkinge¤ortsof itsmembers,i.e.YS=A¢³Pj2Slj ´®;thenfor all i 2Swehave yS i=Ri(Y)=YS XS=A¢³Pj2Slj ´® XS whereRi(:)denotesthequantitythat thedistributiveruleRallocatesto agent i.Inview of this,thepayo¤toanagent i fromacoalitionSwhich he participatedintoisgivenby ¼i (lijS)=A¢³Pj2Slj ´® XS+(1¡li)¢! Itisclear that,oncethelevel ofcooperationPj2Sljisgiven,defectorsare alwaysbetter o¤thancooperatorsinanycoalition.Howeverthe essenceof thesocialgamewearegoingtoexamineinthe…rstpartofthepaperis whether it isworthor not,fromtheindividual point of view,to increasethe level of cooperationconsidering that theworking e¤ort a¤ectsthecoalitional output andthat thisreturnsasabene…t tothesubject viathedistributive rule,but,at thesametime,thisreducesleisure. 3Individualdecisionsandequilibriawithin coalitions Beforedealing withtheproblemof how asocietywill structureitself incoalitions,we examinehow peoplebehavewithinanypossiblecoalitionandwhich kindof equilibriawecanexpect init.Wearelooking for self-enforcing equilibria,i.e.equilibriano onehasanincentivetodeviatefrom.Thestrategy wefollowtoprovethe existenceofsuchkindofequilibriaisthefollowing: …rst weprovethe existenceof Nashequilibria(NE),i.e.equilibriarobust to unilateraldeviations,inthegametakingplacewithinacoalitionSofXS agentswhentheyaretodecidethelevel of their timetodedicatetoproduction,giventhat theyhavealreadycommittedtoS.Thereafter wecheckfor 5 individual rationalityof NEs;thisisaveryimportant stepsinceagentswill alwaysbeabletoimproveuponindividuallynot rational solutionsbywithdrawing fromactual coalitionscreating new ones(remember that production inautharchyisalwaysapossibility). 3.1Incentivecompatibleandindividuallyrational NE withingivencoalitions Inorder topursuetheanalysislet ustakeageneric coalitionS½=andan agent i 2S.Furthermoresupposeagent hascorrect expectationsabout the level of cooperationsuppliedbytheother agentsinSandthat thislevel will not bea¤ectedbyhisownaction.Thenwecanstatethefollowingtheorem concerning existenceof incentivecompatibleNEsinS; asto theterminology full cooperation,partialcooperationandfull defectionstandforsituations inwhichli=1,0<li<1andli=0respectively8i 2S. Proposition1Supposethe economysatis…esAssumption1andthat the distributiveruleRisanequal sharingrule;then 1.inanysingletoncoalition,li=1isthe dominant strategy; 2.if ®·1,partial cooperationisthe onlyNEinanypropercoalition; 3.if ®>1,generalized defectionisalwaysaNEinanypropercoalition; 4.ForanyX¸2there existsavalue®X<2,suchthat full cooperation isaNEinany coalitionwithcardinalityuptoXprovided®¸®X;if ®<®Xat least thosecoalitionswithcardinalitygreaterthanXadmit full defectionasthe onlyNE. 5.if ®>®I´®¤full cooperationisaNEinanypropercoalition. TheproofofthisPropositionisratherlongandispostponedtothe Appendix.Substantiallyitestablishesthatinapropercoalitionofcardinality XSthereare either oneor twoNEdepending onthereturnstoscaleof the coalitional productionfunction. Now weturntoindividual rationalityof incentivecompatibleNE,since individuallynotrational outcomeswill notbeacceptedasequilibriumoutcomes.Resultsaresummarizedinthefollowinglistofremarks,wherewe alwaysassumethe economyto satisfyAssumption1andthedistributiverule Rtobeof the equal sharingtype,asinProposition1. Remark1If ®·1,NEarenot individually rational inany proper coalition. 6 Proof.If returnstoscaleareatmost costant,theonlyNEinanyproper coalitionisapartialcooperationone,inwhichtheworkinge¤ortsupplied byanymember of acoalitionSis ¹ l=¡®¢A !¢1 1¡® X2¡® 1¡® S :(1) Alevel ¹ lof cooperationispro…tableif it grantseachagent at least thepayo¤ hecouldget byhimself actingasasingleton,i.e.if A¢¡¹ l¢XS¢® XS¸A; fromwhich¹ l¢X®¡1 ® S¸1:(2) Substitutingfromequation(1)into(2)weget that theNE,whenreturnsto scaleareat most constant,isindividuallyrational if ¡®¢A !¢1 1¡® X2¡® 1¡® S ¢X®¡1 ® S¸1)XS·µ®¢A ! ¶®·2; wherethelastinequalityfollowsfromAssumption1.2.Hencethepartial cooperationequilibriumisneverindividuallyrationalinanycoalitionwith at least twoagents. Remark2Full defectionequilibria areneverindividuallyrational. Proof.Theproof followsimmediatelyfromequation(2),sinceit can never besatis…edfor ¹ l=0andXS>0. Remark3Full cooperationNE(providedit exists)isindividuallyrational. Proof.>FromProposition1weknowthat full cooperationcanemergein propercoalitionsonlyifreturnstoscaleareincreasing,i.e.®>1.Itcan easilybecheckedthatequation(2)isalways satis…edforforsuchvaluesof ®andanyXS¸1. Summarizing,full cooperationistheonlyindividuallyrational NE,providedit exists.Thepayo¤onegetsinanyother equilibria(i.e.full defection 7 had Vi (S)= 8 > > > > > > < > > > > > > : 3if XS=1 4:2if XS=2 5:2if XS=3 6if XS=4 0if XS=5 0if XS=6 Takethe coalitional structure¾=ff1;2;3g;f4;5g;f6gg andthe coalitional structure¾0=ff1;2;3;4g;f5;6gg.Clearly¾02P (¾)sincethe coalition ofthe …rstfouragentsimproveswithrespect to¾;however¾0=2W(¾) sinceno oneisworseo¤in¾0withrespect to¾.Therefore¾canand will bedisruptedinfavourof¾0.Considernowthe coalitionalstructure ¾00 =ff1;2;3g;f4;5;6gg.It iseasytocheckthat ¾02W(¾00)since agents5 and6 will su¤er aloss in¾0withrespect to¾00 but ¾02P (¾00)sincethe coalitionof the …rst fouragent getsanimprovement.Inthiscasethe coalitional structure¾00 cannot bedisrupted. Wenow turnto the existenceproblemandto thedescriptionof thecharacteristicsofVCEintermsofcoalitions.Themainresultis statedinthe followingproposition. Proposition3Supposethat the economy satis…esAssumption1, ®2(1;®¤) andthe distributionisgoverned byanequalsharingruleR.Thenall the concentratedcoalitional structuresareVCEfor the coalitional structuresformationgame. Proof.First weprovethat inanyequilibriumcoalitional structurethereare exactly¹ N®+1coalitions. 1.Supposethere existsanequilibriumcoalitional structure¾withfewer than¹ N®+1coalitions; thenat least oneof them,saycoalitionS,must possess morethan¹ X®members sothat thevalueoftheircoalition is0.Nowconstructacoalitionalstructure¾0obtainedby¾simply replacingSbytwo ormorecoalitionswithcardinalityless thanor equal to¹ X®,therestofthecoalitionalstructureremainingthesame. Thenewcoalitionalstructure¾wouldbesuchthat¾02P (¾)and ¾0=2W(¾)sincethemembersofnewcoalitions strictlyimprovetheir situations,whiletherest of thepopulationisuna¤ected.Fromthiswe get P (¾)*W(¾),i.e.¾cannot beaVCE. 2.Nowsupposethat the equilibriumcoalitionalstructure¾possesses morethan¹ N®+1coalitions;supposefurther than no oneof themhas 14 morethan¹ X®members,otherwisethesamereasoningasinpoint1 abovecouldbeapplied.Thenarrangethemindecreasing order of cardinality,obtainingthesequenceS1,S2;:::;S¹ N+1;:::;S¹ N®+1+v,v>0, withXS1>::: >XS ¹ N®+1+v.Nowtakethecoalitional structure¾0obtainedfrom¾byredistributingtheagentsinSv j=1S¹ N®+1+joverthe coalitionsS1;:::;S¹ N®+1insuchawaythat XS0·¹ X®for anyS02¾0. Anyof thenew coalitionscontainsat least thesamenumber of agents asin¾so that ¼¾0 i¸¼¾ i8i and¼¾0 i>¼¾ ifor all themembersof at least acoalitionS02¾0.Thismeansthat ¾02P (¾)and¾0=2W(¾).Again wehaveP (¾)*W(¾),i.e.¾cannot beaVCE. ThusfarweprovedthatanyVCEmustpossess exactly¹ N®+1coalitions.Noequilibriumcoalitionalstructurecanpossess acoalitionSwith morethan¹ X®,becausesuchcoalitionalstructurewouldbedominatedby theoneinwhichthemembersofSwithdrawtoformsingletoncoalitions. Noequilibriumcoalitionalstructurecanpossess acoalitionwithless than ¹ Q®agentsbecause,inthiscase,therewouldexistanothercoalitionwith morethan¹ X®agentsandthepreviousreasoningcouldbeapplied.Therefore equilibriumcoalitional structuresmust bemadeupof coalitionswitha cardinalityincludedintheinterval£¹ Q®;¹ X®¤.Ontheotherhandall coalitionalstructureswithsuchcharacteristicsareVCE:indeedtakewhatever ¾¤=nS¤ 1;:::;S¤ ¹ N+1 o suchthat¹ Q®·XS¤ j·¹ N®8j2£1;¹ N®+1¤andlet ¼¾¤=©¼¾¤ 1;:::;¼¾¤ Iªbethecorresponding distributionof individual payo¤s; onlycoalitional structures¾¤¤ suchthat ¼¾¤¤ ¸¼¾¤will not bevetoed.But giventhatVi (S)ismonotonicallyincreasinginthecardinalityofSupto ¹ X®,thecondition¼¾¤¤ ¸¼¾¤canbesatis…edonlyifthereisanincrease inthenumberofcooperatorsinatleastanonmaximalcoalition.Since, byconstruction,cooperationistheobservedbehaviourinanycoalitionin ¾¤,theaboveresultcanbeobtainedjustbyanincreaseinthenumberof agentsformingpopulation=,whichcontradictstheassumptionofa…xed population.Thisendstheproof of theProposition. Itisworth notingthat thesetofPMCEisincludedinthesetofVCE; inotherwordsthepossibilityofane¤ectivevetoincreasesthenumberof possibleself-enforcingagreements. 5Propertiesof coalitional equilibria InthisSectionwecomparethe equilibriumoutcomeswithintheinstitutional settingsexaminedinSections4.1and4.2. 15 5.1Perfectmobilitycoalitional equilibria >FromProposition2weknowthat anyPMCEischaracterizedbyamaximallyconcentratedcoalitional structure.First of all weprovethefollowing: Remark4Inanymaximally concentratedcoalitionalstructure,andhence inanyPMCE,the aggregateproductionisgreaterthaninanyother(not maximally)concentratedcoalitional structure. Proof.Takea¹ N®+1-dimensional vector ¸=(¸1;:::;¸¹ N®+1);0·¸j·1, suchthat the…rst coalitionin¾hascardinality¸1¢ ¹ X®,thesecond¸2¢ ¹ X®, the¹ Nth¸¹ N¢ ¹ X®;forthede…nitionof¸wehavethatP¹ N®+1 j=1¸j¢ ¹ X®=I, whiletheaggregateproductionwill beY¾=P¹ N®+1 j=1A¡¸j¢ ¹ X®¢®.Inorder tomaximizeY¾withrespect to¸wehavetoput themaximumnumberof ¸iequal to1(remember that@Y¾ @¸i>0and@2Y¾ @¸2 i>0and¸i·18i).Because P¹ N®+1 j=1¸j¢ ¹ X®=I,thenImod¹ X®=¹ N®givesthemaximumnumber of¸ithatwecansetequalto1,while¸¹ N®+1=I¡¹ N®¢ ¹ X®,fromwhich ¹ Q®=¸¹ N®+1¢ ¹ X®.Thiscompletestheproof. Thereforeasocialplannerwhowouldmaximizeonlythesocialoutput, independentlyof itsdistribution,shouldopt for aninstitutional setting with perfectmobility,sinceheis surethat the equilibriumcoalitionstructureis maximallyconcentratedandhencetheaggregateoutput ismaximized. 5.2Vetoingcoalitionalequilibria WhenVCEareconcerned,bothmaximallyandnot-maximallyconcentrated coalitional structuresmaybethe equilibriumoutcomeof social interaction; howeversomepropertiescharacterizesthiskindofequilibria.Thenwecan state3 Remark5The total expectedequilibriumoutcomeinasocial settingcharacterized byavetoingpowerisnotgreaterthanthe totaloutputobtainable inthe caseof perfect mobilityi.e. E(Y¾)·Y¾¤¾¤PMCEand¾VCE Proof.Theproofofthestatement is simple:byRemark4weknowthat total output ismaximizedinmaximallyconcentratedcoalitional structures; thereforethetotal output obtainableinaVCEiscertainlyat most asmuch 3Weassumethat probabilityof at least anot maximallyconcentratedVCEispositive. 16 asthatinaPMCE.Itfollowsthat theaveragetotaloutputovertheVCE set isnot greater thantheoutput obtainableinaPMCE. Thenextresultconcernsthedistributionofpayo¤swithinaVCE.Let ¼¾=f¼¾ 1;:::;¼¾ Igbethedistributionof payo¤sinacoalitional structure¾ andlet^¼¾=min¼¾.Noticethat inanymaximallyconcentratedcoalitional structure^¼¾=A¢ ¡¹ Q®¢®¡1;call thisvalue^¼¾¤.Thenwecanstatethe following4: Remark6 The expectedpayo¤of the agent inthe worst positioninaVCE isat least asmuch asthe expectedpayo¤of the same agent inaPMCEi.e. E(^¼¾)¸^¼¾¤¾¤PMCEand¾VCE Proof.First weshow that theagent intheworst positioninaVCEthat is not maximallyconcentrated,getsahigher payo¤that anagent inthesame positiongetsinamaximallyconcentratedcoalitional structure,i.e.^¼¾>^¼¾¤ foranygivennot-maximallyconcentrated¾.Clearlytheagent withtheworst payo¤inamaximallyconcentratedcoalitionbelongsto theresidual coalition of cardinality¹ Q®.Another coalitional structurethat grantstotheagent in theworstpositionthesamepayo¤mustbeagainmaximallyconcentrated, becausethereisno otherwayofarrangingtheremainingI¡¹ Q®agentsin ¹ N®coalitionspreservingindividual incentivestocooperation,i.e.remaining withinthe equilibriumset.Thereforearearrangement of agentsnot leading toamaximallyconcentratedcoalitionalstructuremustgrant theworsto¤ agent apayo¤greater than^¼¾¤.Fromthisit followsthat the expectedpayo¤ of theworst o¤agent ina VCEmust behigherthanthepayo¤thesameagent couldgetinaPMCEsince,inthelattercase,all coalitional structuresare maximallyconcentrated. >FromtheRemarks5and6 wehavethatVCEhavelowerexpected aggregateoutput thanPMCEdo,but preservetheworst o¤agent sincehis expectedpayo¤isgreater inaVCEthaninaPMCE.Fromthisit followsalso thatanegalitariansocialplannermayprefer(andinduce)aninstitutional setting givingrisetoVCE(byallowingvetopower)ratherthanaperfect mobilityistitutionalsetting.So,forexample,aRawlsiansocialplannerwould certainlypreferaVCEtoaPMCE,butevenothersocialpreferencesare compatiblewiththesamechoice.Suppose,forinstance,that thesocial planner’swelfarefunctionisof theformW(¼1;:::;¼I)=PI i=1f(¼i);then, bytheShorrocks’theorem,wecanassert that therealwaysexistsaconcave functionf(f0>0,f00 <0),suchthat VCEarepreferredtoPMCE. 4See thepreviousnote. 17 5.3Agraphical illustration Thefollowingexamplegraphicallyshowsthepropertiesregardingthetwo kindsofequilibria.Supposethat the economyischaracterizedby values of theindividual (!)andtechnological (Aand®)parameters suchthat 1< ¹ X® I<2,i.e.therewill beat most twocoalitionsinanyequilibriumcoalitional structure.Thereforewecanrepresentall possiblecon…gurationsinaplane asinthefollowingpicture E W I I D C 2 S X 1 S X α X U1 U2 Y1 Y2 α X Thehorizontal axis showsthecardinalityof the…rst coalition,S1,while theverticalaxis showsthecardinalityofthesecondcoalitionS2.Theline II representsthelocusofallocationsoftheIagentsinthetwopossible coalitions.Everypointinthesquare¹ X®£¹ X®correspondstoacoalitional structureinwhichthecardinalityof thelargest coalitioniscompatiblewith theincentivestomutualcooperationbyall itsmembers.ForinstanceC (orD)representsapartitionofthewholesocietyintotwocoalitions;the …rstpossessesthemaximal cardinality¹ X®,whilethesecondistheresidual coalition.Apoint asWrepresentsacoalitional structurewithtwo coalitions of cardinality¹ X®inwhicheveryonecooperates;however suchapoint isnot a¤ordablebecausethereshouldbemorethanIagents. ThetwocurveslabelledY1andY2areaggregateisoproduct curves,i.e. anypointoneachofthemshowshowmanycooperatingagents shouldbe inthetwocoalitionsinorder tohavethesameamount of aggregateoutput. SincethecoalitionalproductionfunctionF(:;:)isconvex,andthesumof 18 convexfunctionsisconvexaswell,theisoproduct curvesarebowedout from theorigin.Furthermore,fromthemonotonicityofF(:;:),wehavethat Y1>Y2. ThetwocurveslabelledU1andU2aretheindi¤erencecurvesof asocial plannerwithconvexpreferences(asinthecaseofaconcavesocialwelfare functionof thetypejust seenW(¼1;:::;¼I)=PI i=1f(¼i),f0>0,f00 <0); asitiseasilyseen,ifthesocialplannerpreferencesareconvexenough he wouldchoosetheallocationE,that canbeobtainedonlyif theinstitutional setting istheonegiving riseto aVCE.If thesocial planner wouldmaximize thetotaloutput,thebestresultwouldbeobtainedonthecurveY1,intersecting theset of feasiblecoalitional structuresinthepointsCandDwhere thereismaximal concentration;anyother coalitional structure either would notgetsuchresultorcouldnotbecompatiblewithindividualincentives. Inthiscase,thedesiredresultcouldbeinducedbyaninstitutionalsetting leading onlytoPMCE,i.e.perfect mobility. 6Conclusions Inthispaperwehaveanalyzedtheoutcomesofacoalitionalstructureformationprocess amongidentical agents.Theproblemariseswhenever there areincreasingreturnsintheproductionof thecommodity,sothat thereare incentivesto coalesce,but the external e¤ect isnot so strong to makecooperationalwaysthedominant action.Notwithstandingthat agentscanengage in non-bindingpre-playcommunications,thenon-observabilityof individual actioninducesamoralhazardproblem,sothatpeoplecanengageincooperativebehaviouronlywhenitisindividuallyrationaladself-enforcing. Thepointwestress isthat theoutcomeofthecoalitionalstructureformationgamedependscruciallyontheinstitutional settingsruling the economy. Ifthereisperfectsocialmobilitypeople exploitall opportunitiestoform coalitionsthatareadvantageousforsomeoneandtheresultingequilibrium coalitional structureismaximallyconcentrated,inthesensethat peopleconcentrateasmuchaspossibleinthesmallest number of coalitionscompatible withindividualincentives.Ontheotherhand,amoreconstrainedinstitutionalsetting,inwhichevenasingleagentcancastane¤ectiveveto,gives riseto morecomplexequilibriumcoalitional structuresthat canbenot maximallyconcentrated.Inthiswesee aproblemof e¢ciencyagainst equity:in the equilibriawithperfect mobility,theaggregateoutput ismaximized,but thepayo¤of theagent that isintheworst positioniscertainlylower thanthe expectedpayo¤oftheworsto¤agentwhensocialmobilityisconstrained. Thechoiceofonetarget(eithere¢ciencyorequity)maythereforeinduce 19 thechoiceof aninstitutional settingmoreappropriatetotheobtainment of theformer. Oneofthemainlimitationsofthepresentworkisthatweassumeidentical agents; heterogeneitymayinducefurther problems,since,inthiscase,agents havealsotheopportunitytochoosewhich,andnotonlyhowmany,agents to coalescewith,providedindividual characteristicsarecommonknowledge. Inthiscasealsotheassumptionof aperfectlyegalitariandistributionwould not beappropriate,sinceit seemsunreasonablethat moreableagentsaccept anequaldistributionofaproductwhichtheyeventuallycontributemore. Theseisindeedtheagendafor our further research. 20 A Appendix Proof of Proposition1 Theproofofthe…rststatementistrivial;whenanagentisalonehis payo¤isgivenby ¼i (lijfig)=A¢l® i+(1¡li)¢!;li2[0;1]: SinceA>!byassumption1.2,thebest choicefor asingletonisli=1. Theproof of therest of thepropositiongetsaroundthepropertiesof the payo¤functionfor di¤erent valuesof ®.Denotebyl¡i:=P j2S;j6=i lj(0·l¡i· (XS¡1))thelevel of cooperationsuppliedbyall agentsinSexcept i.Let us start withthecase®·1.Thepayo¤function,conditional tothefact that i isalreadyincoalitionSwithalevel l¡i·XS¡1of expectedcooperation bytheothers,isgivenby ¼i (lijS;l¡i)=A¢[li+l¡i]® XS+(1¡li)¢!: Bythe…rst order conditionwehave ®¢ A¢[li+l¡i]®¡1 XS¡!=0)l¤ i=µ®¢A !¢XS ¶1 1¡®¡l¡i(3) andbythesecondorder conditionweget d2¼i (lijS;l¡i) dl2 i ¯ ¯ ¯ ¯l¤ i =®¢(®¡1)¢ A¢[li+l¡i]®¡2 XS ¯ ¯ ¯ ¯ ¯l¤ i = =®¢(®¡1)¢ A¢·³®¢A !¢XS ´1 1¡®¸®¡2 XS= =®¢(®¡1)¢ A¢³®¢A !¢XS ´®¡2 1¡® XS<0; sothat l¤ iisindeedamaximum.Clearlysuchachoicebyagentiismeaningful providedl¤ i¸0otherwisedefectionwouldbecomethedominantstrategy; thisimplies l¡i·µ®¢A !¢XS ¶1 1¡® 21 sothat thequantity³®¢A !¢XS ´1 1¡®isan upperboundtothenumberofother cooperatorsthatanagentcanexpectinorderto(partially)cooperatehimself. If full cooperationisto beobservedasanequilibriumbehaviourbyall agents, bytheabove equationit must betruethat XS¡1·µ®¢A !¢XS ¶1 1¡®; i.e.(XS¡1)1¡®¢XS·®¢A !: InviewofAssumption1.2and®·1,wehave®¢A !<2andtheaboveequation entailsXS<2,i.e.full cooperationcan never beobservedincoalitionswith at least twoagentsif returnstoscaleareat most constant. Inanyothersymmetricequilibrium,thecooperationlevelsuppliedby anyagent isobtainedfromequation(3) l=µ®¢A !¢XS ¶1 1¡®¡l¢(XS¡1); wherelisthefractionof hisowntimeanyonedevolvestoproduction.Solving for it weget l=µ®¢A ! ¶1 1¡®¢X2¡® ®¡1 S. Supposenow that returnstoscaleareincreasing,i.e.®>1.Inthiscase the…rst order conditionimplies l¤ i=µ!¢XS ®¢A ¶1 ®¡1¡l¡i:(4) Thesecondorder condition now entails d2¼i (lijS;l¡i) dl2 i ¯ ¯ ¯ ¯l¤ i =®¢(®¡1)¢ A¢[li+l¡i]®¡2 XS ¯ ¯ ¯ ¯ ¯l¤ i = =®¢(®¡1)¢ A¢h¡!¢XS ®¢A¢1 ®¡1i®¡2 XS= =®¢(®¡1)¢ A¢¡!¢XS ®¢A¢®¡2 1¡® XS>0 22 sothat thesolutionl¤ iidenti…esaminimumof thepayo¤function.Bythis, agents’choicewill be either tocooperate(li=1)whenever (l¡i+1)®¡l® ¡i>!¢XS A otherwisetheywill defect. Beforeproceeding,weshowthatall NEaresymmetric,i.e.eitherall agentsdefectorcooperate(partial actionsare excludedbythepreviousresult).Acontrario,supposethataNEexistsinwhichthereareC<XS cooperatorsandXS¡Cdefectors;ifthisisanequilibrium,anyoftheC cooperators…ndsanincentivetocooperate,i.e.for him A¢C® XS¡A¢(C¡1)® XS¡!>0; while,for anydefector it holdstruethat A¢(C+1)® XS¡A¢C® XS¡!<0: CombiningthetwoinequalitiesandmultiplyingbyXS Aweget C®¡(C¡1)®>(C+1)®¡C®; thatcan neverbeveri…edfor®>1.Itfollowsthatall equilibriamustbe symmetric,either withfull cooperationor defection. First consider thesituationinwhicheveryonedefect.It will beimmune fromdeviationsif!>A XS ; whichisalwaystrue,inviewofAssumption1.2,inanypropercoalition (XS¸2).Thisprovespoint 3)of Proposition1. Next weshowthat,whenever®¸2,full cooperationisaNEinanyS, thegrandcoalitionincluded.Itis su¢cient toprovethis statementforthe lower boundof theinterval,i.e.for ®=2;generalizedcooperationcanbea NEifA¢X2 S XS¡A¢(XS¡1)2 XS¸!; i.e.if2¡1 XS¸! A: >FromAssumption1.2theright-handsideof theaboveinequalityisalways lower than1,whiletheleft-handsideisalwaysgreater than1inanyproper 23