The role of social and technical excludability for the success of impure public good and common pool agreements

(1)

ContentslistsavailableatScienceDirect

Resource

and

Energy

Economics

j o ur na l h o me pa g e:w w w . e l s e v i e r . c o m / l o c a t e / r e e

The

role

of

social

and

technical

excludability

for

the

success

of

impure

public

good

and

common

pool

agreements

The

case

of

international

ﬁsheries

夽

Michael

Finus

a,∗

_,

_Raoul

_Schneider

b

_,

_Pedro

_Pintassilgo

c

a_University_of_Graz,_Austria_and_University_of_Bath,_Bath,_UK b_Department_of_Economics,_Ulm_University,_Ulm,_Germany

c_Faculty_of_Economics_and_CEFAGE,_University_of_Algarve,_Faro,_Portugal

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received22August2017

Receivedinrevisedform30August2019 Accepted17September2019

Availableonline29September2019 JELclassiﬁcation: C72 F53 H87 Q22 Keywords:

Pureandimpurepublicgoodsandcommon poolresources

Technicalandsociallyconstructed non-excludability

Beneﬁt-costdualityofpublicgoodsand commonpoolresources

Propertyrights Sharedﬁshstocks

Regionalﬁsheriesmanagement organizations

Free-riding

a

b

s

t

r

a

c

t

Wearguethatinternationalﬁsheriesareaprimeexampletostudytheincentivestructureof

formingimpurepublicgoodandcommonpoolagreements.Weconsiderafullyintegrated

multiplezonemodel,inwhichzonesarelinkedthroughdensity-dependentmigration.

TheincentivetoaccedetoRegionalFisheryManagementOrganizations(RFMOs)isrelated tomultiplecharacteristics.Firstly,therelativepatchsizesofthehighseas,whichisthe internationally(publicly)accessibledomain,comparedtoexclusiveeconomiczones,which arestate-owned(privatelyowned).Thiscanberelatedtothedegreeofsociallyconstructed excludability.Secondly,theintensityofﬁshmigrationbetweenvariouszones,whichcanbe relatedtothedegreeoftechnicalexcludability.Thirdly,thegrowthrateoftheresource,which canbeinterpretedasthedegreeofrivalry,withalow(high)degreeofrivalryapproximating publicgood(commonpool)features.Weshowthat,generally,excludabilityreduces free-ridingincentivesbutalsotheneedforcooperation,avariantofthe“paradoxofcooperation”.

Moreover,weshowthatthebeneﬁt-costdualitybetweenpublicgoodsandcommonpool

resourcesgenerallyholdsexceptforsomeextremeparametervaluesforwhichalowdegree ofrivalryfostersthesuccessofcooperation.Finally,throughavariationofthediffusion matrix,wecanalsoanalyzeaclosedaswellasasink-sourcesystem.

1. Introduction

Acentralaspectinthetheoryofpublicgoodsistounderstandtheincentivestructurethatleadstotheunderprovisionof publicgoodsaswellasthepossibilitiesofrectifyingthis.Theincentivestructurecanbebroadlyrelatedtothepropertiesof publicgoods,whichareusuallyassociatedwithtwodistinguishingfeatures:non-excludabilityandnon-rivalry,whichcan betracedbacktotheseminalworkofSamuelson(1954)andMusgrave(1959).Byvaryingthedegreeofexcludabilityand

夽 Weareverygratefultotwoanonymousreviewersandtheeditor,ProfessorKafﬁne,forveryhelpfulcommentsonapreviousversionofthepaper. ∗ Correspondingauthorat:UniversityofGraz,DepartmentofEconomics,Universitätsstraße15,8010Graz,Austria.

(2)

Table1

ClassiﬁcationofImpurePublicGoods.

rivalry,variousmixedformsofimpurepublicgoodsemergeasillustratedinTable1(e.g.,CornesandSandler,1994andKaul

andMendoza,2003).

Intermsofexcludability,theexpectationisthatthehigherthedegreeofexcludability,thecloserarenon-cooperative equilibriumandoptimum,butalsothesmallerarethegainsfromcooperation.KaulandMendoza(2003)emphasizethatthe perceptionofwhatispublicandwhatisprivatehaschangedsigniﬁcantlyovertime.Theydistinguishbetweentheintrinsic propertiesofagood,towhichforinstancetheso-calledtechnicalexcludabilitybelongs,andthepropertiesassignedbysociety tothem,towhichtheyrefertoasso-calledsociallyconstructedexcludability.Whereasthedegreeoftechnicalexcludability canberegardedasgiven(thoughpotentiallysubjecttotechnologicalchangeasinthecaseofencryptioninbroadcasting), sociallyconstructedexcludabilityisdeterminedbytheestablishmentandenforcementofpropertyrights.

Intermsofrivalry,theexpectationappearstobelessclear-cut.Ontheonehand,SandlerandArce(2003)convincingly showthebenefit-costdualityofpurepublicgoods(withstrategiesbeingprovisionlevels)andcommonpoolresources (withstrategiesbeingexploitationlevels).Inthepublicgoodgame,thecostsareprivateandthebenefitsfromprovisionare public.Inthecommonsgame,thereverseistrue,thebenefitsareprivateandthecostsfromexploitationarepublic.Thus, theeconomicsandincentivesshouldbethesameinbothsettingsandhencethedegreeofrivalryshouldnotmatter.Onthe otherhand,despitetheirformalproofofequivalence,theauthorsconcludeinformallythatthereisadifference:inpolitics, itwouldbeeasiertoestablishjointactioninpublicgoodgames(i.e.,cooperativelyprovidingthepublicgood)thanjoint inactionincommonsgames(i.e.,cooperativelyreducingthelevelofharvesting).Thiswouldsuggestthatrivalry,astheonly distinguishingfeaturebetweenpublicgoodsandcommonpoolresources,wouldmatter.

Tothebestofourknowledge,thereisnoformalmodelcapturingthefollowingthreeaspectssimultaneously:1)different degreesofsociallyconstructedandtechnicalexcludabilityaswellasrivalry,2)ananalysisoftheireffectonequilibrium publicgoodprovisionandcommonpoolresource uselevelsand3)a testforthepossibilitytoestablishfullorpartial cooperationinanon-cooperativemodelofcoalitionformation.

Intermsofthefirstaspect,weviewinternationalfisheriesasoneofthefewandparticularlyinterestingexampleswhereall propertiesaresimultaneouslypresent.1_The_degree_of_socially_constructed_and_technical_{excludability}_can_be_{parameterized} alongtheentirehorizontalspectruminTable1.Inourmodel,parameter˛measurestheportionofthetotalfishingground whichispubliclyaccessiblebyallfishingstates(commonproperty),theso-calledhighseas,and1−˛istheportionofthe totalfishinggroundwhichisprivatelyownedbycoastalstates,theso-calledExclusiveEconomicZones(EEZs),asestablished bytheUNConventionontheLawoftheSeain1982.2_Thus,_the_larger_˛,_the_smaller_is_the_degree_of_socially_constructed excludability.Theparameterddeterminestheintensityoffishmigrationbetweendifferentzones.Thelargerd,thesmaller

1 _The_sharing_of_water_resources_has_similar_features._Socially_constructed_{excludability}_can_be_established_through_property_rights_and_technical

exclud-abilitymayvarythroughthediversionofriversandtheerectionofdams.However,manyotherexamplesfeatureonlysomepropertiesmentionedabove thoughnotall.Forinstance,theacidraingameallowscapturingvariousdegreesoftechnicalexcludabilitythroughtheemissiontransportationmatrix(e.g. Mäler,1994andSandler,1998),butsincenationalboundariesaregiven,thedegreeofsociallyconstructedexcludabilityisnotanissue.Thesameapplies totheclassicalexampleofapurepublicgoodgame,climatechangemitigation,evenifwerecognizetheprivatizingeffectsofancillaryorco-beneﬁtsof improvedlocalairqualityfromclimatemitigationasanalyzedforinstanceinMarkandyaandRübbelke(2004).Inthecaseoftheexplorationofthenatural resourcesintheAntarctic(e.g.,oil,gasandminerals),afterpropertyrightswereproperlydeﬁnedandenforced,excludabilitywouldbeperfectastechnical excludabilitycanberegardedasperfect.Intermsofrivalry,allexamplesareonlylocatedatoneextremeofthespectrum:acidrainandclimatechange exhibitnorivalryatallwhereasfornon-renewableresourcesrivalryisperfect.

2 _In_terms_of_terminology,_in_our_setting_“privately_owned”_and_{“privatization”}_means_the_allocation_of_property_rights_to_states_–_the_sole_actors_or_players

inourinternationalfisherygame.Inthefisheryliterature,mainlywithanationalorregionalfocus,thesetermsaresometimesalsousedfortheallocation offishingquotastofishermen.Weabstractfromthesenationaldetails.SeeSection4.4forthequalificationofourassumptions.

(3)

isthedegreeoftechnicalnon-excludability.Alsothedegreeofrivalrycanbeparameterizedalongtheentireverticalaxisin Table1throughthegrowthrateoftheﬁshstock,whichisparameterrinourmodel.Thisallowsustostudyhowtechnical andsociallyconstructedexcludabilityandthegrowthrate(apartfromsomeotherparameters)affectthesuccessofﬁshery agreements.

Intermsofthesecondaspect,wemeasurethelevelofunderprovisionoftheimpurepublicgood(i.e.,“preservationofﬁsh stocks”)asthedifferencebetweenthefullycooperative,partiallycooperativeandnon-cooperativeequilibrium,physicallyin termsofstocklevelsandmonetarilyintermsofpayoffs.Differencesarerelatedtothepropertiesofthepublicgood/common poolresourceandimportanteconomicandbiologicalparametersthatdeterminetheproductionprocess.

Intermsofthethirdaspect,inthetraditionoftheliteratureoninternationalenvironmentalagreements(IEAs)3_and_the literatureoninternationalﬁsheryagreements(IFAs)4_,_we_study_the_formation_of_{self-enforcing}_agreements_as_a_means_to mitigatefree-ridingwithanon-cooperativecoalitionmodel.However,theIEA-literaturehasalmostexclusivelyrestricted attentiontoaglobalemissiongame5_(i.e.,_pure_public_bad)_and_the_{IFA-literature}_considered_a_renewable_common_resource withonlyonejurisdiction(i.e.,purecommonpoolresource).6_In_contrast,_we_allow_for_several_{jurisdictions,}_i.e.,_some_parts oftheoceanmaybecomeprivatelyownedthroughthedeclarationofEEZs.Moreover,amongEEZsandthehighseasthere maybelinksthroughthemigrationofﬁsh.Thus,byvaryingthedegreeofsociallyconstructedandtechnicalexcludability, wevarythedegreeof“purity”,i.e.,wecoverthespectrumfrompure,impuretoprivate.

OurmodelisbasedontheclassicalGordon-Schaefermodel(Gordon,1954)whichisextendedtoaccountformigration betweendifferentfishinggroundsasconsideredforinstanceinSanchiricoandWilen(1999,2005).Wefocusourdiscussion onafullyintegratedmodelwithbilaterallinksamongallzonesandadensity-dependentdiffusionprocess.Aspecialcase istheclosedpatchmodelifthereisnodiffusionbetweenzones.Wefinallyconsiderasink-sourcemodelwithunilateral diffusionprocessasanextension.Weshowthat,generally,establishingRFMOsisdifficultandalwaysfailsinasink-source model.Moreover,excludabilityreducesfree-ridingincentivesbutalsotheneedforcooperation,avariantofthe“paradoxof cooperation”.Moreover,thebenefit-costdualitybetweenpublicgoodsandcommonpoolresourcesgenerallyholdsexcept forsomeextremeparametervaluesforwhichalowdegreeofrivalryfostersthesuccessofcooperation.

Thepaperproceedsasfollows.InSection2,weprovideabriefbackgroundonthehistoricaldevelopmentofthe manage-mentofinternationalfisheriesandtheestablishmentofcooperativeagreements.InSection3,weintroducethebioeconomic modelincludingthetwo-stagecoalitionformationmodel.Section4describesthemodelspecificationsandoursolution pro-cedure.Sections5,6and7discussourresultsforthefullyintegratedmodelwithdensity-dependentdiffusion.Accordingto thesequenceofbackwardinduction,wefirstdiscussresultsofthesecondstage(Section5),thenofthefirststage(Section 6)andfinallypullresultsofbothstagestogetherinSection7.Section8discussesbrieflythreeextensions:i)thepossibility ofexcludingnon-RFMOmembersfromfishinginthehighseas,ii)analternativeagreementformationprocessandiii)a sink-sourcemodelwithuni-directionaldiffusion.Section9concludes.

Wewouldliketopointoutattheoutsetthatalthoughourmodelrelatestointernationalﬁsheries,ouraimistoremainas generalaspossiblewithreferencetotheliteratureonpublicgoodsandhenceweabstractfromtechnicaldetailsinvestigated insomeoftheliteratureonﬁsheries.

2. Historicalbackgroundoninternationalﬁsherymanagement

TheFoodandAgricultureOrganizationoftheUnitedNations(FAO)estimatesthatharvestsfrominternationallyshared ﬁshstocks7_account_for_as_much_as_one_third_of_global_marine_capture_ﬁshery_harvests₍_FAO,₂₀₁₀_and_Munro_et_al.,₂₀₀₄_). Thesestocksareestimatedtobeparticularlyvulnerableandarereportedtobeheavilyoverexploitedorevendepletedin McWhinnie(2009).

Foralongtime,concernmainlyfocusedonthepreservationofcoastalﬁshinggrounds.Somegovernmentsstartedto declareunilaterallyEEZs,thusevictingallforeignﬂeetsfromwhattheyclaimedtobetheirprivateproperty.The1982UN

3_The_literature_on_IEAs_goes_back_to_Barrett₍₁₉₉₄₎_and_Carraro_and_Siniscalco₍₁₉₉₃₎_and_has_grown_immensely_in_recent_years._For_a_recent_survey_and

acollectionofthemostinﬂuentialarticlesseeFinusandCaparros(2015).

4_Stability_of_ﬁshery_agreements_has_been_modelled_as_cooperative_(e.g.,_Kennedy,₂₀₀₃_and_Lindroos,₂₀₀₄₎_or_{non-cooperative}_coalition_games_(e.g.,

Kwon,2006;Lindroos,2008;MillerandNkuiya,2016;PintassilgoandLindroos,2008,andPintassilgoetal.,2010),butalsoasadynamicﬁsherygamewith enforcementthroughpunishment(e.g.,Hannesson,1997andTaruietal.,2008).ForarecentsurveyseePintassilgoetal.(2015).

5_Exceptions_are_for_instance_Mäler₍₁₉₉₄₎_and_Finus_and_Tjøtta₍₂₀₀₃₎_in_the_context_of_a_repeated_acid_rain_game_(i.e.,_impure_public_bad),_though_they

onlyfocusonthestabilityofthegrandcoalitionanddonotexploittherelationbetweentransportationcoefﬁcients(i.e.,measuringthedegreeoftechnical excludability)andthesuccessofcooperationaswedo.

6_Already_Crutchﬁeld_(1964),_p._216,_based_his_call_for_{international}_cooperation_on_the_observation_that_migration_of_ﬁsh_poses_a_natural_limit_to_the

privatizationoffisheryresources:“[...]thefishthemselvesseemindisposedtoacceptsuch[privatizing]solutions.”Spatialexternalitymodelswithout coalitionformationhavebeenstudiedforinstancebyCostelloandPolasky(2008);FlaatenandMjølhus(2005);KvamsdalandSandal(2008);Pezzeyetal. (2000).TheonlyspatialmodelwithcoalitionformationofwhichweareawareisPuntetal.(2013).Theyconsideramarineprotectedarea(MPA)inthe highseas.However,nomigrationisconsideredbetweentheMPAandthefishinggrounds.Theauthorsconcludethatforsymmetricplayersnocoalitionis stableasthisisthecasewithoutMPAs.

7_According_to_FAO’s_{classiﬁcation}_there_are_four_categories_of_shared_ﬁsh_stocks:_{transboundary}_stocks_(resources_that_cross_the_EEZs_of_two_or_more

coastalstates);highlymigratorystocks(foundbothwithintheEEZsandtheadjacenthighseasandhighlymigratoryinnature);straddlingstocks(also coverbothEEZsandthehighseasbutaremorestationary);discretehighseasstocks(foundexclusivelyinthehighseas).

(4)

ConventionontheLawoftheSea(UNCLOS)harmonizedandlegalizedthevariousunilateraldeclarationsinassigningthe righttocoastalstatestoestablishEEZs,comprising200nauticalmiles(Munro,1982).Aftersomeinitialsuccess,itbecame clearthatfurtheractionwasrequiredasthesignificanceofhighseasfisherieshadbeenunderestimated.Inparticular, technologicalprogress,suchastheintroductionoffishcarriersandvesselswithonboardfishprocessingequipment,had madetheresourcesofthehighseasmoreaccessible.Increasingawarenessofoverfishingledtothe1995UNFishStocks Agreement.Underthisagreement,sharedfishstocksaretobemanagedonaregionbyregionbasisbyRegionalFisheries ManagementOrganizations(RFMOs).Therearecurrentlyaround20RFMOsinforceasforexampletheNorthwestAtlantic FisheriesOrganization(NAFO)andtheNorthEastAtlanticFisheriesCommission(NEAFC).8,9

3. Model

3.1. Preliminaries

Theformationofcooperativeagreementsistypicallymodelledasatwo-stagegameinwhichplayerschooseﬁrsttheir membershipandthen theireconomic strategies. (Foranexception, see,e.g.,McCarthyet al.,2001and Sampson and Sanchirico,2019.)Thegameissolvedbybackwardsinduction.Thetypicalassumptioninthesecondstageisthat play-erswhichjoinanagreementchosetheireconomicstrategiescooperativelywhereasnon-membersnon-cooperatively.That is,memberstoanagreementinternalizeallexternalitiesamongtheirgroup.Consequently,ifnoagreementforms,this correspondstotheclassicalNashequilibriumwhereasifallplayersjointheagreement(i.e.,thegrandcoalitionforms),this representsthesocialoptimum.Thetypicalassumptionintheﬁrststageisthat–basedonthepayoffderivedfromthesecond stage–anagreementisstableifnomemberwantstoleaveandnonon-memberwantstojoin.

Thedetailsofthetwostagescanbemodelledinmanyways(see,e.g.,FinusandCaparrós,2015).The“ideal”modelwould allowforadynamiccoalitionformationprocessinbothstages:playerscanrevisetheirmembershipdecisionaswellastheir economicstrategiesovertime.Thatis,(i)currentharvestsaffecttomorrow’sfishstocksacrosstheeconomicszones;(ii) countriesreconsiderregularlytheirfishingquotasandtheirdecisiontovoluntarilyjoinorleaveaRFMO;(iii)suchfishing quotasandmembershipdecisionsarereconsideredcontingentonstocklevels.Suchanidealmodelistechnicallyextremely challenging,andwouldbeevenmoresoinourcontextwithseveralzonesandmigration.Itisforthisreasonthatexceptfor RubioandUlph(2007)inthecontextofclimatechangeandMillerandNkuiya(2016)inthecontextofafishery(without zoningandmigration),allcoalitionmodelsofwhichweareawarehavemadesomesimplifyingassumptions:eitherthefirst orthesecondstageisstatic.

FinusandRundshagen(2006)andFinusetal.(2014)modelasequentialcoalitionformationprocessinstage1,butassume aone-shotpayoffderivedfromeconomicstrategiesinstage2.Wewillbrieﬂyreportontwointerestingextensionstoour modelregardingtheﬁrststageinSection8.

Incontrast,EyckmansandFinus(2006);EyckmansandTulkens(2003)andRubioandCasino(2005)inthecontextof climatechangeandKwon(2006)inthecontextofﬁsheries(withonesinglezone)assumeaone-shotdecisionintermsof membership,basedonthenetpresentvalueoftheequilibriumstrategiesinadynamicgame.Clearly,thisapproachcannot capturetheinterestingfeatureofhowthetransitionfromonesteadystatetoanotherovertimeaffectsstabilityofcoalitions. Transitionsdisappearinthesummationovertime.Hence,maybenotsurprisingly,qualitativeresultsofthenetpresent valueandsteady-statepayoffapproachareverysimilar.Forinstance,inaﬁsherywithsymmetricplayersandstandard assumptions,nostableagreementexistsifthenumberofplayersislargerthantwoasshowninPintassilgoandLindroos (2008)inthesteadystatepayoffmodelandinKwon(2006)inthenetpresentvaluemodel.10_It_is_for_this_reason_that wemodelthesecondstageusingthesimplestaticGordon-Schaefermodel(seealsoFlaatenandMjølhus,2005;Kvamsdal andSandal,2008,Pezzeyetal.,2000,Puntetal.,2013;Lindroos,2008,Lindroos,2008andLongandFlaaten,2011),which basicallyassumesthatthesystemisalwaysinasteadystate.

Takentogether,weassumeasimplecoalitionformationprocessinwhichcountrieschoosesimultaneouslywhetherto joinanRFMOinafirststageandthereaftersimultaneouslychosetheirequilibriumeconomicstrategiesinthesecondstage basedonthesimplemechanicsofthestaticversionoftheclassicalGordon-Schaefermodel(Gordon,1954andSchaefer, 1954).Thismodelisextendedtoaccountfordifferentfishingzonesandthemigrationoffishstocksacrosszones.Inthe following,thebiologicalmodelisdevelopedinSection3.2,theeconomicmodelislaidoutinSection3.3,whichcapturesthe strategicbehavioramongstatesundervariousassumptionsaboutthedegreeofcooperation;italsoincludesthedefinition ofstablecooperativearrangements.

8 _For_an_overview_see_for_instance_Munro_et_al.₍₂₀₀₄₎_and_FAO_online_(2012).

9 _Reports_that_seriously_and_consistently_measure_the_{effectiveness}_of_RFMOs_are_scarce._Some_evidence_is_gathered_for_instance_in_High_Seas_Task_Force

(2006)andLodgeetal.(2007).AsWillockandLack(2006),p.32,write:“Thereappearstobesomereluctanceto,oratleastnervousnessabout,establishing astandardsetofperformanceindicatorsagainstwhichRFMOsmightbeheldaccountableandtheirperformancecompared.”

10 _Similarly,_in_climate_change,_for_a_{linear-quadratic}_payoff_function,_a_stable_agreement_consist_of_three_players_in_the_static_payoff_model_(Barrett,₁₉₉₄₎

(5)

3.2. Biologicalmodel

WeassumethatagivennumberofplayersNexploitasharednaturalresourceofsizek.11_In_the_context_of_biological populations,kiscalledthecarryingcapacityofthebiologicalsystem,whichweinterpretasthegeographicalsizeofthe systemasinFlaatenandMjølhus(2005);Pezzeyetal.(2000)andSanchiricoandWilen(1999).Inourcontext,theresourceis thefishstockandthebiologicalsystemistheocean.Partsofthesystemmayhavebeenprivatizedthroughtheestablishment ofexclusiveeconomiczones.Hence,therearetwotypesofgeographicalzones:thehighseas,abbreviatedHS,thecommon propertywhereallstatescanfish(Art.87,UNCLOS1982),andtheexclusiveeconomiczones,abbreviatedEEZ_i,theprivate propertieswithexclusivefishingrightsofcoastalstatei(Art.56,UNCLOS1982).

Denotingtheentiresizeofthesystembyktotandtheshareoftheresourceforwhichnoprivatepropertyrightshave

beenestablishedby˛,wedeﬁne: kHS=˛ktot and kEEZ=

1−˛

N ktot (1)

assumingforsimplicitythesamecarryingcapacityineachEEZ_i.Henceforth,˛∈[0,1]measurestherelativesizeofthe differentpatches.Itwillbecomeapparentthatinthefullyintegratedmodelwithdensity-dependentdiffusion,thisparameter canbeexclusivelyrelatedtothedegreeofsociallyconstructedexcludabilitywith˛=0implyingperfectsociallyconstructed excludabilityand˛=1perfectnon-excludabilityattherespectivelimits(seeTable1).Inthecontextofasink-sourcemodel, suchsimplificationisnotvalid.We encouragethereadertoconsultAppendix3intheSupplementarymaterial,which explainsunderwhichconditionstherelativepatchsizes˛canbelinkedtothedegreeofsociallyconstructedexcludability. Inourcontext,playersaresovereigncountriesengaginginfishing,i.e.,coastalstates,withexclusiveaccesstotheirown EEZandasharedaccesstothehighseas.WeabstractfromthefactthatEEZscouldbeofdifferentsizeandthatso-called distantwaterfishingstateswithoutEEZengageinfishinginthehighseas.

Thesteady-stateconditionisgivenbyasystemofN+1equations:

G(X)₋H(X,E)₊DX=0, (2)

withX= (X1,...,XN,XHS) thevectorofﬁshstocksinthevariouszones12,thevectorofefforts,E=

EEEZ1,...,EEEZN,EHS

whichisaphysicalmeasureofinput,13_e.g.,_time_spent_fishing,_{G (X) the}_vector_of_growth_functions,_{H (X,}_{E) the}_vector_of harvestlevels,Dadiffusionmatrixaccountingforthemigrationoffishstocksacrosszones,and0avectorofzeroswithsize N+1.Hence,Eq.(2)statesthatinthesteadystate,growthandharvestarebalanced,accountingadditionallyforincoming andoutgoingstockflowsthroughmigration,suchthatthestockineachzoneremainsconstant.Clearly,thehighergrowth, themorecanbeharvestedinequilibriumandhencetheloweristhedegreeofrivalry.

ThecomponentsofG= (G1,...,GN,GHS) describegrowthofthestockineachzone,assumingthatgrowthrequiresan

initialpopulation,Gi

Xi

Xi=0

=0,i=1,...,N,HS,is positiveaslongasthecarrying capacityhasnotbeenreached, Gi

Xi

Xi<ki

>0,andstopsatthecarryingcapacity,Gi

Xi

Xi=ki

=0.14_The_components_of_H₌

_H EEZ1,...,HEEZN,HHS

aretheharvestlevelsineachzonewhichdependbothonthevectorofstocks,X,andthevectorofefforts,E,i.e.,H (X,E). HEEZ1,...,HEEZN aretheharvestlevelsofeachcountryinitsownEEZ;HHSistheaggregateharvestlevelofallcountriesin thehighseas.Duetothemigratorybehaviorofﬁshstocks,harvestfromeachzonegenerallydependsonallﬁshingefforts. Finally,thediffusionmatrixD=

dij

,i,j∈{1,2,..,N,HS}containsallinformationneededtodescribethediffusionprocess; itisnotonlyimportantwhetherzoneiandzonejareconnectedviadiffusion(dij /=0anddji /=0)butalsothestrength

ofinteraction,i.e.,theabsolutevalueofdijanddji,aswellasthesign,withnegativevaluesofdij indicatingnetoutgoing

diffusionfromzoneitojandpositivevaluesnetincomingdiffusion.15

Fromaconceptualpointofview,migrationdeterminesthedegreeoftechnicalnon-excludability.Asitisvirtually impos-sibletoerectfencesintheoceantoseparatefishstocks,itistechnicallynotfeasibleforacountrytoexcludeothercountries entirelyfrombenefitingfromitsfisheryresources.Thus,therecanbesomedegreeofnon-excludability,stemmingfrom migration,evenifsociallyconstructedexcludabilityisperfect,i.e.,allpropertyrightshavebeenallocatedtostates,˛=0, andtheserightsareperfectlyenforceablethroughthedeclarationofEEZs(i.e.,weruleoutillegalfishinginEEZs).

11_Hence,_in_our_setting,_{non-cooperative}_behavior_is_not_identical_to_what_is_called_open_access_in_the_ﬁshery_literature_as_long_as_N_is_ﬁnite._That_is,_rents

arelowerinthenon-cooperativethaninthecooperativeequilibrium,butrentswillnotcompletelydissipatethroughentry.

12_We_talk_about_different_stocks_in_different_zones,_but_one_could_also_talk_about_different_shares_of_the_total_stock_as_we_consider_only_one_species._In_any

case,ifwetalkaboutthetotalstock,wemeanthesumofthecomponentsofthevectorX.Thetotalstockaswellasitsallocationisaresultofequilibrium effortlevelsasdescribedinSection3.3andtheexogenousparametersofthemodel,likeforinstancetheallocation,diffusionandcostparameter.

13_The_aggregate_effort_in_the_high_seas,_E

HS,isthesumofeffortlevelsofallcountries,EHS=

i=1,..,NEHSi.Similarly,forharvestlevels,HHS=

i=1,..,NHHSi

asthehighseasisaccessiblebyallcountriesandisviewedasonezone.AmodiﬁcationofthisassumptionisconsideredinSection8.1.

14_Appendix₃_in_{Supplementary}_material_explains_that_no_such_concept_as_aggregate_growth_over_all_zones_exists,_growth_is_only_deﬁned_in_a_particular

zone.

15_It_would_be_misleading_to_think_of_the_diffusion_matrix_being_similar_to_the_{transportation}_matrix_known_from_{transboundary}_pollution_as_we_explain_in

moredetailSection4.2andinAppendix2intheSupplementarymaterial.Theentriesinthetransportationmatrixarethesharesofemissionsofcountryi whicharedepositedincountryj.Hence,sharesliebetween0and1.Theentriesinthediffusionmatrixreﬂectthespeedanddirectionofmigrationbetween zones.Thus,notonlyunitsaredifferentbutentriescanbepositiveandnegativeandarenotboundedby1.Notethatthesteady-stateconditiondoesnot requirediffusiontovanishbutonlytobebalancedbygrowthandharvestineveryzone.

(6)

3.3. Economicmodel

Eachplayerreceivesaneconomicrentor,aswecallit,payoff˘ithatisobtainedfromtheharvestextractedfromthe

privateandpublicresource:

˘i=p·(HEEZi(XEEZi,EEEZi)+HHSi(XHS,EHSi))−Ci(EEEZi,EHSi) (3) wheretheﬁrsttermcapturesrevenueswithpthe(constant)ﬁshpriceandHEEZ_iandHHS_i theharvestlevelsobtainedby

nationi,i=1,...,N,fromﬁshinginitsownEEZandinthehighseas,andthesecondtermrepresentsthecostfunctionwhich dependsoninputs,i.e.,efforts.Eachplayerihastomaketwostrategicchoices:theﬁshingeffortinhis/herownEEZ,EEEZ_i

andtheﬁshingeffortinthehighseas,EHS_i.

CooperationamongagroupofplayerscorrespondstotheestablishmentofanRFMOwiththepurposeofmanagingand conservingthefishstocksjointly.IfRFMOsareestablished,theyaresingleagreements,i.e.,noRFMOsco-existthatregulate thesamefishstock.ParticipationinanRFMOisvoluntaryandopentoallstatesasreflectedbyArticle8(3)oftheUNFish StocksAgreementin1995.Moreover,weassumethatstates,whichdecideagainstmembershipinanRFMO,cannotbe preventedfromharvestinginthehighseas.16

Inordertocapturetheseinstitutionalfeatures,wechoosefromthesetofcoalitionformationgamesthesinglecoalition openmembershipgameduetod’Aspremontetal.(1983)whichhasbeenfrequentlyappliedintheliteratureonIEAs(e.g., FinusandCaparrós,2015foranoverview)butalsoinotherareasofeconomicinterest(e.g.,Bloch,2003andYi,1997for surveys).Thiscoalitiongameisatwo-stagegame.

Intheﬁrststage,playersdecideupontheirmembership.ThoseplayersthatjointheRFMOformthecoalitionandare calledmembers,thosethatdonotjoinarecallednon-members.Thedecisionsintheﬁrststageleadtoacoalitionstructure

S,1,...,1

whereSisthesetofcoalitionmembersandtheremainingplayersaresingletons.Giventhesimplestructureof thefirststage,acoalitionstructureisfullycharacterizedbycoalitionS.Inthesecondstage,playerschoosetheireconomic strategies,whicharefishingeffortsinourmodel.Ineachstage,strategies(participationandfishingeffort)formaNash equilibrium.Thegameissolvedbackward.

Inthesecondstage,givensomecoalitionShasformedintheﬁrststage,non-membersactnon-cooperativelyand max-imizetheirindividualpayoff,˘i,whilemembers,actingcooperatively,maximizetheaggregatepayoffoftheircoalition,

˘S=

i∈S ˘i:17 argmax (E_EEZj,E_HSj) ˘j(E)

∀

j/∈S (4) argmax (E_EEZS,E_HSS) ˘S(E) (5)

whereE=(EEEZ1,...,EEEZN,EHS1,...,EHSN)denotesthevectorofallﬁshingeffortswhereasEEEZS=

EEEZ_i

i∈SandEHSS=

EHS_i

i∈SdenotethevectorsofﬁshingeffortsofthecoalitionmembersintheEEZsandinthehighseas,respectively.It

isimportanttonotethatwithinanRFMO,onlythememberwhoownsanEEZfishesinthisEEZ,butfishingeffortsin EEZsarecoordinatedacrossRFMOmembers.Inthehighseas,eachmemberfishesandmemberscoordinatetheirefforts. Coordinationmeansthatmembersareawareoftheexternalitiesamongeachother.Thatis,ceterisparibus,fishingreduces thestock,whichforthesameeffortreducesharvestand/orincreasesthecostoffishing.Moreover,fishinginzoneiwill eitherreducetheoutgoingdiffusiontootherzonesorwillincreaseincomingdiffusionfromotherzones,dependingonthe netdirectionofdiffusionifmigrationdependsonstockdensities.Similarly,inasink-sourcemodel,fishinginthesourcewill reducemigrationtothesink.

Thesimultaneousmaximizationof(4)and(5)deliverstheequilibriumﬁshingeffortsE∗(S).Asnotedabovealready,this equilibriumisidenticaltotheNashequilibriumknownfrommodelswithoutcoalitionformationifcoalitionScomprisesonly asingleplayer,S=

i

,orisemptyS=∅.Moreover,ifcoalitionScomprisesallplayers,S=

1,...,N

,i.e.,thegrandcoalition forms,theequilibriumcorrespondstothesociallyoptimalﬁshingvector.Hence,theentirerangefromnocooperation,partial cooperationtofullcooperationcanbecapturedbythisapproach.

Itisworthwhiletomentionthatthesolutionto(4)and(5)willbeidenticalforeverycoalitionS⊆

1,...,N

,i.e.,the degreeofcooperationdoesnotmatter,ifandonlyifboth˛=0(nohighseas)andthereisnodiffusion.Thatis,thereis noexternalityacrossplayersandhencethestudyofRFMOsisnotinteresting.Thiswouldcorrespondtoasystemofclosed patcheswithnohighseas.Incontrast,evenifthereisnodiffusionbetweenanyzone,aslongas˛>0,thereisanareaof commonpropertyresourcethatcanbeexploitedbyallcountries.(Thiswouldbeasystemofclosedpatcheswithhighseas.) Thus,no,partialandfullcooperationimplydifferentvectorsofequilibriumﬁshingefforts.Thisisalsotrueevenif˛=0,i.e.,

16 _The_legal_basis_and_the_implications_of_giving_up_this_assumption_are_brieﬂy_discussed_in_Section_8.1.

17 _The_assumption_that_RFMO-members_choose_their_ﬁshing_efforts_{cooperatively,}_both_in_the_high_seas_and_in_their_EEZs,_is_in_line_with_FAO_(2010),_p.

123,whichstates:“EachRFMOis,interalia,calledupontoensurethatthemanagementmeasuresforthehighseassegmentsoftheresourcesandthosemeasures fortheintra-EEZsegmentsoftheresourcesarecompatiblewitheachother”.

(7)

Table2

FunctionalSpeciﬁcationofModel.

1)HarvestFunctions HEEZi

XEEZi

=qiEEEZi X_EEZi k_EEZi,i=1,...,N;HHS(XHS)= N

i=1 qiEHSi XHS kHS 2)CostFunctions Ci(Ei)=ciEi,i=1,...,N,HS 3)GrowthFunctions Gi(Xi)=riXi

1−Xi ki

,i=1,...,N,HS

4)MigrationProcess:FullyIntegratedModel

EntriesofthedispersalmatrixD: dij=

d

ki/kj ifzonejisadjacenttoi 0 otherwise ∀j/=i dii=−d

j/=i

kj/ki ∀i

5)MigrationProcess:Sink-Source,HS=source

EntriesofthedispersalmatrixD: diHS=_Nd ∀i=1,...,N

dHSHS=−d

0 allotherentries

6)MigrationProcess:Sink-Source,EEZs=source

EntriesofthedispersalmatrixD : dii=−d ∀i=1,...,N

dHSi=d ∀i=1,...,N

0 allotherentries

ri=intrinsicgrowthrateinregioni;XEEZi,XHS=stockinEEZiandHS,respectively;kEEZi,kHS=carryingcapacityinEEZandHS,respectively;qi=efﬁciency

parameterofcountryi;EEEZi,EHSi=effortsinEEZiandHS,respectively;dij=diffusionparameterbetweenregioniandj;ci=costparameterofcountryi.

allpropertyisprivatelyowned,aslongasthereisdiffusionamongatleasttwozonessuchthattheactionofoneplayerhas

animpactonatleastoneotherplayer.Hence,notonlyinafullyintegratedmodelbutalsoinsink-sourcemodel,equilibrium

effortswilldifferwiththedegreeofcooperation.

EquilibriumeffortsE∗(S) derivedfrom(4)and(5)togetherwiththesteady-stateconditionsofstocksin(2)havetobe

insertedintothepayofffunction(3)todetermineindividualpayoffs˘_j/∗_∈S(S) andthecoalitionalpayoff˘_S∗(S).Thecoalitional

payoffwillhavetobedistributedinsomewaysuchthat

_i_∈_S˘_i∗(S) =˘_S∗(S).FordetailsseeSection4.

Havingdeterminedequilibriumpayoffsforeverypossiblecoalitionstructureinthesecondstage,wecannowproceed

tothefirststage.Inthefirststage,acoalitionSisconsideredtobestableifitsatisfiesthefollowingtwoconditions:

Internalstability

Nomemberi∈Sﬁndsitproﬁtabletodeviate,i.e.,thegainfromleavingthecoalitionisnon-positive:˘_i∗(S{i})−˘_i∗(S)≤

0,

∀

i∈S.

Externalstability

Nonon-memberj∈/Sﬁndsitproﬁtabletojointhecoalition,i.e.,thegainfromjoiningthecoalitionisnon-positive:

˘_j∗(S∪{j})−˘_j∗(S)≤0,

∀

j/∈S.

Notethatthegrandcoalitionisexternallystablebydeﬁnition,asthereisnooutsiderleftthatcouldjointhecoalition.

4. Modelspeciﬁcationandsolutionprocedure

4.1. Preliminaries

Asmentionedabove,themodelissolvedbybackwardinduction.Themostcomplexpartrelatestothesecondstagein

whichoptimalﬁshingeffortshavetobedeterminedforagivencoalitionstructure.Forthis,thesystemofEq.(2),which

representsthesteady-stateconditions,andthefirst-orderconditionsderivedfrom(4)and(5)havetobesolved simultane-ouslyinordertoobtainsteady-statestocksandequilibriumfishingefforts.Thesolutiontothe3N+1equationswilldepend onthespecificationofthefunctionalrelationshipbetweenstocks,effortsandpayoffs.Thatis,wehavetospecifygrowth, harvestandcostfunctionsanddefineadiffusionmatrix,whichdescribesthemigrationprocess.ThisisdoneinSection4.2. Moreover,aswefaceahighlynonlinearsystemofequations,which,generally,cannotbesolvedanalytically,wehaveto relyonnumericalsimulations,whichwedescribeinSection4.3.

4.2. Functionalspeciﬁcation

Inthissection,wespecifythefunctionalrelationships(seeTable2).Itwillbeapparentthatthespeciﬁcationsfollowthe mainstreamassumptionsintheliterature.

Regardingtheharvestfunction(Table2,firstrow),wehavetobearinmindthatexceptfortheextensionwhichwe considerinSection8.1,allcountriesareallowedtofishinthehighseaswhereasonlytheownerofanEEZisallowedtofish inthisterritory.Hence,theharvestinthehighseasisthesumoftheindividualharvestlevelsofeachcountry.Ascommonly

(8)

assumed,(total)harvestdependslinearlyon(total)ﬁshingeffortsandstockdensities,withqidenotingthecatchability

coefficient,ameasureoftheefficiencyoffishingfleeti.Hence,forthesameeffort,countrieswillharvestmoreifthestock densityishigh(stockdividedbycarryingcapacity).

Itisacommonassumptionintheliteratureonfisherymanagement(Gordon,1954;Pezzeyetal.,2000andSanchirico andWilen,1999)thatcosts(Table2,secondrow)dependlinearlyonextractionefforts,thoughtheyarestrictlyconvexif expressedintermsofharvestlevels,whereciisthe(constant)marginalcostoffishingeffortofthefishingfleetofcountryi.

Themostcommonly usedgrowthfunction(Table2,thirdrow) isofthelogistictype whereri denotestheintrinsic

growthrateinzonei,whichisourmeasureofrivalrywiththedegreeofrivalryinverselyrelatedtothevalueofri.We

followHannesson(1998);KvamsdalandSandal(2008)andmanyothersandassumethatgrowthdependsonthelocal characteristicsofazone.Hence,evenifthegrowthrateriisthesameineveryzone,growthmaydifferacrosszonesbecause

stocksXiandstockdensitiesXi

⁄

kimaybedifferent.Moreover,notethatgrowthcannotgenerallybeaggregatedacrosszones. SeeAppendix3intheSupplementarymaterialfordetails.

Threeaspectsneedtobeconsideredwhenspecifyingthemigrationprocess.

Firstly,thearrangementofzoneshastobespeciﬁed.WechooseanintuitiveandsymmetricarrangementoftheN+1 zones:theEEZsarearrangedinacirclewiththehighseasatitscenter,asdepictedinFig.1.Thisavoidsboundaryeffectsthat wouldemergewithalineararrangementandrepresentsagoodﬁrst-orderapproximationforthegeographicalsettingof manyexampleswhereanareaofhighseasissurroundedbycoastalzones.Agoodmatchofthisassumptionisforinstance the‘BananaHole’intheNortheastAtlanticorthe‘DonutHole’intheBeringSea(seeMeltzer,1994).

Secondly,wehavetodeterminethedirectionofmigration.(SeeSanchiricoandWilen,1999forageneraldiscussion.) Fig.1arepresentsourbasecase,Fig.1bandcconstituteextensions.InFig.1a,weassumethatdiffusionispossiblebetween alladjacentzonesandinalldirections.Thiscorrespondstothefullyintegratedmodelifdiffusiontakesplacebetweenall zonesandapproachestheclosedpatchmodelifdiffusioniszero.InFig.1band1cweconsideruni-directionaldiffusion,in linewithwhatiscalledasink-sourcemodel.InFig.1b,thehighseasisthesourceandtheEEZsarethesinks.Thereisno diffusionbetweenEEZs.InFig.1c,thisisreversed.TheEEZsarethesourceandthehighseasisthesink.Again,nodiffusion isassumedbetweenEEZs.

Thirdly,theintensityofmigrationbetweentwoneighboringfishinggroundsneedstobespecified(Table2,fourth,fifth andsixthrow).

ForthefullyintegratedmodelinFig.1a,weassumeadensity-dependentdiffusionprocess,i.e.,thestrengthofmigration betweenneighbouringﬁshinggroundsdependsonthedifferenceinstockdensities(e.g.,ArmstrongandSkonhoft,2006and SanchiricoandWilen,1999,2005).Thatis,thechangeofstockXiduetodiffusionbetweenzoneiandjovertimeisgivenby

∂

Xi

∂

t

i↔j =(i,j)

Xj kj− Xi ki

. (6)

Ifthestockdensityinzonejislarger(smaller)thaninzonei,thistermispositive(negative)andthereisincoming (outgoing)diffusioninzonei.Bysymmetry,wehave:

∂

Xi

∂

t

i↔j =

−

∂

Xj

∂

t

i↔j . (7)

Theparameter(i,j)maybeuniformacrosszonesormayreﬂectthecharacteristicsofadjacentzones.Wechoose(i,j)= d

kikj.Thatis,diffusionbetweenlargerzonesislargerthanbetweensmallerzones.Thismeansweusethegeometric

meanofthecarryingcapacities,

kikj,asascalingfactor.Thisseemstobemoreappropriatethansaythearithmeticmean,

ki+kj

/2,whichwouldimplysigniﬁcantdiffusionevenifonecarryingcapacityisverysmall(i.e.,ki→0orkj→0).The

generalintensityofmigration,asacharacteristicofa particularspeciesiscapturedbythediffusionparameterd.This parameterdeterminesthegeneraldegreeoftechnicalnon-excludability.Forhighlymigratoryspecies,parameterdwould beverylarge.ThedetailsofhowdiffusioniscapturedbythediffusionmatrixinFig.1aaredescribedinAppendix2inthe Supplementarymaterial.

Takentogether,(i,j)determinesthespeedatwhichstockdensitiesarebalancedinagiventimespan.Notethatin asteady-state,therecanbenetdiffusionifequilibriumstockdensitiesinneighboringareasaredifferent.Moreover,the diffusionparameterdcanexceedthegrowthraterwithoutresultinginnegativestocksbecausetheabsoluterateofdiffusion inthesteady-statedependsnotonlyondbutalsoonthedifferenceindensities,whichwillbesmallwhendishigh.18_The density-dependentmodelistypicallymotivatedbytheobservationthatﬁshmovefromhighertolowerdensitiesasthere islesscompetitionforfood.Itisalsoinlinewithrandommovements,whicharealsoknownasBrownianmotion.

Thedetailsofthesink-sourcemodelsasdisplayedinFig.1bandcareprovidedinAppendix1intheSupplementary material.Sink-sourcemodelsareconsideredtobeagoodﬁtforﬁshspecieswithaparticularmigratorypatternwhich

18 _For_instance,_think_of_a_very_low_value_of_r_(e.g.,_whales)._In_this_case,_it_may_happen_that_parameter_d_is_larger_than_r,_such_that_differences_in_regional

harvesting(H)arebalancedbydispersal(D),ratherthanbygrowth(G),maintainingthesteady-stategivenbyG−H+D=0.Forsuchahighlymigratory, slow-growingspecies,astockthatisheavilyexploitedinoneregionwouldbereplenishedbyincomingdiffusionfromotherregionsratherthanintrinsic growth.

(9)

Fig.1. MigrationPatternandSpatialAllocationofPropertyRights*.

consistsofmovingfromasourcetoasink,regardlessofthepopulationinthesink.Differentfromthedensity-dependent model,diffusioncannotexceedgrowthinaninteriorsteadystate,whichweassumetohold.

Inordertofocusthediscussion,weassumethefullyintegratedmodelwithdensity-dependentdiffusionifnotmentioned otherwise,andtreatthesink-sourcemodelasanextensioninSection8.3.

4.3. Solutionprocedure

Despiteassumingparticularfunctionalformsasoutlinedintheprevioussubsection,generally,thetwo-stagecoalition gamecannotbesolvedanalytically.Inthecontextofcoalitionformation,thereareplentyofexampleswherethisrelatesto thefirststageofcoalitionformation,thedeterminationofstablecoalitions.Forinstance,thismaybeduetothecomplexity ofthespecificsetting(e.g.,Barrett,1994anddeZeeuw,2008)orduetotheasymmetryofplayers(e.g.,McGinty,2007and Pintassilgoetal.,2010).Inourcontext,thisrelatestothesecondstageofcoalitionformationandisduetothedeparture ofasinglefishingareaandthepossibilityofmigrationoffishstocksacrossareas.Bothfeaturesarerelatedtoourcentral

(10)

Table3

ParameterValuesinSimulations*.

SimulationRuns c r d ˛

A 0.5 0.5 0–1.28 0–1.0

B 0.25-0.75 0.5 0–1.28 0–1.0

C 0.5 0.25-0.75 0–1.28 0–1.0

* _Parameter_variations_for_simulation_runs_are_indicated_in_bold;_p=1,_q=1_and_k

tot=4areassumedthroughout.

ideatocapturevariousdegreesofsociallyconstructedandtechnicalexcludability.Onlyforextremeparametervaluescan

analyticalsolutionsbeobtained.

Forinstance,ifwelet˛=1,thenthisisthemodelwithhighseasonly,andresultsfollowfromPintassilgoandLindroos

(2008)forsymmetricplayersandfromLindroos(2008)andPintassilgoetal.(2010)forasymmetriccostfunctions.Notethat inourmodeldiffusiondoesnotmatterfor˛=1becausediffusionismeasuredbetweenandnotwithinazone.

Bythesametoken,wecanconjecturethatifweletdiffusionparameterdgotoinﬁnity,thenthevalueof˛doesnot matter.Essentially,theallocationofpropertyrightsbecomesirrelevantbecauseveryfastdiffusioneffectivelylinksthemto onezone.Defacto,thiscorrespondstothesituation“onlyhighseas”with˛=1asmentionedabove.

Finally,if˛=0andalld=0,thereisnoexternality.Hence,secondstageﬁshingeffortsaresociallyoptimalregardless ofwhichcoalitionforms.Consequently,thereisnogainfromcooperationbutalsonoincentivetofree-rideandhencethe grandcoalitionisstableasweargueinResult4,Section6,below.

Takentogether,newinterestingresultscanonlybeobtainedintheinterioroftheallocationanddiffusionparameter spaceforwhichsimulationsarerequired.

Itisevidentthatcomputingtimeandcapacityrequirementsincreaseexponentiallywiththenumberofplayers.Forthis reason,weconﬁneourselvestothecaseofN=3players.Thisiscertainlytheminimumnumberofplayersinordertomake theanalysisofcoalitionformationinteresting,butasitturnsout,thisissufﬁcienttoderiveinterestingqualitativeresults.19 ForN=3,wehavetoconsiderthreepossiblecoalitionstructures,namelythegrandcoalition,thetwo-playercoalitionsand theall-singletonscoalitionstructure.Furthermore,wewillrestricttheanalysistosymmetricparametervaluesforallplayers (andthereforewecandroptheindexforparametersqi,ci,andrihenceforth).ThisimpliessymmetricequilibriaintheNash

equilibriumandthesocialoptimum.Moreover,allpossibletwo-playercoalitionsareequivalentwithsymmetricpayoffsfor coalitionmembers(i.e.,equalsplitofthetotalcoalitionalpayoff),thoughtheydifferfromthepayoffofanon-member.20 Moreover,withsymmetry,internalandexternalstabilityarecloselyrelated(CarraroandSiniscalco,1993):ifacoalition withnplayersisnotinternallystable,thenthecoalitionwithn−1playersisexternallystable.

Simulationsrequiretheassumptionofnumericalvaluesfortheparametersofthemodel.Fortunately,acloserlookat thesystemofequationsrevealsthatresultswilldependononlyfewparameters.Thechoiceofparametervaluesfollows goodpractice,covering(almost)theentireparameterspace(undertheassumptionofinteriorsolutions)assummarizedin Table3.

Firstnotethatthetotalcarryingcapacityktotjustrepresentsascalingfactorwhichwenormalizeto4astherearefour

zones.21_Moreover,_all_subsequent_results_do_not_depend_on_c,_p_and_q_{independently}_but_just_on_their_ratio c

pq,withpqkctot beingcommonlyreferredtoasthe‘inverseefﬁciencyparameter’(seeMesterton-Gibbons,1993).Thus,wenormalizepand qto1andhenceonlyvaryc,resulting,ceterisparibus,inavariationoftherelationc

_⁄

_pq_._Since_in_this_setting_prohibitive_costs atwhichcountriesquitﬁshingaregivenbyc≥1irrespectiveofscenarioofcooperation,weneedtoassumec∈[0,1]for interiorsolutions.Inoursimulations,wesetthebasecasevaluetoc=0.5andconsidertwoothervalues:c=0.25and c=0.75.Fortheintrinsicgrowthrater,wechoosethecommonlyusedbasevaluer=0.5andconsidertwoothervalues: r=0.25andr=0.75.22_Recall,_the_growth_rate_approximates_the_degree_of_rivalry.

Forthediffusionparameteroursimulationscovertheranged∈ [0,dmax] inintervalsofd=0.08withtheupper

bounddmax=1.28thatapproximateswellthelimitd→∞.23,24Withrespectto˛,wecoverthewholerange˛∈[0,1],

19 _This_assumption_has_also_the_advantage_that_the_{density-dependent}_model_is_a_fully_integrated_system._Strictly_speaking,_a_fully_integrated_system,_in

whichallzonesarelinked,doesnotexistifthenumberofzonesexceedsfour(duetothefourcolormaptheorem).

20 _Thus,_players_are_ex-ante_symmetric_(before_coalition_formation)_but_may_be_ex-post_asymmetric,_depending_on_whether_they_become_members_or

non-members.Theassumptionofex-antesymmetricplayersiswidespreadintheliteratureoncoalitionformation,notonlyoninternationalenvironmental treatiesbutalsointhecontextofothereconomicproblems(see,e.g.,Bloch,2003andYi,1997foranoverview).

21 _This_is_in_line_with_the_common_{normalization}_k₌₁_in_papers_that_deal_with_only_a_single_zone_(e.g.,_Pezzey_et_al.,_2000)._In_our_model,_assuming_no

diffusionbetweenzoneswithktot=4andsetting˛=0.25resultsinfourisolatedzoneswithcarryingcapacitieski=1.SeeEq.(1).

22 _Our_base_case_values_c₌_0.5_and_r₌_0.5_are_commonly_assumed_in_the_literature_(e.g.,_Hannesson,₁₉₉₇_and_Tarui_et_al.,_2008)._Note_that_a_variation_of

thegrowthrateintherange0.25≤r≤0.75(e.g.,asconsideredinNøstbakken,2006)alreadyhasasigniﬁcantimpactontheoutcomeintermsofpayoffs. Forinstance,inmodelswithonlyasinglezone(e.g.,Pezzeyetal.,2000),whichcorrespondto˛=1inourmodel,aggregatepayoffsintheNashequilibrium atagrowthrater=2/3arealreadyashighasinthesocialoptimumatr=0.5.

23 _Results_for_d₌_d

maxdifferlessthan5%fromtheresultsinthelimitd→∞andconvergetowardsthe‘onlyhighseas’scenario(˛=1),whichcanbe

calculatedanalyticallyaspointedoutabove.

24 _For_the_{density-dependent}_diffusion_model,_d_>>_r_is_possible_since_diffusion_will_always_cease_as_soon_as_differences_in_stock_densities_are_balanced.

Incontrast,inourextensions,whichconsidersasink-sourcemodel,d<rmustholdtoensurethatasteady-stateexistsinwhichintrinsicgrowthcan balanceoutgoingdiffusionandharvest.

(11)

with˛=0implyingthattheentireﬁshingareacomprisesonlystate-ownedexclusiveeconomiczonesand˛=1implying thattheentireareacomprisesonlythecommonpropertyhighseas.25_All_results_are_tested_in_the_entire_interval_in_steps of˛=0.05.Notethatthecarryingcapacities,kEEZandkHS,followfromtheallocationparameter˛andthetotalcarrying

capacityisgivenbyktot=4(seeSection3.2,Eq.(1)).

Byvaryingtheallocationparameter˛andthediffusionparameterd,wemodeldifferentdegreesofsociallyconstructed andtechnicalexcludability,respectively,asshowninTable1.Atthesametime,weareabletocaptureallfourcategories ofsharedﬁshstocksashighlightedinfootnote7:transboundarystocks(˛=0andd>0),straddlingstocksandhighly migratoryﬁshstocks(0<˛<1andd>0)anddiscretehighseasstocks(˛=1).Wealsocapturethe“boundarycases”of non-sharedstocks(˛=0andd=0),i.e.,stationarystockswithinEEZs,andthecaseinwhichtheEEZboundariesbecome irrelevant(d→∞).Inourextensiontoasink-sourcemodelwith0<˛<1,d>0,andd<r,wecanadditionallycapture uni-directionalmigrationpatterns.

Throughthevariationofasingleparameterinacomparativestaticway,wecananalyzehowsucha(ceterisparibus) variationaffectsoutcomes.However,inreality,whencomparingforinstancedifferentfisheries,theywillusuallydifferin morethanoneparameter.Hence,interpretationsrequiresomecaution.Forinstance,increasingtheeconomicparameterc meansthatfishingbecomesmorecostly,andhenceeverythingelseequal,willleadtolowerefforts,lowerharvestlevelsand higherstocks.Increasingthegrowthraterimpliesafasterreproductionrate,whichwillleadtolowerorhigherequilibrium stocksinequilibrium,dependingonhowfishingeffortsareadjusted(seeResult3).

Importantly,parameter˛isaninstitutional/legalparameterdefiningfishingrights.Atthesametime,itdefinesthe relativesizeofhabitats,withpossibledifferentstockdensitiesinthevariouszones.Thequestionarises,whetheritisthe characteristicsofthebiologicalmodelthatdeterminetheimpactof˛onoutcomes(e.g.,equilibriumstocklevels)orrather thechangeinthelegalstatusassociatedwithavariationof˛.Wearguethat,inthedensity-dependentmodelwithsymmetric parametervalues,avariationof˛doesnotaffectthebiologyofthesystemassuch,and,hence,changesinstocklevelsare entirelytheresultofachangeofthelegalstatus.

Toseethis,consideranenlargementofEEZsasimplementedbythe1982UNCLOS(correspondingtoadecreasein˛). AssumingthatstockdensitiesweredifferentinthehighseasandtheEEZareasbeforethechange,thentherewillbea transitionphaseduringwhichanenlargedEEZwillexhibitaspatiallyinhomogeneousdistributionofstock.Aggregationof suchaninhomogeneousdistributionintoonesinglestockparameterXiwouldnotbefeasible,duetothenon-linearityof

thegrowthfunction(seeAppendix3intheSupplementarymaterialfordetails).Yet,ourmodelisnotintendedtocover suchatransitionphase.Inanewsteadystate,however,effortsandtherebythestockwithinonezonewillbedistributed homogeneouslyagain.Hence,intheex-postequilibrium,itistheeconomicimplication(i.e.,changeinﬁshingefforts)ofthe changeinthelegalstatusthatchangesoutcomes,andnotchangesofthebiology.Forinstance,letd=0andconsiderthe extremecases˛=0and˛=1.Non-cooperativeequilibriumstocklevelswillbelowerfor˛=1thanfor˛=0,notbecause stockdisappearsbutbecausenoterritoryispubliclyavailablefor˛=0andallterritoryisavailablefor˛=1.Thesameis trueinapartiallycooperativeequilibrium.Incontrast,inthesocialoptimum,˛willnotmatterfortotalstocks.Thiswillbe apparentfromResult1below.

AlloursimulationrunsareconductedwiththesoftwarepackageMaple18witha10digitprecision.Foreachparameter combination,weletMaplenumericallysolvethesystemofequationsﬁvetimes,eachtimewithdifferentstartingvaluesfor thevariablesstocksandefforts,uniformlydistributedwithintherangeofpossiblevalues.Forstocks,thisrangeisobviously givenby [0,kEEZ] and [0,kHS],respectively.Forefforts,therangeis [0,Emax].Forinstance,theupperlimitinthefully

integratedmodelisgivenbyEmax=

pq−c p2_q2

rktot

3 ,whichcanbecalculatedanalyticallyandcorrespondstotheeffortofa

non-memberifacoalitionoftwoplayershasformedandtherearenoEEZs,i.e.,˛=1.Wedidnotobserveanydependencyon startingvalues.Moreover,solutionsconvergetoanalyticalresultsfortheextremeparametervaluesdiscussedabove.Finally, equilibriumvalueschangesmoothlywithachangeofparametervalues;nojumpsinequilibriumvalueshavebeenobserved. Takentogether,althoughwearenotabletogiveaformalproofofuniqueness,wehaveseveralheuristicindicationsthat multipleequilibriadonotariseinthismodel.ThisgivesusconﬁdencethatMaplecorrectlydeterminestheuniqueinterior equilibriuminoursimulationruns.

4.4. Qualiﬁcations

Whileourmodelisbasedonthemostcommonassumptionsininternationalfisheries(see,e.g.,Stavins,2011),wearewell awarethatsomeaspectsremainneglected(see,e.g.,Clark,2010).Withrespecttoresourcecharacteristics,wedonotdeal withtheagestructureofthestock,possiblepredator-preyrelationsrequiringamulti-speciesapproach,ormigratorypatterns whicharerelatedtothelife-cycleofaspecies.Wealsodonotmodelthemicroleveloffisherypoliciesandproduction,mainly relatedtothenationalimplementationofcooperativeornon-cooperativefisherypoliciesandtheproductionfunctionof individualfishermen.Thus,weneglectissueslikesetuporfixedcosts,policyregulationslikegearrestrictionsorallocation oftradable ornon-tradablefishingquotastoindividualfishermen,effortstoreduceby-catch,andportstatemeasures

25_The_way_we_have_set_up_our_sink-source_model,_neither_˛₌₀_nor_˛₌₁_makes_sense._Thus,_˛_∈_[0.05,_0.95]_in_the_sink-source_model_to_ensure_the

(12)

todeterillegal,unregulatedandunreportedfishing.Essentially,nationalimplementationisassumedtobeefficientand perfectlyenforceableinoursetting.WealsoignoretransactioncostsofimplementingandadministratingRFMOsasfor instanceconsideredinMcCarthyetal(2001)andforcommonpoolresourcesingeneral,inOstrom(1990).Hence,asa tendency,ourmodeloverestimatesthepossibilitiesofcooperation.Concerninginternationalfisheriesmanagement,our crucialassumptionisthatallcountriesfishintheirownEEZandthehighseas.Thisabstractsfromthefactthatsomecoastal statesarenotengagedinhighseasfishingandthatdistantwaterfishingstatesmightoperateinhighseasareasnotadjacent totheircoastalwaters.Italsomeansthatcoastalstatesdonotselltheirfishingrightstootherstates(accessagreements). InlinewithArt.87,UNCLOS1982,weassumethatnon-RFMOmemberscannotbedeterredfromfishinginthehighseas, coveringanalternativescenariowhereexclusionispossibleinabriefdiscussioninSection8.1.Finally,regardingthe sink-sourcemodel,thereremainsomeunsolvedissueswithrespecttotheaggregationofzonesasexplainedinAppendix3in theSupplementarymaterial.

5. Results:secondstageofcoalitionformation

Inthissection,weanalyzehowequilibriumfishingefforts,stocksandpayoffsdependonthedegreeofcooperationand thecrucialparametersofourmodel.Thiswillprovideusefulinformationfortheinterpretationoftheincentivestructureto formstablecoalitionsasanalyzedinthefirststageofcoalitionformationinSection6.Wefocusonthefullyintegratedmodel withdensity-dependentdiffusion,andtreatthesink-sourcemodelasanextensioninSection8.2.Asmentionedabove,the systemoffirstorderconditionsisasystemofnon-linearequations,whichcannotbesolvedanalytically.Nevertheless,itis instructivetoconsiderselectivefirstorderconditionsingeneralform.Supposecountryibehavesnon-cooperatively.Then thefirstorderconditionintermsofeffortlevelsinthehighseasofcountryiisgivenby:

∂

˘i

∂

EHS_i =

∂

p

HHS_i(X,E)+HEEZ_i(X,E)

−C(EEEZ_i+EHS_i)

∂

EHS_i =0. (8)

UsingHHS_i=qEHS_iXHS/kHS,HEEZ_i=qEEEZ_iXEEZ_i/kEEZ_i,andC(EEEZ_i+EHS_i)=c(EEEZ_i+EHS_i),wehave:

(i) (ii) (iii) (i

v

)

pq

XHS kHS + EHS_i kHS

∂

XHS

∂

EHS_i + EEEZ_i kEEZ_i

∂

XEEZ_i

∂

EHS_i

− c=0 (9)

Thus,amarginalincreaseofplayeri’seffortinthehighseas,implies: iamarginalincreaseinplayeri’sharvestlevelfromthehighseas(ﬁrstterm); iiamarginaldecreaseinthestocklevelinthehighseas(secondterm);

iiiamarginaldecreaseintheadjacentstocklevelinplayeri’sEEZviadiffusion(thirdterm)and ivamarginalincreaseinthecost(fourthterm).

Inthepresenceofdiffusion,playerichoosesaﬁshingeffortinthehighseas,whichbalancesthepositiveimpacts(i) andthenegativeimpacts(ii),(iii)and(iv).Forthespeciﬁcfunctions,impact(i)and(ii)willdependonthesizeofthehigh seas,kHS,andimpact(iii)onthesizeoftheexclusiveeconomiczoneofplayeri,kEEZ_iandhenceallthreeimpactsdependon

parameter˛.Forimpact(ii),thenegativevalueonthestockinthehighseas∂XHS

_⁄

∂EHS_iwillalsodependonthegrowthrate r.Forimpact(iii)thenegativevalueonthestockinplayeri’sEEZ∂XEEZ_i

_⁄

_∂_EHS

iwilldependontheintensityofdiffusionand henceonparameterd.Theimpactisalwaysnegativebecauseﬁshinginthehighseaseitherreducesincomingdiffusioninto playeri‘sEEZ,orincreasestheoutgoingdiffusionfromthisEEZtothehighseas,dependingontheequilibriumstocklevels inthedifferentzones.NotethattheFOCcapturesonlytheimpactofeffortsonownpayoffs;externalitiesimposedonother playersarenotinternalizedaslongasthereisnocooperation.

Tomakethingsevenmorecomplicated,wenotethatplayerialsohasaﬁrstorderconditionforhiseffortinhisEEZ,taking careofdiffusionbetweenthehighseasandhisEEZ.Moreover,theﬁrstorderconditionsofacoalitionwouldshowadditional marginaleffectsacrossmembers’EEZsandthoseEEZsandthehighseasascoalitionmembersbehavecooperativelyby maximizingaggregatepayoffs.

Fornotationalconvenience,weskipinthefollowingtheterm“equilibrium”.Unlessotherwisestated,wealwaysreferto efforts,stocksandpayoffsintherespectiveequilibrium:no,partialandfullcooperation,i.e.,allsingletoncoalitionstructure, two-playercoalitionandgrandcoalitionwiththreeplayers.Wemayrecallthatthedegreeofsociallyconstructed(technical) excludability,measuredbytheallocationparameter˛ (diffusionparameterd), isinverselyrelatedtothevalueof this parameter.Thesameholdsforthedegreeofrivalrymeasuredbytheintrinsicgrowthparameterr.

Result1:theroleofsociallyconstructedandtechnicalexcludability

Underfullcooperation,thetotalﬁshingeffort,totalstockandtotalpayoffareindependentofthedegreeofsociallyconstructed excludability(allocationparameter˛)andthedegreeoftechnicalexcludability(diffusionparameterd)wheretotalsreferto

(13)

aggregationoverallplayersandzones.Undernoandpartialcooperation,thetotalﬁshingeffortsoverallplayersincreaseinthe parameter˛andthediffusionparameterd.Accordingly,thetotalstockintheentireﬁshingareaandthetotalpayoffoverall playersdecreasesin˛andd.

Inthesocialoptimum,neitherthedistinctionbetweenhighseasandEEZsmattersforequilibriumstrategiesnorthe levelofdiffusion.Thisisbecauseinthesocialoptimumexternalitiesacrossallplayersarefullyinternalized,i.e.,thesocial plannermaximizestheaggregatepayoffoverallplayersandzones.Effortsaredistributedsuchthateffortdensities,i.e., theeffortsperareaEEEZ,i/kEEZandEHS,tot/kHS areequaleverywhere,irrespectiveofdand˛.Accordingly,stockdensities

XEEZ,i/kEEZandXHS/kHSarethesameineveryzoneandindependentofdand˛.26Thissubstantiatesourclaimabovethat

forthedensity-dependentmodel˛doesnotaffectthebiologyofthesystemassuch.Thisisalsotruefornoandpartial cooperation,thoughachangeof˛changespropertyrightsandhenceequilibriumﬁshingefforts.

Undernoandpartialcooperation,ahighvalueof˛,i.e.,alowdegreeofsociallyconstructedexcludability,aggravates over-exploitationandleadstohigherefforts,whichisreflectedinlowerstocksandpayoffs.Similarly,thehigherthediffusion betweenzones,i.e.,thelowerthedegreeoftechnicalexcludability,themorewillthefishstockbeexploited(highfishing efforts),resultinginlowstocks.Thistranslatesintolowindividualpayoffsandalowtotalpayoff.Forthetotaleffortand totalpayoffundernocooperationthisisillustratedinFig.2,withsimilargraphsforpartialcooperation.27

Thenextresultmeasurestheimportanceofcooperationasafunctionofourmodelparameters.Weconsiderrelative normalizeddifferences(asabsolutevalueshavenosensiblemeaninginastylizedmodel)relatedtothebenchmarkfull cooperation.

Result2:totalstocksandpayoffsunderdifferentdegreesofcooperation

Letthetotalﬁshstockintheentireareaandthetotalpayoffoverallplayersunderfull,noandpartialcooperationbedenoted byXF_,_XN_,_and_XP_,_and_˘F_,_˘N_and_˘P_,_{respectively,}_then

a) XP−XN XF andX F_−XN XF increasein˛andd; b) ˘P_−˘N ˘F and˘ F_−˘N ˘F increasein˛andd.

Result2stressesthattherelativeimportanceofcooperation,eitherpartialorfull,increaseswiththedegreeof intercon-nectednessbetweenplayers,inlinewithSampsonandSanchirico(2019).Thatis,theimportanceincreasesthelowerthe degreeofsociallyconstructedandtechnicalexcludability,i.e.,thehigherthespatialallocationparameter˛andthehigher thediffusionparameterdare.Inotherwords,if˛and/ordarehigh,wewouldhopethatfullcooperationoratleastpartial cooperationisstablewhichistestedinSection6.Incontrastforlowvalues,cooperationdoesnotmattermuch.

Thenextresultlooksattheeffectofavariationofthecostparameterc,reflectingtheunitproductioncostoffishing,and thegrowthparameterr,ourindicatorofthedegreeofrivalry,reflectinghowfastthestockrecoversfromfishing.

Result3:theroleofthecostandgrowthparameterunderdifferentdegreesofcooperation

a)Totalequilibriumeffortsandpayoffsdecreasewhilestocksincreaseinthecostparameterc. Thisholdsirrespectiveofthe allocationparameter˛,thediffusionparameterd,andthedegreeofcooperation.XP−XN

XF andX F_−XN XF aswellas ˘ P_−˘N ˘F and ˘F_−˘N

˘F decreaseincwheneverthereisdiffusion.

b)Totalequilibriumeffortsandpayoffsincreaseinthegrowthparameterr.Thisholdsirrespectiveoftheallocationparameter˛, thediffusionparameterd,andthedegreeofcooperation.Underfullcooperation,equilibriumstocksareindependentofr.Under noandpartialcooperationthetotalstockincreasesinrwheneverthereisdiffusion.XP−XN

XF andX F_−XN XF aswellas˘ P_−˘N ˘F and ˘F_−˘N

˘F decreaseinrwheneverthereisdiffusion.

TheintuitionofResult3aisstraightforward.Withincreasingunitproductioncosts,equilibriumfishingeffortsarereduced, resultinginlowerpayoffs,thoughhigherfishstocks.Thusfromanecologicalpointofview,higherproductioncostshelp topreservefishstocksbutfromaneconomicpointofviewitreduceseconomicrents.Shrinkingrentsunderallscenarios ofcooperationwithincreasingcostsalsoimpliesthattherelativedifferencesintotalpayoffsbetweenthetwocooperative

26_Obviously,_this_result_rests_on_the_assumption_of_symmetric_parameters_and_in_particular_symmetric_dispersal_patterns._For_asymmetry,_an_optimal

ﬁshingpolicy,i.e.,theallocationofefforts,aswellasresultingstockdensitiesdependonthecharacteristicsofﬁshinggroundsanddispersalpatterns(cf. CostelloandPolasky,2008).

27_With_respect_to_Fig.₂_note_that_similar_graphs_are_obtained_for_other_combinations_of_the_parameter_values_of_c_and_r._Those_combinations_follow_from

Table3.Hence,thereasixc-r-parameterconstellationsconsideredinthesimulationsundernocooperationandthesameistrueforpartialcooperation, whichgiverisetoResult1.Consequently,giventhenumberofsubsequentresults,itisevidentthatwecannotdisplayallsimulationresultsingraphs. However,allgraphsareavailableuponrequestfromtheauthors.

(14)

Fig.2.TheEffectofSociallyConstructedandTechnicalExcludabilityonTotalEffortandPayoffunderNoCooperation*.

*Effortsandpayoffsareexpressedinrelationtothesocialoptimumwhichisbeingsetto1.Effortsandpayoffsinthesocialoptimumareindependentof dand˛.Theallocationparameterisvariedfrom˛=0to˛=1(seearrow).Forthecostandgrowthparameterbasecasevaluesareassumed(c=0.5and r=0.5).

scenariosandthenon-cooperativescenariobecomesmaller.Thus,theneedforcooperationdecreasesinthecostparameter c.28

AlsoResult3bisinlinewithintuition.Ahighgrowthrateencouragesfishingandisassociatedwithaneconomic advan-tage.However,higherfishingeffortsdonotnecessarilyimplylowerstocksastheresourcerecoversmorequicklywithahigh growthrater.Onlyifdiffusionisirrelevant,e.g.thereisfullcooperationortheentirefishingareaispublic(˛=1),ahigher

28 _It_may_be_worthwhile_to_recall_that_not_the_absolute_value_of_c_matters_but_the_ratioc

_⁄

_pq_._Thus,_a_higher_c_has_the_same_effect_as_a_lower_price_p_or_a_lower

catchabilitycoefficientq,measuringthetechnologicalefficiencyofharvestingfish.Hence,ahighpriceandtechnologicalefficiencyaredetrimentaltothe ecologicalsystembutareconducivetoeconomicrentsandmakecooperationparticularlyvaluablefromanormativepointofview.