The DART-Europe E-theses Portal

HAL Id: tel-01280780

https://tel.archives-ouvertes.fr/tel-01280780

Submitted on 1 Mar 2016

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Bilateral bargaining and farsightedness in networks : essays in economic theory

Rémy Delille

To cite this version:

Rémy Delille. Bilateral bargaining and farsightedness in networks : essays in economic theory. Eco- nomics and Finance. Université de Bordeaux, 2015. English. �NNT : 2015BORD0356�. �tel-01280780�

TH`ESEPR´ESENT´EE POUROBTENIRLEGRADEDE

DOCTEUR DE

L ’UNIVERSIT´ E DE BORDEAUX

ECOLEDOCTORALEENTREPRISE´ ,´ECONOMIE,SOCI´ET´E(ED42) SP´ECIALIT´ESCIENCES´ECONOMIQUES

ParR´ emy DELILLE

BILATERAL BARGAINING AND FARSIGHTEDNESSIN NET WORKS :

ESSAYSINECONOMIC THEORY .

Sousladirectionde:Nicolas Carayol,ProfesseurdesUniversit´es etJean-Cristophe Pereau,ProfesseurdesUniversit´es

Soutenuele14D´ecembre2015 Membresdujury:

M. Philippe SOLAL

Professeur des Universit´es, Universite´ de ST Etienne, rapporteur M. Tarik TAZDAIT

Directeur de recherche CNRS, Ecole des ponts, ParisTech, rapporteur Mme Ana MAULEON

Professeur des Universit´es, Universite´ de ST Louis, Bruxelles, examinatrice,présidente dujury Mme Noem`ı NAVARRO

Maˆıtre de conferences, Universite´ de Bordeaux, examinatrice

Contents

Introduction 13

1 Bilateralriverbargaining withexternalities 17

1.1 Introduction... 17

1.2 TheSeawallbargaininggame ... 17

1.2.1 LiteratureReview... 19

1.2.2 Thebenchmark... 20

1.2.3 NegotiationProtocols... 22

1.2.4 Example... 26

1.2.5 ConclusionsontheSeawallBargaininggame... 29

1.3 Theriverbargainingproblem ... 29

1.3.1 LiteratureReview... 30

1.3.2 Thebenchmark... 33

1.3.3 Negotiationprotocols... 36

1.3.4 Example... 43

1.3.5 ConclusionsontheRiverbargainingproblem... 46

1.4 AppendixA... 46

1.4.1 ProofofProposition1... 46

1.4.2 ProofofProposition2... 48

1.4.3 ProofofProposition3... 50

1.4.4 SummarizedResultsfortheSeawallbargaininggame... 50

1.4.5 ProofofProposition5... 53

1.4.6 ProofofProposition6... 55

1.4.7 ProofofProposition7... 57

2 Level-KFarsightednessinaverticallyrelatedeconomy 59 2.1 Introduction... 59

2.1.1 RelatedLiterature ... 61

2.1.2 Roadmap ... 64

2.2 Modelingnetworksof ManufacturersandRetailers... 64

2.3 FarsightedStability... 68

2.3.1 Thepairwiserationale... 68

2.3.2 Farsightedimprovingpaths ... 69

2.3.3 Pairwisefarsightedstability... 70

2.3.4 Level-K farsightedstability ... 72

2.4 Thealgorithm... 75

2.5 Theresults ... 80

2.5.1 The myopiccaseG^∞₁ ... 80

2.5.2 Greaterlevelsoffarsightedness ... 84

2.6 Welfareanalysis... 98

2.6.1 Paretooptimality...100

2.6.2 Strongefficiency ...103

2.7 Extensions...107

2.7.1 Moreonthereliabilityofthealgorithmicresults...107

2.7.2 Relationtothepairwisefarsightedstability ...108

2.7.3 Efficiencyoftheindustrialsegment...110

2.8 Conclusion...114

2.9 AppendixB...117

2.9.1 Payoffs...117

2.9.2 Pseudocode...118

3 Allocatingvalueamongfarsightedplayersinnetworkformation 123

3.1 Introduction...123

3.1.1 LiteratureReview...125

3.1.2 Roadmap ...127

3.2 Allocatingvalueamongfarsightedplayers ...128

3.3 VonNeumann-Morgensternfarsightedstabilitywithbargaining ...132

3.4 Pairwisefarsightedstabilitywithbargaining...138

3.5 Extensions...143

3.5.1 Moreontheroleoftopconvexity...143

3.5.2 Moreontheroleofequalbargainingpower...145

3.5.3 Moreontheroleofcomponentadditivity...146

3.6 Conclusions...147

3.7 AppendixC...147

3.7.1 ProofofProposition23...147

3.7.2 ProofofProposition25...149

Conclusion 151

L istofF igures

2.1 TypologyofOrganizationalStructure. ... 66

2.2 StabilityforK =1... 82

2.3 StabilityforK =2... 85

2.4 StabilityforK =3... 88

2.5 StabilityforK =4... 91

2.6 StabilityforK =5... 93

2.7 StabilityforK ≥6... 95

2.8 TypesofPareto-optimalregions...101

2.9 Thestronglyefficientnetworks...104

2.10Strongcorrespondencebetweenstablesetsandstronglystablenetworks. .105 2.11 PairwiseFrasightedstablesetsandthetransitivecorrespondence...109

2.12 Paretooptimalityoftheindustrialsegment...111

2.13StrongParetooptimalityoftheindustrialsegment...112

2.14Stronglyefficientnetworksoftheindustrialsegment...113

2.15StongcorrespondencebetweenstablesetsandStronglyefficientnetworks intheindustrialsegment. ...115

3.1 The Myersonvalue(Theplayer-basedflexiblenetworkallocationrule). ..132

3.2 Theplayer-basedflexiblenetworkallocationrule. ...137

3.3 Topconvexityandfarsightedstabilitywithbargaining...139

3.4 Valuefunctionnottopconvexandfarsightedstabilitywithbargaining...144

L istof Tab les

1.1 Theseawallbargaininggame... 28

1.2 SolutionsunderATS... 44

1.3 SolutionsunderUTI... 45

1.4 Efforts(ine)andpayoffs(inbe²)inthecooperativeandnon-cooperative cases.γ=c/bande=a/b. ... 51

1.5 Efforts(ine)andpayoffs(inbe²)inthetwo-by-twonegotiationprocess... 52

1.6 Efforts(ine)andpayoffs(inbe²)indoublenegotiation... 52

2.1 Equilibriumvaluesforg^N... 67

2.2 Payoffsfor(d,k)=(0.7,0.005)... 81

2.3 Adjacenciesofthesupernetwork. ... 83

2.4 Level-2adjacencies... 86

2.5 Level-3adjacencies... 87

2.6 Level-4adjacencies... 90

2.7 Level-5adjacencies... 92

2.8 Level-6adjacenciesandover... 94

2.9 Payoffsfornetworksg^Nandx1...117

2.10 Payoffsfornetworksed1,m1,r1,s1andg^∅ ...117

2.11 Thecompare(g1,g2,g3,d,k)function...118

2.12 Thematfip(d,k)function...119

2.13 Thedevdet(G,g,g)function...120

2.14 Theinstab(G,MK)function...121

2.15 Theexstab(G,MK)function...122

3.1 Afarsightedallocationruleforvaluefunctionv ...135 3.2 Anotherfarsightedallocationruleforvaluefunctionv...136 3.3 Allocationssatisfyingw-weightedbargainingpowerforγ=2 ...145

Abstract[EN ]

Thethesisconsistsinfouressaysthatdealwithbargainingandnetworksinnoncoop- erativegametheory. Thefirstchapterintroduceriverbargaininggamesinthecontext ofexternalities. ThefirstsectionentitledTheseawallbargaininggamedealswithanon cooperativeapproachofaninvestmentgameinacontextofpositiveexternalities.In thissection,westudythesetofbilateralbargainingprocedureswhichcanbeundertaken amonggeographicallyrelatedplayers. The mainresultshowsthatthepositioningofthe agentsimpactstheirincentivestositatthebargainingtable,leadingtoachickengame. Resultsshowthatanintermediaryplayershouldleadthenegotiationstoimprovetheso- cietalwelfare. ThesecondsectiondealswiththeRiverbargainingproblem,thatis, more precisely,anoncooperativebargainingonaflowingresourceinthepresenceofnegative externalities. Thepurposeofthisstudyis:(i)tofindthebargainingequilibriagiven anexogenousbilateralbargainingprotocol,and(ii)tostudythesocietaldesirabilityof suchequilibriainthepresenceofheterogeneousplayers. Resultsshowthatdependingon theinstigatorofthebargainingsequences,thereareanalogiesbetweensolutionsunder the Absolute TerritorialSovereigntyandthe Unlimited TerritorialIntegrityprinciples. Theresultsalsoshowthat,dependingontheprotocol,theimpassepointreachedinthe firstnegotiationcaneitherstrengthenorweakentherelativepositionoftheagentsin theforthcomingnegotiations. Thesecondchapterdealswiththeformationofnetworks of manufacturersandretailersinthepresenceofnegativeexternalitieswhenplayersare level-K farsighted. Theaimofthechapterare,(i)tocharacterizethelevel-K farsighted stablesetsastheintensityofcompetition(theintensityoftheexternality),soasthecost oflinkingvary;(ii)toformalizeadefinitionofthesocialoptimaandtoconfrontthese

optimatothestablesets.Theresultsshowthat,(i)arelativelylowleveloffarsightedness issufficienttoreachtheinfiniteleveloffarsightedness;(ii)usualdefinitionsofoptimality orefficiencyfindlimitationswhenitcomestobeconfrontedtoaset-baseddefinitionof stability.(iii)Ifthereistransitivecorrespondencebetweenthepairwisefarsightedstable setandthelevel-∞ farsightedstableset,thenthissetislikelytobestronglyefficient.The lastchapterisentitledAllocatingvalueamongfarsightedplayersinnetworkformation.

ThistheoreticchapterproposestheconceptofavonNeumann-Morgensternfarsighted stablesetwithbargaining. Underthissolutionconcept,thestablenetworkssoasthe componentwiseegalitarianallocationruleemergeendogenously. Thischapterprovides necessaryconditionsunderwhichavonNeumann-Morgensternfarsightedstablesetwith bargainingsustainsthestronglyefficientnetworks.

Abstract[FR ]

Cetteth`eseconsisteenquatreessaisquitraitentden´egociationetder´eseauxenth´eorie desjeuxnoncoop´erative. Lepremierchapitrepr´esentedesjeuxden´egociationsdans uncontexted’externalit´es. Lapremi`eresectionintitul´eelejeuden´egociationsurla diguetraited’uneapprochenoncoop´eratived’unjeud’investissementdansuncontexte d’externalit´espositives. Danscettesection,nous´etudionsl’ensembledeproc´eduresbi- lat´eralesden´egociationquipeuventˆetremisesenplaceentredesagentsg´eographiquement li´es.Ler´esultatprincipaldelasectionmontrequelasituationdesagentsimpacteleursin- citations`aprendrepartauxn´egociationsetsesynth´etiseenun”jeudelapoulemouill´ee”. Dupointdevuesoci´etal,lesresultats montrentqu’ilestsocialementplusefficacequ’un joueurinterm´ediaire m`enelesn´egociations. Lasecondesectionduchapitretraitedu probl`emeden´egociationsurlarivi`ere.Ils’agitd’unjeuden´egociationnoncoop´eratif surl’utilisationdelaressourcefluvialeenpr´esenced’externalit´esn´egatives. Lebutde cette´etudeestde(i)d´eterminerles´equilibresden´egociation´etantdonn´eunproto- coleden´egociationexog`ene;(ii)d’´etudierl’int´erˆetsoci´etaldetels´equilibresenpr´esence d’acteursh´et´erog`enes.Lesr´esultatsmontrentqu’enfonctiondel’instigateurdess´equences den´egociations,ilexistedesanalogiesentrelessolutionsobtenuesdanslescasdeSou- verainet´eTerritorialeAbsolueetd’Int´egrit´eTerritorialeIllimit´ee.Lesr´esultatsmontrent aussique,enfonctionduprotocoleretenu,lepointd’impasseobtenulorsdelapremi`ere n´egociationpeutrenforcerouaffaiblirlapositionrelativedesagentsdanslesn´egociations

`

avenir. Ledeuxi`emechapitretraitedelaformationder´eseauxdeproducteursetde d´etaillantsenpr´esenced’externalit´esn´egativeslorsquelesjoueurssontclairvoyantsde degr´e-K. Lebutdecechapitreest(i)ded´eterminerlesensemblesder´eseauxstable

dedegr´e-K lorsquel’intensit´edelaconcurrence(l’intensit´edel’externalit´e)etlescoˆuts deliaisonvarient;(ii)dedonneruned´efinitionformelledesoptimasociauxafindeles comparerauxensemblesder´eseauxstables. Lesr´esultats montrentque(i)undegr´ede clairvoyancerelativementfaibleestsuffisantpouratteindrelaclairvoyanceabsolueou infinie;(ii)lesd´efinitionshabituellesdel’optimumoudel’efficienceneconviennentpas parfaitement`aunconceptdestabilit´eensembliste.(iii)S’ilexisteunecorrespondance transitiveentrelastabilit´eclairvoyanteparpairesetlastabilit´eclairvoyantededegr´ein- fini,alorsl’ensemblestablepeutˆetreefficient.Ledernierchapitreestintitul´eattribution delavaleurentrejoueursclairvoyantsdansleprocessusdeformationder´eseau.Ils’agit d’unchapitreth´eoriquequiproposeleconceptdestabilit´evonNeumann-Morgensternavec n´egociation. Dansceconceptdesolution,lesensemblesder´eseauxstables,ainsiqu’une r´epartition´egalitaireauseindescomposantsdur´eseausontd´etermin´esconjointement,et demani`ereendog`ene.Cedernierchapitremeten´evidencelesconditionsn´ecessairespour quelesr´eseauxvonNeumann-Morgensternavecn´egociationsoientefficients.

Acknow ledgments

IwanttothankmytwothesissupervisorsProfNicolasCarayolandProfJean-Christophe Pereaufortheircollaboration,helpfulcomments,supportandpatience. Igratefully acknowledgethe membersofthejury,PhilippeSolal,TarikTazdait,Ana Mauleonand Noem`ıNavarro.IalsoadressmygratitudetoVincentVannetelbosch,JosephHannaand St´ephaneLambrecht. Mybelovedfamilyandfriendsforsupportalongthislongjourney.

Introduct ion

Thethesisconsistsinfouressaysin microeconomics. Whiletheinternalcomposition ofthedocumentdealswithvarioustopics,thesefouressayshaveacommonpoint. The thesisstudiesbilateralrelationshipsinregardstoanetworkedstructure.Thefirstchapter addressesthequestionofthenegotiationinanexogenousnetworkstructureframework. Thesecondchapterdealswiththequestionofanendogenousnetworkformationwith farsightedplayers. Thelastchapterproposesanendogenousnetworkformationprocess thatintegratesabargainingprocedureandfarsightedplayers. Thestudyremainswithin thescopeofthenoncooperativegametheory.

Thefirstchapterdealswiththe managementofanaturalresource.Economicagents aregeographicallyscatteredalongariver.Inthisalignednetworkstructure,thegeo- graphiclocationdeterminestheaccesstothepublicgood.Indeed,regardingthenature ofthevaluationoftheestate,thestrategicaladvantagesdependsonthegeographiclo- cationoftheplayers. Traditionally,thestudyofpublicgoodshasbeenundertakenusing anidenticalaccesstotheresourceandhasidentifiedthesourcesofsocietalinefficiencies suchasfree-ridingorthetragedyofthecommons.Intheriversetting,theheterogeneous accesstotheresourceleadtoreconsidertheusualissuesandtherefore,itssolutions.The questionofthepropertyrights,ortherighttogetapriorityaccessquicklyarises.Indeed, giventheorientationofthestream,anex-antepropertyrightsdivisionorthefactofhav- ingapriorityaccessovertheresourceinducesdifferentformsofinefficienciesandthus, differentsolutionstodecentralizeasecond-bestequilibrium.Resultsfromthecooperative gametheoryusuallyleadtotheefficientoutcomebutoftenfailsatexplainingtheinner processleadingtothatsolution.Itappearsthatabargaining mechanismleadstothe

efficientoutcomebutthesequencesofnegotiationsneedtobefullydescribed. Thefirst chapteraimsatopeningthe”black-box”oftheriverbargainingusingthenoncoopera- tiveapproachoftheRubinsteinalternatingoffers modelinthecontextofexternalities. ThefirstsectionentitledTheseawallbargaininggameisinspiredfromapaperconjointly writtenwithmyPhD.supervisor:JeanChristophePereau.Inthispaper,theagentsare locatedfromdownstreamtoupstreamalonganestuaryandareexposedtoaflooding risk. Theplayershavetheabilitytoinvestinfacilitiestoprotectthemselvesfromthis hazard(forinstanceaseawall,adikeoraditch). Theagentslocatedupstreamtheriver benefitfromtheeffortsinprotectionoftheplayersdownstream. Asthebenefitsofthe localpublicgoodincreasealongtheestuary,upstreamagentshavetobargainfor mone- tarycompensationwiththemostdownstreamagentsinexchangeforagreaterprotection effort. Thesectionanalysesdifferentbargainingprotocolsanddeterminestheconditions underwhichagentsarebetteroff.Theresultsshowthatnobargainingprocedureisobvi- ouslypreferredbyalltheplayers.Theresultsshowthattheupstreamagentsareinvolved inachickengamewhenitcomestobargainwiththe mostdownstreamagent.

ThesecondsectionentitledTheriverbargainingproblem isbasedonanarticlecon- jointlywrittenwithJeanChristophePereauanddealswithariverbargaininggameinthe contextofnegativeexternalities.Inthis model,theplayersbenefitfromextractingthe resource.Thenatureoftheexternalityliesinthefactthattheresourceisscarceandthat thesumoftheindividualoptimalextractionsexceedstheamountofavailablewater.The sectionanalysestheoutcomesofdifferentnegotiationproceduresbetweenthreeagents locatedalongariverinaRubinsteinalternating-offers modelwhereagentsbargainover transfersandwaterconsumptionlevels.Twodifferentkindofsequentialproceduresgiven theex-antedivisionofpropertyrightsareinvestigated.Thedownstreambargainingpro- cedureisinstigatedbythe mostdownstreamplayerandtheupstreamprocedurebythe mostupstream. Resultsshowthatundersomeassumptionsonthenoncooperativelevel ofextractionsandgiventheAbsoluteTerritorialSovereigntyortheUnlimitedTerritorial Integrityapplies,theproceduresyieldanalogoussolutions. Unlikeforsimultaneousbar- gaining,agreementsinsequentialbargainingproceduresarenotefficientforsociety,even

iftheperiodbetweenstagesbecomesinfinitelysmall. Thisinefficiencyresultsfromthe player’sinsideoptions,whicharegivenbytheirtemporarydisagreementpayoffs. Results alsoshowthatdependingonthesequenceof moves,insideoptionsandimpassepoints canstrengthenorweakentherelativepositionoftheplayersinvolvedinthenegotiation process.

Whilethefirstchapterassumesthatthestructureofthenetworkisexogenous,driven bygeographicalconstraints,thesecondandlastchaptersreleasetheconstraintofan exogenoussocialstructureastheydealwithendogenousnetworkformation.

Thesecondchaptertakesplacethecontextofaverticallyrelatedeconomy.Thestudy ofverticalformsofeconomieshasdeephistoricalrootsinIndustrialOrganization.Indus- trialOrganizationhasraisedvariousformsofquestionsregardinganexogenousstructure ofverticalrelationships(incentivesfortheverticalintegration,theeffectsoffranchising andexclusivezones...). Newtheoretictoolssuchas”thestrategicnetworksanalysis”

formalizedtheendogenousformationofnetworksandthusshednewlightontheques- tioninginIndustrialOrganization.Iinvestigatethequestionofthestabilityofcommercial agreementsbetween manufacturersandretailersinregardtotheirabilitytoforecastthe likelyevolutionofthenetwork.Inthiscontext,Iquestionthetypologiesofnetworks thatmightbeformedwhentheproductionofasetofdifferentiatedmanufacturershasto bedistributedthroughasetofretailers. Thus,thescopeofthepaperencompassesthe bilateralformationofcommerciallinks,ie.theconsentofa manufacturerandaretailer isneededtobuildaconnection,whiledistributionchannelscanbeunilaterallysevered.

Commercialcontractsbetweenfirmscanbecostlytoendorseandeconomicagents havetoanticipatetheaftermathsinducedbytheirnewpartnershipsandalsotheconse- quencesofnewand/orrevokedpartnershipsofthirdparties. Thedecisiontodiscussa newcommercialcontractortoabolishanexistingrelationshipisdrivenbytheabilityto anticipate. Thisanalysisisthereforebasedonthedepthofreasoningoftheplayers. On thelowestlevelofreasoningplayersbilaterallyaddorunilaterallyseverexistinglinksat random,whileatthehighestlevelofreasoning,theplayersanticipateallthedeviations thattheirinitialactioninduce.

Computationaltoolsareusedtoestablishthesetofnetworksthatsatisfiesapairwise networkformationtogether withanadjustableleveloffarsightednessforanycostof linkingandanyintensityofcompetitionbetween manufacturers.

Beyondbeingastrategicasset,anticipationiscostlyintermsofgatheringinformation orintermsoftimeneededtotreattheinformation. Theresultsshowthatanintermedi- arylevelofanticipationoftheplayerissufficienttoachievethesameoutcomeastheone obtainedwhentheyarefullyfarsighted.Ithenconfrontthestablesetstoseveralopti- malityandefficiencycriteria.Itappearsthat,forlowvaluesofproductsdifferentiation andhighvaluesoflinkingcoststherecanbearelationbetweenstabilityandefficiency.

Thequestionoftheendogenousnetworkformationaccordingtoabargainingpro- cedure withfarsightedplayersisthepurposeofthethirdchapter. Thislastchapter isbasedonanarticleconjointlywrittenwith myPhDsupervisor NicolasCarayoland VincentVannetelbosch.Inthischapter,weproposeaconcepttostudythestabilityof socialandeconomicnetworkswhenplayersarefarsightedandallocationsaredetermined endogenously. AsetofnetworksisavonNeumann-Morgensternfarsightedlystableset withbargainingifthereexistsanallocationruleandabargainingthreatsuchthat(i) thereisnofarsightedimprovingpathfromonenetworkinsidethesettoanothernetwork insidetheset,(ii)fromanynetworkoutsidethesetthereisafarsightedimprovingpath tosomenetworkinsidetheset,(iii)thevalueofeachnetworkisallocatedamongplayers sothatplayerssufferorbenefitequallyfrombeinglinkedtoeachothercomparedtothe allocationtheywouldobtainattheirrespectivecrediblebargainingthreat. Weshowthat thesetofstronglyefficientnetworksistheuniquevonNeumann-Morgensternfarsightedly stablesetwithbargainingiftheallocationruleisanonymousandcomponentefficientand thevaluefunctionistopconvex. Moreover,thecomponentwiseegalitarianallocationrule emergesendogenously.

Chapter1

B i latera lr iverbarga in ing w ith externa l it ies

1 .1 Introduct ion

Thefirstchapterdealswithbargainingproceduresintheframeworkofexogenousnetwork andisdividedintwo mainsections. ThefirstsectiondealswiththeSeawallbargaining game.Inthisgame,playersareorderedalongariverfromdownstreamtoupstream.The playerslocatedupstreambenefitfromtheeffortsoftheplayersdownstream.Thecontext ofpositiveexternalitiesmakecoordinationhardertoachieveanddifferentbargainingpro- tocolarestudied. Abriefintroductionofthe motivationforthis modelareexposedin Section1.2.Thesection1.3handlestheriverbargaininggameinthepresenceofnegative externalities.Inthissection,playersareorderedfromupstreamtodownstreamalong ariver. Giventheprinciplethatprevails(Absolute TerritorialSovereigntyor Unlim- itedTerritorialIntegrity),twotypesofsequentialbargainingprotocolsarestudied(the downstreamandtheupstreamprocedure).

1 .2 TheSeawa l lbarga in inggame

Hirshleifer[26]shows withhis“AnarchiaIsland”fablethatcitizenshavesuccessfully agreedtobuildseawalls(ordikes)toprotectthemselvesfromstormsthreateningtoflood

thecoastlinedespitetheweakest-linkstructureofthatlocalpublicgood. Theseawall isknownasaparticularpublicgoodin whichthelevelofeffectiveprotectionforthe wholeislanddependsonthecitizenwhohasconstructedthelowestseawall.Thischapter analysesasimilarproblemofflood-protectionwhenagentsarelocatedsubsequentlyfrom downstreamtoupstreamalonganestuaryandexposedtoafloodinghazard.Inthatcase, floodprotectiondoesnotonlyconsistinbuildingthehighestseawall,italsorequiresthe constructionofotherfacilities,suchasafirstseawalltobreakthewave,theuseofwetland asfloodwaterretentionland,anetworkofditchestocontrolthefloodorasecondseawall. Thisestuarinegeographyfeatureimpliesthatwhentheseaenters,theprotectioneffort implementedbyanagentwillbeapublicgoodfortheagentslocatedupstreamfromhis position. Atthetwoendsofthespectrum,themostdownstreamagent(theclosesttothe sea)doesn’tgetadditionalbenefitsfromtheupstreamfacilities,whilethemostupstream agentbenefitsfromalltheeffortsexertedbytheplayersdownstream. Thus,theseawall bargaininggameisnolongeraproblemofpublicgoodwithaweakest-linkaggregation technology,butisapublicgoodwithpositiveexternalitiesthatincreasealongtheestuary. Sincethebenefitofthepublicgooddependsonthegeographicalposition,itappearsthat cooperationishardtoachieve.Inordertosustainahighlevelofeffortfromthe most downstreamagent,upstreamagentshavetheabilitytobargain monetarytransfers. The modelingofthenegotiationsisthusofagreatpartinthedeterminationoftheresults.The mostobviousapproachistobasetheanalysisonwhatisknowasbargainingtheorywith theRubinsteinalternating-offers model. Rubinstein[49]describestheprocessthrough whichnegotiatingagentscanreachanagreement. Theagentopeningthenegotiations makesanoffer.Theotheragentcaneitheraccepttheoffer,inwhichcasethenegotiation ends,orrejectitandmakeacounter-offer,whichmayalsobeacceptedorrejectedwitha newcounter-offer. Theinterestofthenoncooperativeapproachisthatitfullyspecifies thebargainingsequences.Inourframework,anofferwillcovertwovariablesrelatedto theeffortofsea-floodprotectionanda monetarytransfer. Ourframeworkalsoassumes aRubinsteinbargainingwith3playersandanalysesseveralbargainingprotocols. The chapterdeterminesthelikelyprotocolsonecouldexpecttobeimplementedgiventhenon

cooperativebehaviorsoftheplayers.

1 .2 .1 L iterature Rev iew

Theseawallbargaininggamecanbe modeledasaparticularcaseofglobalpublicgood in whichallagentsbenefitfromtheactionofthedownstreamplayers, whatevertheir location.Intheliteratureoninternationalenvironmentalagreements,forinstance,results showthatforidenticalagents,onlyaverysmallnumberofplayerswillformacoalition. SeminalpapersusingthisapproachareCarraroandSiniscalco[11]andBarrett[6],anda surveycanbefoundinFinus[19]. Ourframeworkshares,incommonwiththe“sharing river” model,thedownstream/upstreamagentstructurein whichdownstreamagents createexternalitiestoupstreamagents. Thisliteratureisbasedoncooperativegame theoryand, moreprecisely,onthecore(seeAmbecandSprumont[1]andBealetal.[7] forasurvey).Theobjectiveistosetupaburden-sharingruleabletofavorthecooperation ofall. Thesharingruleaimsatpreventinganyindividualagent,butalsoanysub-group ofagents,fromhavingnoincentivetoleavetheagreement.

However,thecoalitionalapproachoftenignoresthenegotiatingprocessintermsof offersandcounter-offersthatcharacterizeallnegotiationsbetweenself-interestedagents. ThenoncooperativesolutionconceptisfirstintroducedbyStaahl(1972)[51]andRubin- stein(1982)[49]foratwoplayersbargaining¹. AspointedoutbyCarraroetal.(2007) [10],itiscrucialtothoroughlydescribethebargainingprocessthatdesignsthemostlikely burden-sharingrulesoastoimplementanoncooperativeprotocol.

Thenextsubsectionspresentthecooperativeandnon-cooperativeoutcomes. The nextfollowingsubsectionisdevotedtotheanalysisofsingleanddoublenegotiations. Insubsection1.2.4,aspecificexampleshowsthe mainresultsofthe modelandthelast subsectionpresentstheconcludingremarksontheSeawallbargaininggame.

1Foranadvancedandcomprehensiveoverviewofthealternatingoffersmodelssee Muthoo(1999)[40].

1 .2 .2 Thebenchmark

Consideringanorderedsetofplayers N = {1,2,...,n}. Everyagentislocatedfrom downstreamtoupstreaminalexicographicordering². Wenoteei,theeffortofprotection realizedbyagenti.Therespectivepayoffsaredefinedasthedifferencebetweentheconcave benefitfunction(withB_i> 0andB_i< 0)andtheconvexcostfunction(C_i> 0and C_i>0). Wedenotebyi<jthefactthatagentiisdownstreamofagentj. Following notationofAnsinkand Weikard(2009)[3]Ui={j∈N :j>i}standsforthesetofagents locatedupstreamofagentiandreciprocallyDi={k∈N :i>k}.Thefirstplayeronly benefitsfromhisowneffort,hispayoffisπ1(e1)=B1(e1)−C1(e1). Thesecondagents benefitsfromhisowneffortandfromtheeffortsinprotectionfromthepreviousplayer. Intheend,nbenefitsfromeveryone’sefforts. Thegenericpayofffunctionisthus:

πi({ej}_j≤i)=Bi {ej}_j≤i −Ci(ei)

Theequationsshowthatcostsareprivatewhiletheagentsbenefitfromtheeffortsofthe previousplayers.

Cooperativeoutcome

Thecooperativesolutionisgivenbytheprogram:

maxei

n i=1

πi(e)

Effortlevelsaresolutionof:

j≥i

∂Bj k≤j

ek =∂Ci(ei)∀i∈N,k≥1,j≤n.

2Theoppositeoftheclassicalorderingisforconvenienceworries.

Thatisinthe3-playerssetting³:

B₁(e1)+B₂(e1+e2)+B₃(e1+e2+e3)=C₁(e1) (1.1) B₂(e1+e2)+B₃(e1+e2+e3)=C₂(e2) (1.2) B₃(e1+e2+e3)=C₃(e3) (1.3)

Playeritakestheimpactofhislevelofeffortontheplayersupstreaminhismarginal benefit.Itreturnsauniquevectorofeffortse^c={e^c_i}_i∈N. Anadditionalunitofeffortby agentiexertsanadditionalbenefitforhimandfortheagentsupstreamofhisposition. At theequilibrium,the marginalcostofthatunitequalizesthesumofhis marginalbenefit andthe marginalbenefitoftheupstreamagents. Thevectorofeffortsreturnsaunique vectorofcooperativepayoffs{π_i^c}_i∈N.

Non-cooperativeoutcome:

Whenevertheplayersactnoncooperatively,theoptimalityconditionsare:

∂Bi k≤i

ek =∂Ci(ei)∀i∈N,k≥1.

Thatisinthe3playerssetting:

B₁(e1)=C₁(e1) (1.4) B₂(e1+e2)=C₂(e2) (1.5) B₃(e1+e2+e3)=C₃(e3) (1.6)

Theprevioussystemreturnsauniquevectorofnon-cooperativeeffortse^nc= {e^nc_i}_i∈N. Eachagentequalizeshis marginalbenefittohis marginalcost. Thevectorofefforts returnsauniquevectorofpayoffs{π^nc_i}_i∈N. Thedifferencebetweenthecooperativeand thenoncooperativeeffortincreasesaswegodownstream.Indeed,downstreamplayers takethepositiveimpactoftheiractiononagentslocatedupstreamintoaccount. For

3Consideringathree-agentframeworkisenoughtoputforwardthe mainresultsofthepaper.

anintermediaryplayer2,twoeffectsareatstake. Theadditionaleffortofagent1in thecooperativecasereducestheincentivesof2todolikewise,butduetothepositive impactthat3exertson3,theagent2tendstoincreasehiseffortinthecooperativecase. Thislattereffectdominatestheformer,sincee^nc₂ <e^c₂. However,thisistheoppositefor the mostupstreamagent. Hiseffortisreducedinthecooperativecase,sincehebenefits fromtheeffort madebythepreviousdownstreamagents,andanadditionaleffortdoes notexertadditionalbenefittoanyplayer,thisleadingtoe^c₃<e^nc₃. Theaggregateeffort ishigherinthecooperativecasethaninthenon-cooperativecase.Intermsofpayoffs, agents2and3arebetteroffinthecooperativecase,unlikethe mostupstreamagent, whosepayoffonlydependsonhiseffort.Itturnsthatagent1hastomakeagreatereffort inthenon-cooperativecase,andthissubstantiallyreduceshispayoff.Ityieldsπ_i^c>π_i^nc fori=2,3butπ^c₁<π₁^nc. Thisisthe mainfeatureoftheestuarinegeography. The dominantstrategyofagent1istoimplementhisnon-cooperativeeffortregardlessofthe actionoftheupstreamagents. Hence,decentralizingthethenon-cooperativecasetothe cooperativecasewillneverbefoundprofitableforthe mostdownstreamagent,evenif theaggregatepayoffinthecooperativecaseishigherthaninthenon-cooperativecase.

1 .2 .3 Negot iat ion Protoco ls

Ifsocietalgainsofcooperationarepositive,thenthenegotiationmayimprovethepayoffs ofsomeagents. Inparticular,the mostdownstreamagentcouldfindit worthitto increasehiseffortiftheotherplayerscanfindamechanismtourgehimindoingso.The negotiationcantakeplacebetweenagentsoveranextraamountofeffortthatanagent willimplementinexchangefora monetarytransferorcompensation. However,several protocols mustbeconsideredgiventhe multiplicityofpairsofagents. Thus,bargaining betweentwoagentsconsistsinalevelofeffortandatransfer.

Anegotiationbetweenagentsiandjhastwoarguments. A monetarytransfersor side-paymentsinexchangefora modificationofthevariablethataffectsbothagents.In theestuarine model,theagentiproposesthe monetarytransferτijtotheagentjin exchangeforhiscommitmenttoincreasehisefforttoaej>e^nc_j. Alternatively,theagent

jcouldproposetheplayeritoincreasehisprotectionleveltoej>e^nc_j inexchangeforthe subventionτji. Notethatτij=−τjiandbyconventionthefirstplayerinthesubscript paysthetransfertothesecond(Ifipays−τijthenhereceivesτji). Anofferoraproposal isacoupledenotedbyoij=(e,τij)iftheofferis madebyitojandisaboutavariation ine=eiinexchangeforatransferofvalueτjiifi<jandaboute=ejandthetransfer τijifj<i.Bargainingroundsassumesperfectinformation.

Thefirstandsimplestnegotiationonlyconcernstwoplayers,eitheragent1with2 or3orbetween2and3.Inthisprotocol,oneplayerneversitsatthebargainingtable. Inanycases,regardlessoftheplayerwhodoesn’tbargain,theplayer3 maximizeshis individualpayoffbysettinghisefforte3.ThesetofavailableoffersisthusO1={o12,o13} andO2= {o23}. ThepayoffsaredenotedbyV1(e1) =B1(e1)−C1(e1),V2(e1,e2) = B2(e1+e2)−C2(e2)andV3(e1,e2,e3)=B3(e1+e2+e3)−C3(e3).

Inathree-agentcase,thegeneralbargainingframeworkisgivenbythefollowingnet payofffunction:

π1(O1,t)=π^nc₁ +δ^t₁(V1(e1)−π^nc₁ −τ12−τ13) π2(O1,O2;t)=π^nc₂ +δ^t₂(V2(e1,e2)−π^nc₂ +τ12−τ23) π3(O1,O2,e3;t)=π^nc₃ +δ^t₃(V3(e1,e2,e3)−π₃^nc+τ13+τ23)

where0<δi<1standsforthediscountfactorofagenti.

Proposition1 Whenthenegotiationisoverasinglepair (ei,τij)ofeffortandtransfer betweenagentsi(theproposer)andj(theresponder)fori=1,2,j=2,3,j= iand k=j,theoptimalvectorofeffortsesatisfies:

B_i(e)+B_j(e)=C_i(ei) B_j(e)=C_j(ej) B_k(e)=C_k(ek)

andtheassociatedpayoffsare:

π_i^∗(e)=π_i^nc+ (1−δj) (1−δiδj)Π π_j^∗(e)=π_j^nc+δj(1−δi)

(1−δiδj)Π π_k^∗(e)=π_k^nc+(V_k^∗−π_k^nc)

whereΠ=(V_i^∗−π^nc_i)+V_j^∗−π_j^nc >0standsforthecreatedsurplus. Proof. SeetheproofinSection1.4.1.

Singlenegotiationsrefertothefollowingcases:1bargainswith3over(e1,τ13)or1 bargainswith2over(e1,τ12)or2bargainswith3over(e2,τ23).Ineachcase,theoutsider ofthenegotiationactsnon-cooperatively. When2or3bargainswith1,thenegotiation impliesahighereffortfor1,e^c₁>e^∗₁>e^nc₁ andlowereffortsfor2or3e^c_j<e^∗_j<e^nc_j, j=2,3. Whenthenegotiationisbetween2and3,2increaseshiseffortsuchthat e^∗₂>e^c₂>e^nc₂ ande^c₃<e^∗₃<e^nc₃,whiletheeffortof1remainsunchangede^∗₁=e^nc₁. When agentsbargain,theygetashareofthegeneratedsurpluswhiletheoutsideractsasa free-riderandbenefitsfromthepublicgood.(Thereisnoadditionalbenefitsforthemost downstreamagent,sincehispayoffequalshisnon-cooperativeoutcome). Thisparticular casereferstothemaximizationoftheaggregatepayoffundertheconstraintthatthemost downstreamagentgetshisnon-cooperativepayoff.Inthelimitwhenthetimebetween bargainingroundsvanishesδ=δi→ 1∀i,thecreatedsurplusissharedequallyamong thetwoinvolvedplayers.Inthatcase,the Rubinsteinsolutionconvergestothe Nash solution.Itturnsoutthatagent2isbetteroffwhen3bargainswith1(theadditional effortsof1substitutetotheeffortinprotectionof2),andbysymmetry,3isbetteroff when2bargainswith1(sameeffect,theadditionaleffortsof1and2substitutetothe effortinprotectionof3).

Expectingthatthecreatedsurpluscanbehigherwhenallagentsareatthenegotiation table, weassumethatthethreeagentsnegotiateintwobilateralnegotiations. This structureensurestheuniquenessofthesubgameperfectequilibrium(SPE)⁴.Twocases

4AsshownbyShakedandreportedbySutton[52],theuseof Rubinstein modelina multilateral

areconsidered. First,3bargainstwiceover(e1,τ13) with1andover(e2,τ23) with2 (settingτ12=0).Second,2bargainstwiceover(e1,τ12)with1andover(e2,τ23)with3 (settingτ13=0).

Thefirstnegotiationyieldsthefollowingproposition.

Proposition2 When 3bargainsasaproposer with1over(e1,τ13)and with2over (e2,τ23),theRubinsteinbargainingsolutionshowsthat:

1. Theoptimalvectorofeffortsesatisfies:

B₁(e1)+B₃(e1+e2+e3)=C₁(e1) B₂(e1+e2)+B₃(e1+e2+e3)=C₂(e2)

B₃(e1+e2+e3)=C₃(e3)

2. Equilibriumpayoffsaftertransfersare:

π^∗_i=π_i^nc+δi(1−δj)(1−δ3)

η Π3,i=1,2andj=i π^∗₃=π₃^nc+(1−δ1)(1−δ2)

η Π3

whereΠ3= ³_i=1(V_i^∗−π_i^nc)>0standsforthecreatedsurpluswhen3isinvolved intwonegotiationsandη=(1−δ1δ3)(1−δ2δ3)−δ1δ2(1−δ3)²>0.

Proof. SeetheproofinSection1.4.2.

Thesecondnegotiationyieldsthefollowingproposition.

Proposition3 When 2bargainssimultaneouslyasaproposerwith1over(e1,τ12)and with3over(e2,τ23),theRubinsteinbargainingsolutionshowsthat:

bargainingframework mayyield multipleequilibriaundertheunanimityrule.

1. Theoptimalvectorofeffortsesatisfies:

B₁(e1)+B₂(e1+e2)=C₁(e1) B₂(e1+e2)+B₃(e1+e2+e3)=C₂(e2)

B₃(e1+e2+e3)=C₃(e3)

2. Equilibriumpayoffs(aftertransfers)are:

π^∗_i=π_i^nc+δi(1−δ2)(1−δj)

φ Π2,i=1,3andj=i π^∗₂=π₂^nc+(1−δ1)(1−δ3)

φ Π2

Where Π2= ³_i=1(V_i^∗−π^nc_i)>0standsforthecreatedsurpluswhen2isinvolved intwonegotiationsandφ=(1−δ1δ2)(1−δ2δ3)−δ1δ3(1−δ2)²>0istheadditional generatedsurplus.

Proof. SeetheproofinSection1.4.3.

Inbothnegotiations,agents1and2willincreasetheirefforts,suchthat:e^c₁>e^∗₁>e^nc₁ ande^∗₂>e^c₂>e^nc₂,implyingadecreaseintheeffortforthe mostupstreamagente^nc₃ >

e^∗₃>e^c₃withrespecttohisnon-cooperativeeffort. Agentsgetashareofthegenerated surplusbuttheproposerstillbenefitfromthefirst moveradvantage(capturedthrough theleveloftransfers). However,ininstantaneousnegotiationswithdiscountfactorsset tothelimitδi= δj= δk= δ→ 1,everyplayergetonethirdofthesurplus. Thetwo negotiationsshowthatagentsarebetteroffcomparedtothenon-cooperativeoutcome π₁^∗>π^nc₁ >π^c₁andπ^c_i>π_i^∗>π_i^ncfori=2,3. However,inbothcases,evenifthedefinition ofthegeneratedsurplusisidentical,Π3differsfromΠ2,sinceequilibriumeffortsdidnot satisfythesameoptimalityconditions.

1 .2 .4 Examp le

Weknowturntoacomparisonbetweenthetwobargainingprotocols.Soastomakethe comparisontractable,weassumethatplayershaveidenticalcostandbenefitfunctions

regardlessoftheirlocationalongtheestuary:

B(z)=az−b

2z²,a,b>0 C(z)=c

2z²,c>0

Itisalsoassumedthatδ=δi→ 1∀i. Theoutcomeforallbargainingprotocolsunder thepreviousassumptionsaresummarizedinTables1.4.4,1.4.4and1.4.4inSection1.4.4. Thefollowingassertionscanbedrawnfromtheresultsonthebargainingprotocols.

Result1: Thecooperativeoutcome(highestaggregateeffortandpayoff)cannotbe reachedbyaparticularnegotiationprotocol. Thisresultcomesfromthestructureofthe model. Actingnon-cooperativelyisadominantstrategyforthe mostdownstreamagent. Exertinganefforte3>e^nc₃ alwaysdecreasesπ3. Thesetofbilateralbargainingreturns aconstrainedcooperativesolution(anintermediatesolution). Thissolutionconsistsin maximizingtheaggregateoutcomeundertheconstraintthanthemostdownstreamagent alwaysexertse^nc₃. However,theresultsshowthatagentsarealwaysbetteroffwhenaset ofnegotiationsovereffortsassociatedtoasetoftransfersexist.

Result2:Inthesinglenegotiationsscheme,ifanagreementisreachedbetweenagents 1and2(respectively,1and3)theremainingplayerpreferstofreerideandtostandoff thebargainingtable.Itappearsthatplayers2and3bothprefertheotherplayerto bargainwith1sincethebenefitfromthepublicgood. Thisconflictofinterestbetween agents2and3comesfromthepresenceofthepositiveexternality. Eachagentwould ratherbenefitfromtheeffortsrealizedbytheotheratnocost,asshowninTable1.2.4. Thethree-agentSeawallbargaininggamecanbesummarizedinnormalformwherethe spaceofstrategiesofeachagentconsistsineithertheacceptanceAortherefusal(R)of negotiations.Si={A,R}fori={1,2,3}. Notationi↔ jmeansthat inegotiateswith j.Itfollows:

Ifplayer1refusestobargain(S1=R)thenthebestresponseof2and3istoplayS2= S3=A.however,asseenbefore,thepayoffofplayer1isgreaterwhenhefindapartner tobargainwithcomparedtohisnon-cooperativepayoff,π₁^{1↔2}>π^nc₁ andπ^{1↔3}₁ >π₁^nc.

S1=A

2\3 S3=A S3=R

S2=A π₁^{2↔1,3},π₂^{2↔1,3},π₃^{2↔1,3} π^{1↔2}₁ ,π^{1↔2}₂ ,π^F₃ S2=R π₁^{1↔3},π^F₂,π^{1↔3}₃ π₁^nc,π₂^nc,π^nc₃

S1=R

2\3 S3=A S3=R

S2=A π₁^nc,π^{2↔3}₂ ,π₃^{2↔3} π₁^nc,π₂^nc,π^nc₃ S2=R π₁^nc,π^nc₂,π₃^nc π₁^nc,π₂^nc,π^nc₃

Table1.1: Theseawallbargaininggame.

ItappearsthatS1= Aisadominantstrategy. TheSeawallbargaininggameexhibits two Nashequilibriainpurestrategies.(S1,S2,S3) =(A,R,A)and(A,A,R).Inboth equilibria,agents2and3arebetteroffwhentheyactasfreeriders,π_i^F>π_i^{i↔1,j}and π_i^{1↔i}>π_i^ncfori=2,3,j=i.ThestructureoftheSeawallbargaininggameisachicken game,asinCarraroandSiniscalco[11].agent2(respectively,3)prefersthathisopponent bargainswith1andbenefitsfromtheoutcomeofthenegotiationwithoutbearingany cost.

Indoublenegotiation,theconflictisoverthepositionoftheproposer,butthisisa directconsequenceoftheRubinsteinalternatingoffer model. Whenthetimebetween bargainingroundsvanishes,thisfirst moveradvantagedisappears.

Result3:Itissociallyoptimaltoaskagent2to managethetwonegotiationswith the mostdownstreamandthe mostupstreamagents. Thesurplusgeneratedindouble negotiationisgreaterthanthesurplusofsinglenegotiations. Thesizeofthecreated surplusincreases withthenumberofplayersatthebargainingtable. Anegotiation between2and1yieldsagreatereffortofagent1incomparisonwithanegotiationbetween 3and1(Player3wouldacceptalowereffortfrom1inexchangeforalowtransferas heanticipatesapositiveeffortfromplayer2). Thisgreatereffortincreasesthebenefit

ofplayersupstream. Anegotiationbetween3and2yieldsagreatereffortofagent2 incomparisonwithanegotiationbetween2anda4^thplayer(this4^thplayeraskingless effortsfrom2sinceheanticipatesapositiveeffortfromplayer3). Thisresultcanbe generalizedtonagents. Asetofgeographicallyrestrictedbilateralnegotiationsimproves individualpayoffs.Inthissetting,eachplayerdirectlybargainwithhispredecessor,n negotiateswithn−1,n−1withn−2,...,uptothenegotiationbetween2with1.

1 .2 .5 Conc lus ionsontheSeawa l l Barga in inggame

Hirshleifer[26]showsthatcooperationoverthebuildingofaseawallcanbeachievedeven iftheseawallisknownasaweakest-linkpublicgood. Theseawallexamplehasbeenre- visitedforanalternativegeographicstructurewhereagentsarelocatedfromdownstream toupstreamandhavetoexertacostlyefforttoprotectthemselvesfromseafloods. This featureimpliesthatthebenefitofthepublicgoodincreasesalonganestuary.Inasim- plifiedthree-agentframework,ourresultsshowthattheredoesnotexistabargaining protocolthatcanbepreferredbyalloftheagents. Agentslocatedafterthe mostdown- streamagentalwaysprefertofreerideratherthanenteringinsinglenegotiationsover anadditionaleffortofthe mostdownstreamagent. Thiscasereferstoachickengame. Whenthenegotiationinvolvesalloftheagents,ourresultsshowthatitismoreprofitable forsocietytogivetherighttotheagentlocatedinthe middleoftheestuarytoconduct negotiationswithboththe mostdownstreamandthe mostupstreamagent.

Thenextsectionintroducesthe modelwithnegativeexternalities.

1 .3 Ther iverbarga in ingprob lem

Gametheoreticanalysisoftheriversharingproblemhasbeenaveryactiveresearcharea overthepasttwodecades(Barret,(1994)[6]; Kilgourand Dinar,(2001)[34]; Ambec andSprumont,(2002)[1]and Bealetal,(2013)[7]forarecentsurvey). The main motivationforthisresearchreliesontheconflictingnatureofwaterusesbetweenvarious agentsorcountriesforresidential,industrialoragriculturalpurposes. Analyzingwater

allocationrulesamongagentsorcountrieswhoarelocatedalongariveralsoraisessome interestingquestionsintermsofefficiencyandequitywhenpropertyrightsarenotwell defined. Thesharingproblemreferstoasituationinwhichagentshaveunequalaccess totheresourcedependingtheirlocationontheriver. The mostupstreamagenthavea fullaccessbutastheriverflowgoesdowndownstreamagentsgettheremainingwater leftbyupstreamagentswhoarelocatedinfrontofthem. Gametheoretic modelsandin particularbargaining modelsarerelevanttodealwhichsuchaproblem. Theliterature onwaterallocationcanbeclassifiedintotwobroadapproaches.

Thefirstapproachisbasedoncooperativegametheoryand, moreprecisely,onthe core. Theobjectiveistosetupaburden-sharingruleabletofavorthecooperationof all,ensuringthattherulepreventsthatanyindividualagent,butalsoanysub-group ofagents,fromleavingtheagreement. Basedona modeltakingexplicitlyintoaccount directionalflows,AmbecandSprumont(2002)[1]showthattheconvexityofthecooper- ativegameensuresanon-emptycore. Theyanalyzehowacompromisesolutionbetween thetwointernationallawprinciplesofwatersharingintransboundaryriverbasinscanbe reachedinacooperativegamewithutilitytransfers.Thesetwoprinciplesaretheabsolute territorialsovereignty(ATS)thatprescribesthateachagentisfreetouseallthewater hecontrolsonhisterritoryandtheunlimitedterritorialintegrity(UTI)thatstatesthat theamountofavailablewatertoanagentcannotbealteredbyalltheagentswhoare locatedupstreamfromhislocation. Anupstreamagentisonlyallowedtoconsumewater ifhehastheexplicitconsentofallhisdownstreamagents.

1 .3 .1 L iterature Rev iew

Houbaetal.(2014)[28]considerastrictinterpretationoftheUTIrulestatingthatonly themostdownstreamagentmayclaimallthewaterandcanrestrictallhispredecessorsto zeroextractionaslongasnoagreementhasbeenreached. Ansinkand Weikard(2012a)[4] modeledtheriversharingproblemasasequentialbankruptcygameinwhichthesumof theclaimsofalltheagentsexceedtheavailabilityoftheresource(Aumannand Maschler, (1985)[5]andThomson(2013)[54]forarecentsurvey). Theyanalyzeseveralsharing

rulesspecifiedintermsofamountof watergiventoeachagent. Whenallthe water originatesattheheadoftheriver,thesharingrulestatesthateachagentgetsthesame proportionofhisclaimandthelinearorderoftheagentsdoesnot matter. Otherwise boththedistributionofclaimsandwaterendowmentsneededtobeconsidered. Houba etal.(2014)[28]showthatthemostdownstreamagentalwayspreferstheUTIprinciple butatleastoneoftheotheragentpreferstheATSrule.

Thesecondapproachisbasedonnon-cooperativegametheory,alsowithseveralcat- egories. Ansinketal.(2012b)[2]modeledtheriversharingproblemasatwo-stageopen- membershipcartelgameusingtheconceptofinternalandexternalstability(d’Aspremont etal.,(1983)[15]). Assumingthatariveragreementisagroupofagents(acoalition)who havemergedandmaximizetheirwelfarejointly,theirgoalistodeterminewhichcoalitions arestable,inthesensethatnoagentwantstoleaveitorjoinit. Assumingthatagents haveidenticalbenefitfunctionsandonlydifferintheirlocationalongtheriver,results showthatacoalitionofatleastfouragentsisnotstable. Ansinkand Weikard(2009) [3] modeledtheriversharingproblemasacontestedgameoverthepropertyrightsin whichtwoagentscandecidetobargainornot.Inthelattercaseagentsusetheiroutside optionsbyaskingathirdagenttoimplementanequitablesolution. Resultsshowthat agentscanendupinaninefficientequilibriuminsteadofbargaininganefficientoutcome. Carraroetal.(2007)[10]reviewseveralbargaining modelstowaterissuesinorderto showhowanagreementisreachedamongsectorsorcountries. Theyemphasizecomplex negotiationproblemsdealingwithmultilateralandmulti-issuesfeaturesthatcanonlybe solvedbyuseofcomputersimulations.Inanalternating-offerRubinstein model,Houba (2008)[27]interpretsthemodelofAmbecandSprumontinabargainingperspectivebut onlyinabilateralcasewithoneupstreamagentandonedownstreamagent.

Mostoftheseapproachesand modelsabove mentionedhaveincommontopartly abstractfromthenegotiatingprocess. Thispaperusesasimplified Rubinstein model withthreeagents whocanbargainsequentially withendogenousdisagreementpoints. UndertheATSrule,themostdownstreamagentisconstrainedinhiswaterconsumption whileunderthe UTIrule,the mostupstreamplayercannotextracthisoptimallevel

ofconsumption. Hencenegotiationcantakeplacebetweenagentsbutisnotlimitedto neighboringagentsasin Wang(2011)[56].In modelingterms,thecontributionofthis paperistoemphasizetheroleofinsideoptionsinaRubinsteinframeworkwithseveral sequentialbargainingprocedures.Insideoptionsrefertothepayoffsthatagentsobtain whentheytemporarilydisagree.Byassumingineachcasethatdelaybetweennegotiation roundsvanishes,thegoalsofthepaperaretoexplainthesourceofpossibleinefficiencies andthedifferentnegotiationoutcomesthatcanresultfromsequentialbargaining. We considerthreedifferentnegotiationprocedures.Thefirstprocedurereferstoasimultane- ousnegotiationinwhichthemostdownstreamagentbargainswitheachupstreamagent overwaterextractionandtransfers(sidepayments).Thesecondprocedureassumesthat thetheconstrainedagentbargainssequentiallywithbothupstreamagents.Thelastpro- cedureconsidersthattheanagentwhodoesn’tsufferfromscarcity(the mostupstream underATS,themostdownstreamunderUTI)bargainssequentiallywiththeotheragents accordingtotheirlocation.Ineachcase,theoutcomeintermsofwaterextractionand transfersisanalyzedandcomparedtothesocialoptimum,showingthatwithbothsequen- tialprocedures,andregardlessofthepropertyrightsdivision,theoutcomeisinefficient fromthepointofviewofsociety. Theintuitionforthisinefficiencycomesfromthefact thatinourgameagentsbargainovertransfersandwaterconsumptionlevels. Thus,not onlythedistributionofthenetsurplusisdifferentunderdifferentprotocols,thewater extractionleveland,hence,thenetsurplusitself,isalsodifferent. Wealsoshowthatthe inefficiencyinthesequentialprocedurescomesfromtheinsideoptions.Finally,weshow that,dependingonthesequenceof movesinthesequentialnegotiation,insideoptions canstrengthenorweakentherelativepositionoftheagentsinvolvedinthenegotiation.

Thefollowingsubsectionspresentthenoncooperativeandthecooperativeoutcomesso asthesimultaneousnegotiation.Subsection1.3.3isdevotedtotheanalysisofsequential negotiationsaccordingtothetwoprotocols.Insection1.3.4aspecificexampleshowsthe mainresultsofthepaper. Thelastsubsectionpresentstheconcludingremarksonthe Riverbargainingproblem.

1 .3 .2 Thebenchmark

ConsiderasetofN ={1,2,...,i,...n}agentslocatedalongariverinalexicographicor- dering. Agent1isthemostupstreamagentwhileagentnislocatedatthemouthofthe river.Eachagentextractsanamountofwaterxi≥0fromwhichheearnsBi(xi),∀i∈N.

Thebenefitfunctionisassumedtobeincreasinguptoamaximumvalueequaltoxisolu- tionofB_i(xi)=0anddecreasingforgreaterextractions,formally:xi=argmaxiBi(xi) isthesatiationpointoftheagenti. Theplayersaresaidtobehomogeneousifevery benefitfunctionsareidentical,thatis∀{i1,i2}∈N,Bi1(xi1)=Bi2(xi2). Theamount of wateravailableisE. Weconsidertheriver merelyflowsatitsownsourceandthe absenceofintermediaryinflows. Wedenotebyi1<i2thefactthatagenti1isupstream ofagenti2. FollowingnotationofAnsinkand Weikard(2009),Ui1= {i2∈N :i2<i1} standsforthesetofagentslocatedupstreamofthelocationofplayeri1andreciprocally Di1={i3∈N :i1<i3}.

Noncooperativeoutcome

UndertheAbsoluteterritorialsovereignty principle,eachagenticontrolstheamountof waterE− _j∈Uixjthathasn’tbeenconsumedbytheplayersupstreamofthelocationofi.

Inthenoncooperativecase,themostupstreamagentchooseshowmuchtoextract,under theconstraintthatthisleveldoesnotexceedtheavailableamountE.Then,thefollowing agentchoosesalevelofextractionfromtheremainingwaterE−x1.Thisprocessgoes uptothemostdownstreamagentn.Thesub-gameperfectequilibria(SPE)ofthisgame showsthateachagentiextractsthe maximumbetweenhisnoncooperativelevelxiand theamountofwaterhecontrols:E− _j∈Uixj. Thenoncooperativesetofstrategiesfor i∈N undertheATSprincipleisthefollowing:

Si=















xi if xi≤E−

j∈Ui

xj

E−j∈Ui

xj if xi>E−

j∈Ui

xj

0otherwise.

Fromupstreamtodownstream,theplayersextracttheirbestresponseuptoaplayerwho suffersfromscarcity.Ifxi2≥E−

i1∈U_i2xi1,thesubgameperfectequilibrium(SPE)ofthe gamereturnsauniquevectorofextractions:

X_ATS^nc =















xi1 if i1<i2

E−i1∈U_i2xi1 for i2

0 if i3>i2

ThebenefitsB_i^ncforalli∈N canbederivedfromthewaterextractionslevels. UndertheUnlimitedTerritorialIntegrity principle,the mostdownstreamplayercan claimtoconsumeanunalteredwater.Themostdownstreamagentscanclaimtoconsume orextractanylevelofavailablewater. The mostdownstreamagentchooseshislevelof extraction,iexn=argmaxxnB(xn)ifxn< E . Thus,apartoftheresourceisclear ofpropertyrights. Theplayersupstreamcanextractapartoftheresourceprovided thattheUTIprinciplestillapplies,thatis:x^∗_nisatleastflowingontheterritoryofthe mostdownstreamagent. Thus,upstreamplayersapplythefollowingrationale: Player n−1playshisbestresponseregardingtheamountofwaterhisallowedtoextract.If xn−1≤ E−xn,thenx^nc_n−1= xn−1isexactlypumpedatleveln−1. Thisextraction behaviorisenforcedbytheplayersallalongtheriveruptothe mostupstreamagent. ThenoncooperativesetofstrategiesundertheUTIprinciplefori∈N isthefollowing:

Si=















xi if xi≤E−

j∈Di

xj

E−j∈Di

xj if xi>E−

j∈Di

xj

0otherwise.

Fromdownstreamtoupstream,theplayersextracttheirbestresponseuptoaplayer whosuffersfromscarcity.Ifxi2≥E−

i3∈D_i2xi3,thesubgameperfectequilibrium(SPE)

ofthegamereturnsauniquevectorofextractions:

X_UTI^nc =















0 if i1<i2

E−i3∈D_i2xi3 for ε2

xi3 i3>i2

AnoncooperativegameΓ^nc_P undertheprincipleP isa mappingoftheorderingof theplayersontotheindividuallevelsofextraction. Thelevelofextractions(andthus thelevelsofindividualbenefits)dependsontheorderingofplayersandthedefinition ofpropertyrights. LetN bethereversedorderingofN. Thefollowingpropositionis straightforwardandispresentedwithoutproof:

Proposition4 IfB1=Bn,Γ^nc_ATS(N)=Γ^nc_UTI(N)andΓ^nc_UTI(N)=Γ^nc_ATS(N).

Iftheplayerswhosufferfromscarcityareidentical(player1underUTIandplayer nunderATS),thelevelofnoncooperativeextractionsareidenticalfortheorderingN under ATSandforthereversedorderingunder UTI.Ifplayersareheterogeneous,the solutiondiffersconformingtotheconcavityofthebenefitfunctions.

Cooperativeoutcome

Thecooperativesolutionisgivenbytheprogram:

xmax1,...,xn

i

Bi(xi) sc

n i

xi≤E

Waterextractionlevels X^c={x^c₁,...,x^c_n}aresolutionof

B₁(x1)=...=B_n(xn)

withxn=E− _i∈Uixi. EfficiencyrequirestheestateE tobesharedsoastoequalize the marginalbenefitsamongagents.Itisstraightforwardthatthecooperativeoutcome doesn’tdependontheprincipleandthatΓ^nc_ATS(.)=Γ^nc_UTI(.)holdsforanyordering. Note

obverseoccursfortheremainingplayers:x^c_i3≥x^nc_i3 fori3∈Di2∪{i2}underATSandfor i1∈Ui2∪{i2}underUTI.Thishasdirectimplicationsintermsofprofit,wenotethat B_i^nc₁ ≥B_i^c₁fori1∈Ui2underATSandfori3∈Di2underUTI.TheCooperativeoutcome hasincreasedthenetbenefitofplayersi3∈Di2∪{i2}undertheATSandi1∈Ui2∪{i2} undertheUTIprinciple.

Soastodecentralizethenoncooperativeoutcometowardthecooperativesolution undertheATSprinciple(respUTIprinciple),aincentive-based mechanism mustbeim- plemented. Amonetarycompensationisnecessarytoencouragethemostupstream(resp downstream)agentstorefrainfromoverconsumingwater. This monetarycompensation canbegeneratedbythehigherlevelofbenefitassociatedtoahigherwaterconsump- tionofthe mostdownstream(respupstream)agents. Thestrictconcavityofthebenefit functionuptothe maximumpointensuresthatthelossincurredbytheupstream(resp downstream)agentwillbelowerthanthegainobtainedbythedownstream(respup- stream)agent. Thus,theremustexistsabargainingschemethatensuresthattheagents willindividuallybenefitfromdecentralizingthenoncooperativeoutcometowardsanal- ternativesolution. Wenowdetailinthenextsectionthreebargainingprotocols.

1 .3 .3 Negot iat ionprotoco ls

Werestrictourattentiontogamesforwhichthereisascarcityconstraint,thatisgames forwhich∀j∈{1,n}, _i∈N\{j}xi≤ E ≤ _i∈Nxi. Weassumethatnegotiationsare bilateral.Tosimplifynotationwewrite:Bi=Bi(xi). Anegotiationbetweenagentsiand jhastwoarguments. Amonetarytransfersorside-paymentsinexchangeforareduction ofwaterconsumption. Theagentjproposesthe monetarytransferτijtotheagentiin exchangeforhiscommitmenttoreduceisconsumptiontoalevelxi<x^nc_i. Alternatively, theagenticouldproposetheplayerjtoreducehisextractiontoalevelxi<x^nc_i in exchangeforthesubventionτji. Notethatτij=−τjiandbyconventionthefirstplayer inthesubscriptpaysthetransfertothesecond(Ifipays−τijthenhereceivesτji). An

offeroraproposalisacoupledenotedbyoij=(x,τ)with

(x,τ)=







(xi,τji) if −^∂(B_∂x^i+B_i^j)>0 (xj,τij) if −^∂(B_∂x^i+B_j^j)>0

Iftheofferismadebyitojandisaboutanabatementinxi,thenplayerjincreaseshis consumptioninexchangeforthetransferτji.Bargainingroundsassumesperfectinforma- tion.

Wealsoassumethatthemostupstreamandthemostdownstreamagentsaretheonly agenda-settersinthenegotiationprocess. Dependingontheprinciple,eitherthe most upstreamplayerorthemostdownstreamplayersuffersfromthescarcityconstraint.Ifnot mentionedotherwise,weassumethattheplayerwhosufferfromthescarcityconstraint entersnegotiationsasthefirstproposer. Thatisplayer1andnhavetherightto make offersasproposertoalltheotheragents. Playersinbetweenonlyactsasrecipientsof theproposals. Thus,a3playerssettingissufficienttoputforwardthe mainideaofthis paperwith1or3actingasproposersand2canthusbeseenasrepresentativeofplayers inbetween.

Ifplayer3istheproposer(asinthe ATSprinciple),thenthegeneralnetpayoff function(aftertransfers)forthe3agentsofanagreementreachedatperiodtis:

π1(o31;t) = B₁^nc+δ₁^t(B1−B^nc₁ −τ13) (1.7) π2(o32;t) = B₂^nc+δ₂^t(B2−B^nc₂ −τ23) (1.8) π3(o31,o32;t) = B₃^nc+δ₃^t(B3−B^nc₃ +τ13+τ23) (1.9)

where0≤δi≤1fori={1,2,3}standsforthediscountrate,andt={0,1,..}arethe periodsatwhichtheoffersandcounter-offersareformulated. Apermutationof3and1 intheequations1.7to1.9returnsthepayoffundertheUTIprinciplewhenplayer1is thefirstproposer. Negotiationscanoccursimultaneouslyorinasequential manner.