Endogenous Coalition Formation With Identical Agents

Endogenous CoalitionFormationwith

IdenticalA gents

1

D avideFiaschi

_{- P ierM arioP acini}

U niversità di P isa

D ipartimentodi ScienzeEconomiche

September29 , 19 9 8

1_{Inspiteofthis paperderivesfrom ajointworkofthetwoauthors, thesections} 1, 3 and 4 can be attributed toP ierM arioP acini, while the sections 2 and 5 to D avideFiaschi.

2_{d…aschi@ ec.unipi.it} 3_{pmpacini@ ec.unipi.it}

1 Introduction

Cooperativebehaviouroftenemerges atagroup, ratherthansociallevel. In manyinstancesweobservetheformationofindependentandsometimecom-petinggroups, teams, clubs, cooperatives (coalitions forshort) eachofthem persecutingthesamegoal(in turn provision ofcommodities, maximization ofpro…ts, raising ofpublic funds, standards ofbehaviouretc.). Examples ofthis behaviourarenumerous both atmicroandmacrolevel: scienti…cre-search groups, universitydepartments, consumers’associations, …rms as or-ganizations, consumption and production cooperatives, industrialdistricts, internationalcommercialtreatises amongcountries are allinstances ofvol-unteeragreementsamongindependentpartiesthatcoalescetoobtainasame goal. O nce coalitions form, society is partitioned in acoalitionalstructure. T his facthas alreadybeenobservedandstudied in anumberofworks from di¤ erentpoints ofview: amongthem [7 ], [1],[11]dealtwith theproblem of coalition formation forthe provision ofpublicgoods, [8]with the problem oftheformation ofcooperatives ofworkers with di¤ erentworkingcapacity, [6],[2],[3],[4]dealtmoredirectlywiththeproblem ofcoalitionalstructures formationinmoregeneralsettings (see[10]foranexcellentreview). Inthese worksithasbeenstressedthattheformationofcooperatinggroupsisa¤ ected by amoralhazard problem wheneverprivate actions cannotbe monitored; this factmay contribute todetermine the extenttowhich cooperation can besustainedasaself-enforcingequilibrium. T hequotedworksshowthatthe cooperativeagreementis self-sustainedifthecoalitionsizeis underacertain threshold, whereas itwould notbeincentivecompatibleinlargercoalitions; theseconclusions aremuchinthespiritof[13](seealso[12])

T his work takes a similarapproach butwith a di¤erentstrategy: …rst we …nd a setofconditions underwhich cooperation can emerge in groups thataresubsetsofthewholepopulation;thenwedeterminethelimitstothe sustainabilityofcooperativebehaviourwithincoalitionsand…nallywestudy howa society willform a coalitionalstructure. W e approach the problem in two steps; …rstwe examine agents’ behaviourwithin each ofthe possi-ble coalitions can form and we investigate the kind ofequilibria that we can expecttoemerge, coalitions taken as given: we showthatthe equilib-ria ofthe infra-coalitionalinteraction depend on individualcharacteristics (outsideopportunities) ofagents, returnstoscaleoftheavailabletechnology forthe transformation ofcooperative e¤ orts in outputand the distributive rule governing the allocation ofthe obtained output. O nce we knowhow people behave in acoalition, we proceed backward and study whatkind of coalitionalstructures can form in such away thatnoone has an incentive todeviate from. In this stage we shallsee thatthe equilibrium coalitional

structures depend on the institutionalsettings ruling the degree ofsocial mobilitywithin population. W eexaminetwoextremeinstitutionalarrange-ments: in the …rstwe assume there tobe perfectmobility and capacity to change coalitions, while, in the other, we assume an extremely limited mo-bility due tothe factthateven a single vetocan forbid the formation ofa coalitionalstructure. A s weshallsee, thesetwoextremecases willgiverise totwodi¤erentsets ofequilibrium coalitionalstructures.

Inoursettingagents canengageinpre-playnon-bindingcommunication amongthemselves (see [8]). B y this they can agree on pareto-dominating, self-enforcingequilibriainanyoccasionaselectionproblem isconcernedwith. H owever, justbecausesuchpre-playtalksareunbinding, noonecanbeforced toperform actions notcompatiblewithindividualincentives and, forexam-ple, cooperate when only group rationality would callforit. T herefore we adoptamixedapproachinwhichwesupposeagentscandealeachotherand constraintthemselves inabindingmanner, butinfullrespectoftheindivid-ualfreedom non toparticipate in any agreementwhich provides individual (andcoalitionalaswell) incentivestodeviatefrom. T hisapproachisnotnew intheliteratureandwerefertotheextensiveworkof[9 ]and[10]forfurther reference.

T he work is clearly limited by some strongassumptions thatwe make; amongthem ²thefactthatallindividuals areidentical ²thefactthatthedistributionruleis …xed ²thefactthatwedonotexamineintermediateinstitutionalsettings, in whichsocialmobilitymaybejustpartiallyinhibited. A throughoutworkshouldgetridoftheseassumptions;however, in this work, wekeep them as simplifyingdevices in ordertocarryoutan analysis capable ofmanageableresults. Section 2 describes the basiccharacteristics ofthe economy;Section 3 …nds whatwillbe the equilibrium self-enforcing outcomes within any possible coalition;Section 4 analyzes the equilibrium outcomes ofthe coalitionalstructure formation process underthe two dif-ferentinstitutionalsettings mentionedabove;…nallySection5 examines and compares theproperties ofthetwodi¤ erentkinds ofequilibrium coalitional structures. Conclusions closethepaperwithsomeinsightintofurtherwork.

2 D escriptionofthe model

T hereisapopulation=ofI agentsindexedbyi;theyaresupposedidentical in all characteristics and endowed with one unit oftime and one unit of a physicalcapital. Capitalis used with labourto produce a consumption commodityY bymeans ofapubliclyandfreelyavailabletechnology. T here arenoalternativeuses forcapital.

A gents can produce the commodity Y either in isolation or within a group. Ittakes place in agroup when agents in asubsetS µ=pooltheir capital units; then agents independently decide the amount oftheirown timetospend on production. T hecontribution ofthecapitalendowmentis a necessary and su¢cientcondition to have the righttoreceive a share of theproducedoutput.

G ivenacoalitionSfortheproductionanddistributionofthecommodity Y , letX SdenotethecardinalityofS. A coalitionSis proper if2 ·X S·I ;

ifX S= 1, S is a singleton. A coalitionalstructure is a partition of=into

subsets and is denotedby¾ = fS1;S2;:::;Sj;:::;SJg, whereJ 2 N + is the

numberofnon-emptygroups in thecoalitionalstructure¾. A nycoalitional structuremustsatisfythefollowingproperties:_S J j= 1Sj= = 8(h ;j), h 6= j, Sh \ Sj= ;, Sh;Sj2 ¾ . L et§ bethesetofallpossiblecoalitionalstructures in =.

2.1 Individualcharacteristics

Ifanagentchooses toparticipateintoacoalitionS hemust…rstcontribute his own capitalendowment; then he has to decide the amountoftime li

todedicate to production;hence (1 ¡li) is the leisure time. T he more he

spends his timeonproduction, themorethetotaloutputincreases;butone unitofleisuregives him anutilityof!1_{. A nagentcan always decidenotto}

participate in acoalition;in such acase hekeeps his capitalendowmentto produceinautharchy.

A gents’ utility depends on consumption ofcommodity and on leisure. Forsimplicitypurposes, theutilityfunctionis assumedtobelinearinallits arguments and takes theform

Ui= yi+ (1¡li)¢!

whereyiis thequantityagentireceives ofcommodityY .

A n agentspendinghis time on production willbecalled acooperator, a defector otherwise. A gents can choosetocontributeeven afraction oftheir working capacity, i.e. 0 · li· 1, butthey are notallowed to play mixed

strategies.

Finallyweassume that, whereas individualcapitalcontributions can be publiclyobserved, liisavariableknownonlytoagentiandcannotbedirectly

monitored byanyoneelse.

2.1.1 Technologicalcharacteristics

T he technology available forthe production ofY can be represented by a production function F (K ;L), where K is the capital stock and L is the amountoflabour; clearly, in any coalition, 0 · L · K since no one will contribute his own working e¤ort in a coalition in which he has no right toreceive the produced output. W e assume F (:;:) tosatisfy the following properties.

A xiom 1 T heproduction function F (K;L) is suchthat 1. F (K ;L) = A ¢minfK®_;L®_g

2. 2 ¢! > A > ! 3. ® > 0

N otice that A xiom 1 togetherwith 0 · L · K yield to F (K;L) = A_¢L®_{. T heparticularform oftheproductionfunctionisasimplifyingdevice,}

allowingforreturnstoscaletolabourinvarianttotheamountoftheworking e¤ ortputintotheproductionprocess;themagnitudeofthereturns toscale depends on the value of® . T he factthatthe scale parameterA is greater than the outside opportunity serves only to induce the performing ofthe workinge¤ortwhenanagentacts alone, whilethefactthat2 ¢! > A serves toexcludethatcooperationmayariseinpropercoalitionsforreasonsthatare notjuststrategic. Itis worthnoticingthattheproductionwithinacoalition ofsize K has no externale¤ecton the production ofothercoalitions (see [10]).

2.1.2 D istributive characteristics

O ncethetotaloutputY isproducedinagivencoalitionbytheworkinge¤orts ofsome ofits members, itis then redistributed within the same coalition. T hedistributionofY is governedbyanequalsharingruleR thatweassume

tobe the same in allcoalitions; in otherwords allmembers ofa coalition thatcontributedtheircapitalendowments havetherighttoreceiveansame shareofthetotaloutputY , independentlyoftherespectivecontribution in terms ofworkinge¤ort. T his assumption is motivated by the factthat, li

beingunobservable and thecapitalassets havingequalvalues, thereareno means todiscriminateamongtheparticipants toacoalition.

T obemorepreciseon theform ofR , takeagenericcoalitionS ofcardi-nalityX Sand saythatYSis thetotaloutputproduced within S bymeans

oftheworkinge¤ orts ofits members, i.e. Y S_{= A¢}³P j2Slj ´® ;then forall i_{2 S wehave} y_iS= Ri(Y ) = YS X S = A_¢³P_j2Slj ´® X S

where Ri(:) denotes the quantity thatthe distributive rule R allocates to

agenti. Inviewofthis, thepayo¤ toanagentifrom acoalitionSwhichhe participatedintois given by ¼i(lijS) = A¢³Pj2Slj ´® X S + (1_¡li)¢!

Itis clearthat, once thelevelofcooperationP_j2Sljis given, defectors are

always bettero¤ than cooperators in anycoalition. H owevertheessenceof the socialgame we are going to examine in the …rstpartofthe paperis whetheritis worthornot, from theindividualpointofview, toincreasethe levelofcooperationconsideringthattheworkinge¤orta¤ectsthecoalitional outputand thatthis returns as abene…ttothesubjectviathedistributive rule, but, atthesametime, this reduces leisure.

3 Individualdecisions and equilibria within

coalitions

B eforedealingwiththeproblem ofhowasocietywillstructureitselfincoali-tions, weexaminehowpeoplebehavewithinanypossiblecoalitionandwhich kindofequilibriawecanexpectinit. W earelookingforself-enforcingequi-libria, i.e. equilibrianoone has an incentivetodeviatefrom. T hestrategy we followtoprove the existence ofsuch kind ofequilibria is the following: …rstweprovetheexistenceofN ashequilibria(N E), i.e. equilibriarobustto unilateral deviations, in the game taking place within a coalition S ofX S

agents whentheyaretodecidetheleveloftheirtimetodedicatetoproduc-tion, given thattheyhavealreadycommitted toS. T hereafterwecheckfor

individualrationalityofN Es;this is averyimportantstep sinceagents will always beabletoimproveupon individuallynotrationalsolutions bywith-drawingfrom actualcoalitionscreatingnewones(rememberthatproduction in autharchyis always apossibility).

3.1 IncentivecompatibleandindividuallyrationalN E

withingivencoalitions

In ordertopursuetheanalysis letus takeagenericcoalition S½=andan agenti2 S. Furthermoresupposeagenthas correctexpectations aboutthe levelofcooperationsuppliedbytheotheragentsinSandthatthislevelwill notbea¤ected byhis own action. T henwecanstatethefollowingtheorem concerningexistenceofincentivecompatibleN EsinS;astotheterminology fullcooperation, partialcooperation and fulldefection stand forsituations in whichli= 1, 0 < li< 1 and li= 0 respectively8i2 S.

P roposition1 Suppose the economy satis…es A ssumption 1 and thatthe distributive rule R is an equalsharingrule;then

1. in anysingleton coalition, li= 1 is the dominantstrategy;

2. if® ·1, partialcooperation is theonlyN E in anypropercoalition; 3. if® > 1, generalizeddefection is always aN E in anypropercoalition; 4. ForanyX ¸2 thereexists avalue ®X < 2 , such thatfullcooperation

is a N E in anycoalition with cardinalityup toX provided® ¸®X ;if

® < ®X atleastthose coalitions with cardinalitygreaterthan X admit

fulldefection as the onlyN E.

5. if® > ®I ´®¤fullcooperation is aN E in anypropercoalition.

T he proof of this P roposition is rather long and is postponed to the A ppendix. Substantiallyitestablishesthatinapropercoalitionofcardinality X SthereareeitheroneortwoN E dependingon thereturns toscaleofthe

coalitionalproductionfunction.

N owwe turn toindividualrationalityofincentive compatibleN E, since individually notrationaloutcomes willnotbe accepted as equilibrium out-comes. R esults are summarized in the following listofremarks, where we alwaysassumetheeconomytosatisfyA ssumption1 andthedistributiverule R tobeoftheequalsharingtype, as in P roposition 1.

P roof. Ifreturns toscaleareatmostcostant, theonlyN E in anyproper coalition is a partialcooperation one, in which the workinge¤ortsupplied byanymemberofacoalitionS is ¹ l = ¡_®¢A ! ¢1 1¡® X 2¡® 1¡® S : (1) A level¹lofcooperationispro…tableifitgrantseachagentatleastthepayo¤ hecould getbyhimselfactingas asingleton, i.e. if

A¢¡¹l¢XS ¢® XS ¸A; from which ¹ l_¢X ®¡1® S ¸1: (2)

Substitutingfrom equation(1) into(2) wegetthattheN E, whenreturns to scaleareatmostconstant, is individuallyrationalif ¡_®¢A ! ¢1 1¡® X 2¡® 1¡® S ¢X ®¡1® S ¸1 ) X S· µ ® _¢A ! ¶® ·2 ;

where the lastinequality follows from A ssumption 1.2. H ence the partial cooperation equilibrium is neverindividually rationalin any coalition with atleasttwoagents.

R emark2 Fulldefection equilibriaareneverindividuallyrational.

P roof. T heprooffollows immediatelyfrom equation(2), sinceitcannever besatis…ed for¹l= 0 and X S> 0 .

R emark3 Fullcooperation N E (provideditexists) is individuallyrational. P roof. >From P roposition 1 weknowthatfullcooperationcan emergein propercoalitions only ifreturns toscale are increasing, i.e. ® > 1. Itcan easily be checked thatequation (2) is always satis…ed forforsuch values of ® and anyX S¸1.

Summarizing, fullcooperation is theonlyindividuallyrationalN E, pro-videditexists. T hepayo¤ onegets inanyotherequilibria(i.e. fulldefection

and sometimes partialcooperation) can be improved upon by withdrawing from thecorrespondingcoalitionandproducinginautharchyas asingleton. T hesubstanceofthestorystylizedintheP roposition1 andR emarks1, 2 e3issimple. Ifreturnstoscalearenotincreasing, thereisnoincentivetopool togetherresources, since theallocation rule grants anyonenomore than he candobyhimself. Incentivetocoalesceexistonlywhenthereareincreasing returns from cooperation. O nce this condition is met, cooperation is easier to emerge in groups of limited size (the extent of the maximal coalition capableofsustainingcooperation depends positively on the strength ofthe returnstoscale). T hefactthatfullcooperationcanbesustainedincoalitions notgreaterthan acertain threshold can be explained in the followingway. Supposeallagents in acoalition S ofcardinalityX Sarecooperatingatfull

extent: theirpayo¤ is A ¢X ®¡1

S . Suppose furtherthatX Sis a continuous

variable;ifan agentoutsidethe coalition bids forenteringand cooperating intoS, each ofthe preexistingX Sagents willreceive an increase in payo¤

equaltotheper-capitamarginalproductofthenewagent, i.e. (® ¡1)¢A ¢ X ®¡2

S . T henewcoalition willcontinuesustainingfullcooperation provided,

inthenewsetting, defectionwillnotbemorerewardingthancooperation, i.e. provided (® ¡1) ¢A ¢X ®¡2

S ¸!. If® < 2 , theleft-hand sideis adecreasing

function ofthe coalitionalcardinality, so thatthere willbe a value ofX S

constituting an upper bound to the cardinality of the coalition that can sustaincooperation.

3.2 T he value ofacoalition

IntheP roposition1 sectionweshowedthat, when® > 1, insomecoalitions therecanbetwopurestrategiesN ashequilibria: afullcooperationandafull defectionone. H oweveraseriouscoordinationproblem neverarises, sincethe fulldefectionequilibrium willneverbeadheredtobyagents foratleasttwo reasons: …rstofallitis notanindividuallyrationalequilibrium (seeR emark 1), sinceagents havealways theopportunitytoleavethecoalitionandform asingletongrantingmoretothemselves(A ratherthan!). Secondly, insuch asituation, there is always the possibility forasingle oragroup ofagents toagreetoa(joint) deviationformingacooperatingsubgroup suchthatthe bestreplyofeveryoneinthecomplementarycoalitionistocooperateaswell, thus increasing coalitionaland individualwelfare. H ence, by allowing for a pre-ply communication stage, we can conclude thatthe only meaningful N ashequilibrium, whenamultiplicityis possible, is thefullcooperationone thatis both individuallyrationalandcoalition-proof (as de…nedin[5]).

T herefore we can say that, given returns to scale as speci…ed by the parameter® , people willaccepttocooperate atfullextentin allcoalitions

in which such behaviouris incentive compatible. B y this wecan de…ne the valuetoanagentiofacoalitionSwhichhebelongsto;inordertodothis, let

¹

X ® bethecardinalityofthegreatestcoalition thatcan sustain cooperation

whenreturns toscalearegivenby®2_{. T hen wecan posethefollowing:}

D e…nition1 T he value Vi(S) ofa coalition S to an agentiis the payo¤

thathe can getin the N E forthatcoalition, i.e. 8i2 S2 ¾ 2 § ® 2 [0 ;1]) Vi(S)·A ® _{2 (1;®}¤_{]) V} i(S) = ½ A_¢X ®¡1 S ifX S· ¹X® 0 ifX S> ¹X® ® > ®¤) Vi(S) = A¢X S®¡1

In words the value function Vi(S) simply says thatallagents recognize

and agree thatmutualcooperation willbe the outcome oftheirinteraction wheneveritis possible, where possible means thatin playingcooperatively noone, neitherin a group norin isolation, willbe incentivated to deviate from the reached agreement. O utofthese situations, the impossibility to make binding commitments tocooperate compels agents torecognize that thevalueoftheirparticipationtothecoalitionislowerthanthepayo¤ ofthe surealternativethattheyhavealwaysthepossibilitytochoose, i.e. withdraw and actas singletons. Forthe sake ofconvention the value ofthe coalition in this caseis setto0.

4 Equilibrium coalitionalstructures

R egardingthe analysis ofcoalitionalstructure formation we con…ne tothe case1 < ® < ®¤_{. T hereasonforthis is simple;ifweallow® ¸®}¤_{, therewill}

be no disagreementto agree to mutualcooperation in the grand coalition =. Indeed, by P roposition 1, we knowthat, when returns toscale are sub-stantial(® > ®¤_{), cooperation is an equilibrium outcome in any coalition,}

= included; then, in the pre-play communication stage, agents willrecog-nizetheadvantages ofcooperatingtogethersincethis behaviourmaximizes allindividualutility and is incentive compatible and individually rational. T hereforethewholesocietywillstructureas asolegrandcoalitioninwhich everyonecooperates.

T he mostinterestingbehaviours can be observed when returns toscale, given with the distributive rule, donotallowforaggregation ofthesociety as awhole. In this casetherewillbecon‡icts amongtheagents abouthow

2_{In otherwords, ¹X}

topartition societyand which coalitions toparticipateinto. A s wewillsee, theequilibrium outcomesofthecoalitionalstructureformationgamedepend crucially on the institutionalsettings rulingthe capacity ofagents toform newcoalitions and disruptalreadyexistingones.

H erewetakeintoaccounttwoextremeinstitutionalsettings. Inthe…rst agentsareabsolutelyfreetoform anddisruptcoalitions, inordertomaximize individualpayo¤ . In the second, individualmobility is constrained by the vetoes thatsomeagents inacoalitioncancastagainstthetransferof(some of) theothermembers when this would inducealoss in thepayo¤ levels of theformerones. W eshallanalyzethetwocases separately.

4.1 P erfectmobility

Firstweexaminethecaseinwhichagents arecompletelyfreetoform coali-tions and abandon previouslyestablished ones, provided theyobtain again forthemselves, independentlyofwhathappenstotheothers. A nequilibrium within this institutionalsettingwillbe acoalitionalstructure with respect towhichnofurtherrearrangementofcoalitions (i.e. transferofpeoplefrom acoalitiontoanother) canturnoutadvantageous tothemembers ofatleast one newly formed coalition. In orderto make this de…nition rigorous, let ¼¾

i bethepayo¤ thatagentireceives when hebelongs toacoalition in the

coalitionalstructure¾;wesaythatacoalitionalstructure¾0_{is preferredby}

acoalition S0_{2 ¾}0_{tothecoalitionalstructure¾ if¼}¾0

i > ¼¾i8i2 S0, i.e. ifa

coalitionin¾0_{existssuchthatallitsmembersarebettero¤ thanin¾. L etus}

denotebyP(¾ ) = ©¾0_{2 § j9S}0_{2 ¾}0_:¼¾0

i > ¼¾i8i2 S0

ª

thesetofcoalitional structures thatare preferred byatleastacoalition tothecoalitionalstruc-ture¾. Ifperfectmobilityis allowed, thenacoalitionalstructure¾ can, and indeed will, be disrupted ifP(¾) 6= _{;, i.e. ifatleasta newcoalition (and} henceadi¤ erentcoalitionalstructure) canform inwhichallits members ob-tainapositiveincrementinpayo¤. T hedisruptionoftheexistingcoalitional structure ¾ takes place independently ofwhathappens tothe otheragents in ==S. T heseremainingagents havenocountermovebutarearrangement amongthemselves;as aconsequenceofthis rearrangementsomeoftheorig-inaldeviators could beinduced tocomebackformingadi¤erentcoalitional structureandsoon. W henthis process admits nofurthermove, i.e. whena positionisfoundsuchthatnogroup ofpeoplecanimproveupontheirpayo¤, wehavereachedanequilibrium. T his leads tothefollowingde…nition: D e…nition2 A coalitionalstructure ¾¤_{2 § is an equilibrium with perfect}

mobility(P M CE) ifP(¾¤_{) = ;.}

Example 1 Supposewehaveapopulationofsixagents, i.e. == f1;2 ;3;4 ;5;6g. T he technologicaland individualparameters are such thatA = 3, ® = 1:5 and! = 2 . Simple computation shows that

Vi(S) = 8 > > > > > > < > > > > > > : 3 ifXS= 1 4 :2 ifX S= 2 5:2 ifX S= 3 6 ifXS= 4 0 ifXS= 5 0 ifXS= 6

Take the coalitionalstructure ¾ = ff1;2 ;3g;f4 ;5g;f6gg; clearly P(¾) 6= ; since any coalitional structure ¾0_{containing a coalition S}0_{with X}

S0 = 4 will be preferred to ¾ by S0 itself. A lso the coalitional structure ¹¾ = ff1;2 ;3;4 g;f5g;f6gg can be disrupted, since the coalitionalstructure ¾00₌

ff1;2 ;3;4 g;f5;6gg 2 P(¹¾); indeed, in the lastcoalitionalstructure, both agents 5 and 6receive 4.2 ratherthan 3 and hence they increase theirown welfare. Finallythe coalitionalstructure ¾¤_{= ff1;2 ;3;4 g;f5;6gg cannotbe}

disruptedbyanycoalitionsincenoothergroup canform atmutualadvantage ofallits participants, sothatitis a P M CE.

N otethat, inthede…nitionofP M CE, agentsareconcernedwiththeirown payo¤ maximization and arefreetoform anycoalition theywant, buthave nopowerinblockingtheformationofacoalitionalstructureinwhichaneven negligiblegroup gets an increasein payo¤ , whilealltheothers su¤ eraloss. In otherwords theformation and theacceptanceofcoalitionalstructures is subjecttogroup consensus, butthereis novetopower.

B efore dealing with the main resultofthis section, letus pose the fol-lowingnotation: ¹N ® = I mod ¹X ® and ¹Q® = I ¡ ¹N ® ¢¹X®. In ordertoavoid

trivialsituationsletussuppose ¹Q® > 0 . T henwecanintroducethefollowing

de…nition:

D e…nition3 A coalitionalstructure ¾ is concentratedwhen itis formed byexactly ¹N ® + 1 coalitions suchthat ¹Q® ·X S· ¹X ®;acoalitionalstructure

¾ ismaximally concentrated when itis concentrated and ¹N ® coalitions

has the maximalcardinality ¹X ® while the lastone has cardinality ¹Q®.

W ecannowstateandprovethefollowingproposition.

P roposition2 Supposethattheeconomysatis…esA ssumption1, ® 2 (1;®¤_]

and the distribution is governed by an equalsharing rule R . T hen allthe maximally concentrated coalitionalstructures are P M CE forthe coalitional structures formation game.

P roof. Firstwe prove that, in equilibrium, allcoalitionalstructures must beformedbyexactly ¹N ® + 1 coalitions.

1. Ifacoalitionalstructure¾ werecomposedbylessthan ¹N ®+ 1 coalitions,

thentherewouldexistatleastonecoalition, sayS, withmorethan ¹X®

agents; in this case allmembers ofS would withdrawfrom S since, so doing, they get a higherpayo¤ (at leastA rather0 ). T herefore any coalitionalstructure ¾0_{containing a coalition S}0_{½ S such that}

X S0· ¹X_® would be preferred to ¾ by S0, so thatP(¾) 6= ;. T his shows thatinequilibrium theremustbeatleast ¹N ® + 1 coalitions.

2. Ifacoalitionalstructure¾ werecomposedbymorethan ¹N ® + 1

coali-tions, thanatleasttwoofthem, sayS1andS2, shouldcontainlessthan

¹

X ® agents. B utthen acoalition S0could form such thatS0µS1[ S2

and ¹X ® ¸X S0> m infX _S1;X _S2g;allthemembers ofS0 willgetapay-o¤ Vi(S0) > maxfVi(S1);Vi(S2)g. T hereforeanycoalitionalstructure

¾0_{3 S}0_{wouldbepreferredto¾, sothatP(¾ ) 6= ;. T hisshowsthatany}

coalitionalstructurewithmorethat ¹N ® + 1 coalitions canbedisrupted

andhence, inequilibrium, therecan beatmost ¹N ® + 1 coalitions.

>From the above points, an equilibrium coalitionalstructure mustbe concentrated. N owitis immediate tosee thatin any P M CE there cannot existacoalition with acardinalitygreaterthan ¹X ®, since, would itexist, a

newsubcoalition could form disruptingthegiven coalitionalstructureas in point1 above. O n the otherhand, in aP M CE notwocoalitions can have fewerthan ¹X ® members, since, in this case, the same reasoningas in point

2 abovecould beapplied. Combiningthesefacts together, thestatementof theP ropositionis proved.

A ccordingtotheP roposition, thereis amultiplicityofequilibriaeachof them corresponding to one possible way ofcombining I agents in ¹N ® + 1

coalitions, with ¹N ® havingthemaximalcardinality ¹X ®. H

oweverthestruc-ture ofallP M CE is always the same (maximally concentrated), only the identityofthemembers in coalitions change. In Section 5 weshallusethis facttoexaminesomebasicproperties ofthis kindofequilibria.

4.2 V etopower

H ereweanalyzeadi¤erentinstitutionalsettingcharacterizedbythefactthat agents arestilltryingtoexploitallopportunities toincreasethecoalitional outcome, butnowtheyareendowedalsowithavetopower, i.e. eachofthem

cansuccessfullycastanoppositiontotheformationofcoalitionalstructures in which hegets alowerpayo¤ . In otherwords itis as ifagents, in forming coalitions, signed contracts with the following clause: anyone ofthem will be able toleave the coalition he belongs toonly ifthe othermembers give him theirconsensus notsu¤ering any loss. O fcourse, when some agents can getan advantage from the formation ofa certain coalitionalstructure with no othersu¤ ering a loss (a P areto improvementforthe society as a whole), thennovetoeswillbeopposed;however, whenthepayo¤ maximizing attempts bysomeonewillinducealoss tosomeoneelse, thelatterones will oppose theirvetoes and willnotallowthe formation ofthe newcoalitional structure. W henwe…ndacoalitionalstructureinwhichanyfurtherattempt toincrease someone’s payo¤ encounters the veto ofsomeone else, then we havereachedanequilibrium;weshallcallsuchapositionavetoingcoalitional equilibrium (V CE). A s in the case ofP M CE, …rst we have to make this de…nitionrigorous. Supposethatsocietyisarrangedinacoalitionalstructure ¾ ;we say thatacoalitionalstructure ¾02 § , ¾0_{6= ¾, is vetoed by someone}

ifthere exists atleastan agentwhosu¤ erareduction in his payo¤ passing from ¾ to ¾0_{, i.e. 9i 2 = such that¼}¾0

i < ¼¾i. L etWi(¾) be the setof

coalitionalstructures thatarevetoed ifsocietyhappens tobestructured as in ¾, i.e. W (¾) = ©¾0_{2 § j9i2 =:¼}¾0

i < ¼¾i

ª

. Finally letP(¾) have the samemeaningas in Section 4.1, i.e. itis the setofcoalitionalstructures in which acoalition could form with anetpositive bene…tforallits members withrespecttowhattheygetin¾. T henwecanposethefollowingde…nition: D e…nition4 A coalitionalstructure¾¤_{2 § isaV CE ifP(¾}¤_{)nW (¾}¤_{) = ;.}

T he interpretation ofthis de…nition is simple: consider¾¤_{and take a}

coalitionalstructure in ¾ 2 P(¾¤_{);this means thatin ¾ there is atleasta}

coalition thatwas notin ¾¤_{and in which allmembers getahigherpayo¤.}

Since P(¾¤_{)nW (¾}¤_{) = ;, then ¾ belongs toW (¾}¤_{) as well, i.e. there will}

besomeagentsu¤ eringfrom therearrangementleadingto¾ ;ifavetopower is allowed and accepted, the latterpeople willobjectto the formation of ¾ and this coalitional structure willnot form. W hen vetoes are opposed toanycoalitionalstructure thatprovides individualand group incentiveto its formation, then we have reached a position which no one, willingly or unwillingly, moves from, i.e. we are in an equilibrium. A s itcan be easily checked, thesetofV CEcorrespondstothesetofcoalitionalstructuresgiving risetoP aretoe¢cientdistributions ofpayo¤s, giventhedistributiveruleR . A n examplecanclarifyconcepts.

had Vi(S) = 8 > > > > > > < > > > > > > : 3 ifXS= 1 4 :2 ifX S= 2 5:2 ifX S= 3 6 ifXS= 4 0 ifXS= 5 0 ifXS= 6

Take the coalitionalstructure ¾ = ff1;2 ;3g;f4 ;5g;f6gg andthecoalitional structure ¾0_{= ff1;2 ;3;4 g;f5;6gg. Clearly ¾}0_{2 P(¾ ) since the coalition}

of the …rstfour agents improves with respectto ¾ ; however ¾0 _{2 W (¾ )=}

since no one is worse o¤ in ¾0_{with respectto ¾. T herefore ¾ can and}

will be disrupted in favour of ¾0_{. Consider now the coalitional structure}

¾00_{= ff1;2 ;3g;f4 ;5;6gg. Itis easytocheckthat¾}0_{2 W (¾}00_{) sinceagents 5}

and6willsu¤ eraloss in¾0_{withrespectto¾}00_but¾0_{2 P(¾}00₎

sincethecoali-tion ofthe …rstfouragentgets an improvement. In this case the coalisincethecoali-tional structure ¾00_{cannotbe disrupted.}

W enowturntotheexistenceproblem andtothedescriptionofthechar-acteristics ofV CE in terms ofcoalitions. T he main resultis stated in the followingproposition.

P roposition3 Supposethattheeconomysatis…esA ssumption1, ® 2 (1;®¤₎

and the distribution is governed by an equalsharing rule R . T hen allthe concentratedcoalitionalstructures areV CE forthecoalitionalstructures for-mation game.

P roof. Firstweprovethatinanyequilibrium coalitionalstructurethereare exactly ¹N ® + 1 coalitions.

1. Supposethereexists an equilibrium coalitionalstructure¾ with fewer than ¹N ®+ 1 coalitions;thenatleastoneofthem, saycoalitionS, must

possess more than ¹X® members so that the value of theircoalition

is 0. N ow constructa coalitionalstructure ¾0_{obtained by ¾ simply}

replacing S by two or more coalitions with cardinality less than or equalto ¹X ®, therestofthe coalitionalstructure remainingthe same.

T he new coalitionalstructure ¾ would be such that¾0_{2 P(¾ ) and}

¾0_{2 W (¾) since themembers ofnewcoalitions strictly improvetheir=}

situations, whiletherestofthepopulationis una¤ected. From this we getP(¾ )_{* W (¾), i.e. ¾ cannotbeaV CE.}

2. N ow suppose that the equilibrium coalitional structure ¾ possesses morethan ¹N ® + 1 coalitions;supposefurtherthan nooneofthem has

more than ¹X ® members, otherwise the same reasoning as in point1

abovecouldbeapplied. T henarrangethem indecreasingorderofcar-dinality, obtainingthesequence S1, S2;:::;S_{N + 1}¹ ;:::;S_N¹_{®+ 1+ v}, v > 0 ,

with X S1 > :::> X S_{N ® + 1+ v}¹ . N owtakethecoalitionalstructure¾0 ob-tained from ¾ by redistributing the agents inSv

j= 1SN¹®+ 1+ joverthe coalitions S1;:::;SN¹®+ 1 in such awaythatX S0· ¹X ® foranyS02 ¾0. A nyofthenewcoalitions contains atleastthesamenumberofagents as in¾ sothat¼¾0

i ¸¼¾i8iand¼¾

0

i > ¼¾iforallthemembers ofatleast

acoalitionS0_{2 ¾}0_{. T his means that¾}0_{2 P(¾) and¾}0_{2 W (¾ ). A gain=}

wehaveP(¾ )_{* W (¾), i.e. ¾ cannotbeaV CE.}

T hus farwe proved thatany V CE mustpossess exactly ¹N ® + 1

coali-tions. N o equilibrium coalitionalstructure can possess a coalition S with more than ¹X®, because such coalitionalstructure would be dominated by

the one in which the members ofS withdrawto form singleton coalitions. N o equilibrium coalitionalstructure can possess a coalition with less than

¹

Q® agents because, in this case, there would exist anothercoalition with

more than ¹X ® agents and the previous reasoningcould be applied. T

here-foreequilibrium coalitionalstructures mustbemadeup ofcoalitions with a cardinality included in the interval£¹Q®; ¹X ®

¤

. O n the otherhand allcoali-tionalstructures with such characteristics are V CE: indeed take whatever ¾¤= nS₁¤;:::;S_{N + 1}¤¹ o such that ¹Q® · X S¤ j· ¹N ® 8j2 £ 1; ¹N ® + 1 ¤ and let ¼¾¤ = ©¼¾¤ 1 ;:::;¼¾ ¤ I ª bethecorrespondingdistributionofindividualpayo¤ s; onlycoalitionalstructures ¾¤¤_{such that¼}¾¤¤_¸¼¾¤_{willnotbevetoed. B ut}

given thatVi(S) is monotonically increasing in the cardinality ofS up to

¹

X ®, the condition ¼¾¤¤ ¸ ¼¾¤ can be satis…ed only ifthere is an increase

in the numberofcooperators in atleasta non maximalcoalition. Since, by construction, cooperation is the observed behaviourin any coalition in ¾¤_{, the above resultcan be obtained justby an increase in the numberof}

agents forming population =, which contradicts the assumption ofa …xed population. T his ends theproofoftheP roposition.

Itis worth notingthatthe setofP M CE is included in the setofV CE; in otherwords the possibility ofan e¤ ective veto increases the numberof possibleself-enforcingagreements.

5 P roperties ofcoalitionalequilibria

InthisSectionwecomparetheequilibrium outcomeswithintheinstitutional settings examinedinSections 4.1 and4.2.

5.1 P erfectmobilitycoalitionalequilibria

>From P roposition 2 we knowthatanyP M CE is characterized byamaxi-mallyconcentratedcoalitionalstructure. Firstofallweprovethefollowing: R emark4 In anymaximallyconcentrated coalitionalstructure, and hence in any P M CE, the aggregate production is greater than in any other (not maximally) concentratedcoalitionalstructure.

P roof. Takea ¹N ® + 1-dimensionalvector¸ = (¸1;:::;¸_N¹_®+ 1); 0 ·¸j·1,

such thatthe…rstcoalition in¾ has cardinality¸1¢¹X ®, thesecond¸2 ¢¹X®,

the ¹N th _¸ ¹

N ¢¹X ®;forthe de…nition of ¸ we have thatP ¹ N®+ 1

j= 1 ¸j¢¹X ® = I ,

while the aggregate production willbe Y¾ =

P_N¹_{®+ 1} j= 1 A ¡ ¸j¢¹X ® ¢® . In order tomaximize Y¾ with respectto¸ we have toputthe maximum numberof

¸iequalto1 (rememberthat@Y_@¸¾_i> 0 and@

2_Y ¾

@¸2_i > 0 and ¸i·1 8i). B ecause

P_N¹_{®+ 1}

j= 1 ¸j¢¹X ® = I , then I mod ¹X® = N¹® gives the maximum number

of¸ithatwe can setequalto 1, while ¸_N¹_{®+ 1} = I ¡ ¹N ® ¢¹X ®, from which

¹

Q® = ¸N¹®+ 1¢¹X ®. T his completes theproof.

T herefore a socialplannerwhowould maximize only the socialoutput, independentlyofits distribution, shouldoptforaninstitutionalsettingwith perfectmobility, since he is sure thatthe equilibrium coalition structure is maximallyconcentrated andhencetheaggregateoutputis maximized.

5.2 V etoingcoalitionalequilibria

W henV CE areconcerned, bothmaximallyandnot-maximallyconcentrated coalitionalstructures maybethe equilibrium outcomeofsocialinteraction; howeversome properties characterizes this kind ofequilibria. T hen we can state3

R emark5 T he totalexpectedequilibrium outcome in a socialsettingchar-acterized by a vetoingpoweris notgreaterthan the totaloutputobtainable in thecase ofperfectmobilityi.e.

E (Y¾)·Y¾¤ ¾¤P M CE and¾ V CE

P roof. T heproofofthestatementis simple: byR emark4 weknowthat totaloutputis maximized inmaximallyconcentrated coalitionalstructures; thereforethetotaloutputobtainableinaV CE is certainlyatmostas much

as thatin a P M CE. Itfollows thatthe average totaloutputoverthe V CE setis notgreaterthan theoutputobtainablein aP M CE.

T he nextresultconcerns the distribution ofpayo¤ s within a V CE. L et ¼¾ _{= f¼}¾

1;:::;¼¾Ig bethedistribution ofpayo¤ s in acoalitionalstructure¾

and let^¼¾ = m in¼¾_{. N oticethatinanymaximallyconcentratedcoalitional}

structure ^¼¾ = A_¢¡¹Q®

¢®¡1_{; callthis value ^}

¼¾¤. T hen we can state the following4_:

R emark6T heexpectedpayo¤ oftheagentin theworstposition in aV CE is atleastas muchas theexpectedpayo¤ ofthe same agentin aP M CE i.e.

E (^¼¾)¸^¼¾¤ ¾¤P M CE and¾ V CE

P roof. FirstweshowthattheagentintheworstpositioninaV CE thatis notmaximallyconcentrated, gets ahigherpayo¤ thatan agentin thesame positiongetsinamaximallyconcentratedcoalitionalstructure, i.e. ^¼¾ > ^¼¾¤ foranygivennot-maximallyconcentrated¾. Clearlytheagentwiththeworst payo¤ inamaximallyconcentratedcoalitionbelongstotheresidualcoalition ofcardinality ¹Q®. A nothercoalitionalstructurethatgrants totheagentin

the worstposition the same payo¤ mustbe again maximally concentrated, because there is nootherway ofarrangingthe remainingI ¡ ¹Q® agents in

¹

N ® coalitions preservingindividualincentives tocooperation, i.e. remaining

withintheequilibrium set. T hereforearearrangementofagents notleading toa maximally concentrated coalitionalstructure mustgrantthe worsto¤ agentapayo¤ greaterthan^¼¾¤_{. From thisitfollowsthattheexpectedpayo¤}

oftheworsto¤ agentinaV CEmustbehigherthanthepayo¤ thesameagent could getin aP M CE since, in the lattercase, allcoalitionalstructures are maximallyconcentrated.

>From the R emarks 5 and 6we have that V CE have lowerexpected aggregateoutputthan P M CE do, butpreservetheworsto¤ agentsincehis expectedpayo¤ isgreaterinaV CEthaninaP M CE.From thisitfollowsalso thatan egalitarian socialplannermay prefer(and induce) an institutional setting giving rise to V CE (by allowing veto power) ratherthan a perfect mobilityistitutionalsetting. So, forexample, aR awlsiansocialplannerwould certainly prefera V CE to a P M CE, buteven othersocialpreferences are compatible with the same choice. Suppose, for instance, that the social planner’s welfarefunction is oftheform W (¼1;:::;¼I) = PI_{i= 1}f (¼i);then,

bytheShorrocks’theorem, wecan assertthattherealways exists aconcave function f (f0_{> 0 , f}00_{< 0 ), such thatV CE arepreferred toP M CE.}

5.3 A graphicalillustration

T he following example graphically shows the properties regarding the two kinds ofequilibria. Suppose that the economy is characterized by values oftheindividual(!) and technological(A and ® ) parameters such that1 <

¹ X®

I < 2 , i.e. therewillbeatmosttwocoalitionsinanyequilibrium coalitional

structure. T herefore we can representallpossiblecon…gurations in aplane as inthefollowingpicture E W I I D C 2 S X 1 S X α X U1 U2 Y1 Y2 α X

T hehorizontalaxis shows thecardinalityofthe…rstcoalition, S1, while

the verticalaxis shows the cardinality ofthe second coalition S2. T he line

I I represents the locus of allocations of the I agents in the two possible coalitions. Every pointin the square ¹X ® £ ¹X® corresponds toacoalitional

structureinwhichthecardinalityofthelargestcoalitionis compatiblewith the incentives to mutualcooperation by allits members. Forinstance C (orD ) represents a partition ofthe whole society into two coalitions; the …rstpossesses themaximalcardinality ¹X®, whilethe second is the residual

coalition. A pointasW representsacoalitionalstructurewithtwocoalitions ofcardinality ¹X ® inwhich everyonecooperates;howeversuch apointis not

a¤ ordablebecausethereshould bemorethan I agents.

T hetwocurves labelled Y1 _{and Y}2 _{areaggregateisoproductcurves, i.e.}

any pointon each ofthem shows howmany cooperatingagents should be inthetwocoalitions inordertohavethesameamountofaggregateoutput. Since the coalitionalproduction function F (:;:) is convex, and the sum of

convexfunctions is convexas well, theisoproductcurvesarebowedoutfrom the origin. Furthermore, from the monotonicity of F (:;:), we have that Y1 > Y2.

T hetwocurves labelledU1 _andU2 _{aretheindi¤erencecurves ofasocial}

plannerwith convex preferences (as in the case ofa concave socialwelfare function ofthetypejustseen W (¼1;:::;¼I) = PI_{i= 1}f (¼i), f0> 0 , f00< 0 );

as itis easily seen, ifthe socialplannerpreferences are convex enough he wouldchoosetheallocationE, thatcanbeobtainedonlyiftheinstitutional settingis theonegivingrisetoaV CE.Ifthesocialplannerwouldmaximize the totaloutput, the bestresultwould be obtained on the curve Y1,

inter-sectingthesetoffeasiblecoalitionalstructures inthepoints C andD where thereis maximalconcentration;anyothercoalitionalstructureeitherwould notgetsuch resultorcould notbe compatible with individualincentives. In this case, the desired resultcould be induced by an institutionalsetting leadingonlytoP M CE, i.e. perfectmobility.

6 Conclusions

In this paperwe have analyzed the outcomes ofacoalitionalstructure for-mation process amongidenticalagents. T heproblem arises wheneverthere areincreasingreturns intheproductionofthecommodity, sothatthereare incentivestocoalesce, buttheexternale¤ectisnotsostrongtomakecooper-ation always thedominantaction. N otwithstandingthatagents can engage innon-bindingpre-playcommunications, thenon-observabilityofindividual action induces a moralhazard problem, so thatpeople can engage in co-operative behaviouronly when itis individually rationalad self-enforcing. T he pointwe stress is thatthe outcome ofthe coalitionalstructure forma-tiongamedepends cruciallyontheinstitutionalsettingsrulingtheeconomy. Ifthere is perfectsocialmobility people exploitallopportunities to form coalitions thatare advantageous forsomeone and the resultingequilibrium coalitionalstructureismaximallyconcentrated, inthesensethatpeoplecon-centrateas muchas possibleinthesmallestnumberofcoalitions compatible with individualincentives. O n the otherhand, amore constrained institu-tionalsetting, in which even asingle agentcan castan e¤ective veto, gives risetomorecomplexequilibrium coalitionalstructures thatcanbenotmax-imallyconcentrated. Inthis weseeaproblem ofe¢ciencyagainstequity: in theequilibriawithperfectmobility, theaggregateoutputis maximized, but thepayo¤ oftheagentthatisintheworstpositioniscertainlylowerthanthe expected payo¤ ofthe worsto¤ agentwhen socialmobility is constrained. T he choice ofone target(eithere¢ciency orequity) may therefore induce

thechoiceofaninstitutionalsettingmoreappropriatetotheobtainmentof theformer.

O neofthemainlimitationsofthepresentworkisthatweassumeidentical agents;heterogeneitymayinducefurtherproblems, since, inthiscase, agents have alsothe opportunitytochoose which, and notonly howmany, agents tocoalescewith, providedindividualcharacteristics arecommonknowledge. Inthis casealsotheassumptionofaperfectlyegalitariandistributionwould notbeappropriate, sinceitseemsunreasonablethatmoreableagentsaccept an equaldistribution ofa productwhich they eventually contribute more. T heseis indeedtheagendaforourfurtherresearch.

A

A ppendix

P roofofP roposition1

T he proofofthe …rst statementis trivial; when an agentis alone his payo¤ is given by

¼i(lijfig) = A ¢l®i+ (1¡li)¢!; li2 [0 ;1]:

SinceA > ! byassumption1.2, thebestchoiceforasingletonis li= 1.

T heproofoftherestofthepropositiongets aroundtheproperties ofthe payo¤ functionfordi¤ erentvaluesof® . D enotebyl¡i:= P

j2S;j6= i

lj(0 ·l¡i·

(X S¡1)) thelevelofcooperationsuppliedbyallagentsinSexcepti. L etus

startwith thecase® ·1. T hepayo¤ function, conditionaltothefactthat iis alreadyin coalition S with alevell_¡i·X S¡1 ofexpectedcooperation

bytheothers, is givenby ¼i(lijS;l¡i) = A¢[li+ l¡i]® X S + (1¡li)¢!: B ythe…rstorderconditionwehave ® _¢A¢[li+ l¡i] ®¡1 XS ¡! = 0 ) l¤_i= µ ® _¢A ! ¢XS ¶ 1 1¡® ¡l¡i (3)

and bythesecond orderconditionweget d2_¼ i(lijS;l¡i) d l2 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 ; sothatl¤ iisindeedamaximum. Clearlysuchachoicebyagentiismeaningful provided l¤

i ¸ 0 otherwise defection would become the dominantstrategy;

this implies l¡i· µ ® ¢A !_¢X S ¶ 1 1¡®

sothatthe quantity³®¢A !¢XS

´ 1 1¡®

is an upperbound tothe numberofother cooperatorsthatanagentcanexpectinorderto(partially)cooperatehimself. Iffullcooperationistobeobservedasanequilibrium behaviourbyallagents, bytheaboveequation itmustbetruethat XS¡1 · µ ® ¢A ! _¢XS ¶ 1 1¡® ; i.e. (X S¡1)1¡® ¢X S· ® _¢A ! :

InviewofA ssumption1.2 and® ·1, wehave®¢A

! < 2 andtheaboveequation

entails X S< 2 , i.e. fullcooperationcan neverbeobservedin coalitions with

atleasttwoagents ifreturns toscaleareatmostconstant.

In any othersymmetric equilibrium, the cooperation levelsupplied by anyagentis obtained from equation(3)

l = µ ® _¢A ! _¢XS ¶ 1 1¡® ¡l¢(X S¡1); wherelisthefractionofhisowntimeanyonedevolvestoproduction. Solving foritweget l = µ ® _¢A ! ¶ 1 1¡® ¢X 2¡® ®¡1 S .

Supposenowthatreturns toscaleareincreasing, i.e. ® > 1. Inthis case the…rstordercondition implies l_i¤= µ !_¢XS ® ¢A ¶ 1 ®¡1 ¡l¡i: (4) T hesecondordercondition nowentails d2_¼ i(lijS;l¡i) d l2 i ¯ ¯ ¯ ¯ l¤ i = ® ¢(® ¡1) ¢A¢[li+ l¡i] ®¡2 X S ¯ ¯ ¯ ¯ ¯ l¤_i = = ® _{¢(® ¡1) ¢}A¢ h¡_!¢XS ®¢A ¢1 ®¡1i ®¡2 XS = = ® _{¢(® ¡1) ¢}A¢ ¡_!¢XS ®¢A ¢®¡2 1¡® XS > 0

sothatthesolution l¤

iidenti…es aminimum ofthepayo¤ function. B ythis,

agents’choicewillbeeithertocooperate(li= 1) whenever (l_¡i+ 1)® _¡l® ¡i> ! _¢XS A otherwisetheywilldefect.

B efore proceeding, we show thatallN E are symmetric, i.e. eitherall agents defectorcooperate(partialactions areexcluded by theprevious re-sult). A contrario, suppose thata N E exists in which there are C < X S

cooperators and X S¡C defectors; ifthis is an equilibrium, any ofthe C

cooperators …nds anincentivetocooperate, i.e. forhim A¢C®

X S ¡

A¢(C ¡1)®

XS ¡! > 0 ;

while, foranydefectoritholds truethat A_{¢(C + 1)}®

X S ¡

A_¢C®

XS ¡! < 0 :

Combiningthetwoinequalities and multiplyingbyXS

A weget

C® _{¡(C ¡1)}®

> (C + 1)® _¡C®_;

thatcan neverbe veri…ed for® > 1. Itfollows thatallequilibria mustbe symmetric, eitherwith fullcooperation ordefection.

Firstconsiderthesituation in which everyonedefect. Itwillbeimmune from deviations if

! > A X S

;

which is always true, in view ofA ssumption 1.2, in any propercoalition (X S¸2 ). T his proves point3) ofP roposition1.

N extweshowthat, whenever® ¸2 , fullcooperation is aN E in anyS, thegrand coalition included. Itis su¢cienttoprove this statementforthe lowerbound oftheinterval, i.e. for® = 2 ;generalizedcooperation can bea N E if A¢X 2 S X S ¡ A¢(X S¡1)2 X S ¸!; i.e. if 2 _¡ 1 XS¸ ! A:

>From A ssumption1.2 theright-handsideoftheaboveinequalityis always lowerthan1, whiletheleft-handsideis always greaterthan1 inanyproper

coalitionandincreasingwithX S. Itfollowsthattheconditionforcooperation

is satis…edforanyX S.

Finally we prove thatforany S with X S· I , there is avalue ®Ssuch

that, for® > ®S, fullcooperation is a N E in S. In a generic coalition of

cardinalityX Sfullcooperationis aN E if A_¢X ® S XS ¡ A_¢(X S¡1)® X S ¸!:

>From the previous point, we knowthatitis certainly satis…ed for® ¸2 ; moreovertheleft-handsideis anincreasingfunctionof® , since

d ³A¢XS® XS ¡ A¢(XS¡1)® XS ´ d ® = A X S¢[X ® S¢lnX S¡(X S¡1)® ¢ln(X S¡1)]> 0 : T hereforeitwillexistavalue®Ssuchthat A¢X ®S S XS ¡ A¢(X S¡1)®S X S = !:

Clearly forall® > ®I ´ ®¤< 2 cooperation willbe an equilibrium in any

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