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.Bargainingroundsassumesperfectin forma-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.

Simultaneousnegotiation

Weset N ={i,j,k}withoutordering. Weassumethattheagentwhosuffersfromscarcity initiatesthenegotiationswiththetworemainingplayer. Theproposerisidentifiedby iandxi= E−xj−xk. Thusibilaterallynegotiatesatthesametime withkover oik=(xk,τik)andwithjoveroij=(xj,τjk). Weassumethatkisthecentralplayer,thus theabsenceofbargainingbetweenjandk,impliesthatτjk=0.

Leto⁽ⁱ⁾_ij=(x⁽ⁱ⁾_j,τ_ji⁽ⁱ⁾)betheoffer madebyitoj. TheRubinsteinBargainingSo lu-tion(RBS)thatwearelookingforistheuniqueSPEgivenbythefollowingconditions (Rubinstein,1982; Muthoo,1999):

πi(o^(j)_ij,oik;0)=πi(o⁽ⁱ⁾_ij,oik;1)andπj(o⁽ⁱ⁾_ij,oij;0)=πj(o^(j)_ij,oik;1).

andbetweeniandk:

πi(o^(k)_ik,oij;0)=πi(o⁽ⁱ⁾_ik,oij;1)andπk(o⁽ⁱ⁾_ik,oij;0)=πk(o^(k)_ik,oij;1).

Thesetwosystemsofindifferenceequationsstatesthateachagentisindifferentbetween acceptingthecurrentofferofhisopponentand makingacounterofferinthenextperiod whichwillbeaccepted.

WecannowstatethefollowingProposition:

Proposition5 Thesolutionofthegames Γ^sim_ATS(j,k,i)andΓ^sim_UTI(i,k,j)returnsthe uniqueoptimalvectorofextractionsx^c= x^c_i,x^c_j,x^c_k ,thehighestlevelofwealthΠ =

i∈N(B_i^c−B_i^nc)>0andthepayoffsaftertransfersare: π_j^∗ = B_j^nc+µjΠ

π_k^∗ = B_k^nc+µkΠ

π_i^∗ = B_i^nc+(1−µj−µk)Π

withµj=_(1−δjδi^(1−δ_)(1−δ^{i)(1−δk)δj}

kδi)−δjδk(1−δi)²andsimilarlyfork.

Proof. seeproof1.4.5.

Inbothcases,theplayersiwhosuffersfromthescarcityconstraintsbeg insthene-gotiationswithplayersjandk.Inthelimitcaseδ→ 1,eachagentgetshisimpasse point(permanentdisagreementpayoff)plusaequalshare(1/3)ofthetota lcreatedsur-plus. Thissimultaneousnegotiationaccountsforthebenchmarkcaseforthesequential bargainingprocedures. UndertheATSprinciple,{1,2,3}= {j,k,i}andplayer3pays atransfertoagents1and2inexchangeforareducedwaterextraction. UndertheUTI principle,{1,2,3}={i,k,j}andplayer1paysatransfertoagents2and3inexchange foranabatementinwaterconsumption.

Onceagain,ifBi=Bj,theProposition4stillapplies.Thesolutionissensitivetothe concavityofthebenefitfunctionsotherwise.

Sequentialnegotiation

Basedonthesamesetofpayoffs(1.7)-(1.9),weanalyzedtwobargainingproceduresthat wecalledthe downstreamandtheupstreamprocedure.Inthedownstreamprocedure, themostdownstreamplayerisassumedtobargaintwice(regardlessheistheplayerwho suffersfromscarcity). Inthefirstroundofnegotiation⁵,ifj <k <i,theniandk bargainover(xk,τik).Inthesecondround,ibargainswithjover(xj,τij).Still,thereis nonegotiationbetweenjandk.Intheupstreamprocedure,the mostupstreamplayer bargainswiththetwootherplayersintwodifferentrounds.Ifj<k<i,thenjbargains withkover(xj(xk),τik)inthefirstroundandwithiinthesecondroundover(xj,τij).

Thesolutionsarefoundusingbackwardinductioninbothcases.Inbothprocedures,the secondroundofnegotiationsrunsknowingthatanagreementhasbeenreachinthefirst stage.Thenegotiationsremainbilateral. Wealsoassumethatthediscountfactorforthe intra-negotiationsisidenticalinthetworounds. Wesettheinter-negotiationsdiscount factortounity. Howeverasweconsiderresultsinthelimitcasewhenthedelaybetween negotiationroundsvanishes,thisassumptionisnotrestrictive.

5Itiscrucialtodistinguishtheintra-negotiationsbetweentwoplayers(sequencesofbilateraloffers andcounteroffer)andtheinter-negotiations(sequencesofnegotiationsinvolvingdifferentplayers).In eachroundofnegotiations,theproposerbargainswithadifferentplayer.

Askingfor water Inthisfirstsequentialprotocol, weassumethattheplayer who suffersfromscarcity(i)bargainstwiceasaproposer.Inroundtwo,ibargainswithj knowingthatanagreementhasbeenreachedinroundonewithkandthisagreement willbeeffectiveeveninthecaseofadisagreementinthiscurrentround.Thisfirstround agreementreferstotheinsideoptionforagenti.Itdetermineshisimpassepoint.

Proposition6 TheRubinsteinbargainingsolutionshowsthatwhenibargainswithkin round1andwithjinround2,

1. Waterextractions(x^∗_j,x^∗_k)aresolutionof:

B_j(xj) = B_i(xi)

B_k(xk) = B_i(xi)+δj(1−δi)

(1−δjδi)B_ix^d_i −B_i(xi) withx^d_i=E−x^nc_j −xk

2. Equilibriumpayoffsaftertransfersare:

π^∗_j = B_j^nc+µjΠ −(1−ψk−ψi) B_i^d−B^nc_i −(B_k^nc−B_k^∗) π^∗_k = B_k^nc+µkΠ +ψk B_i^d−B^nc_i −(B_k^nc−B_k^∗)

π^∗_i = B_i^nc+(1−µj−µk)Π+ψi B_i^d−B_i^nc −(B^nc_k −B_k^∗)

withΠ =B_i^∗−B_i^nc+B_j^∗−B_j^nc+B_k^∗−B_k^nc>0,istheadditionalsurplusgenerated thedownstreamprocedure,B_i^d= Bi(x^d_i)andthecoefficientsµj=^δ_(1−δ^j(1−δ_jδⁱ_i⁾₎,µk=

δ_k(1−δi)(1−δj)

(1−δkδi)(1−δjδi),ψk=_(1−δ^δjδ_j^k_δi^(1−δ_)(1−δⁱ⁾²_kδi)andψi=_(1−δ^δj(1−δ_kδi^j_)(1−δ^)(1−δ_jⁱ_δ⁾_i). Proof. seeproof1.4.6.

Waterextractionsgivenbyproposition6donot maximizesocial welfare(x^∗_i= x^c_i

∀i).Itimpliesthatthenetsurplusislowerinthesequentialcasethaninthecooperative case.Theintuitionforthisinefficiencycomesfromthefactthatanalternativebargaining protocoldoesnotonlymodifiesthedistributionofthenetsurplus,butalsothenetsurplus itself.Indeed,agentsarebargainingovertransfersandwaterextractions,thustheseizeof

thenetsurplusisalsodeterminedbythenegotiations.Theoptimalityconditionssuggest thattheplayerjextractsanintermediarylevelofwaterbetweenthecooperativesolution andthenoncooperativesolution:x^nc_j >x^∗_j>x^c_j. Theintermediaryplayerreachedhis lowestlevelforthisprotocol:x^∗_k<x^c_k<x^nc_k whilethenegotiationsalloweditogethis highestlevelofextractionsx^∗_i>x^c_i>x^nc_i.

Inthefirstround,ibargainedalow waterextractionlevelorequivalentlyahigh reductionwithrespecttohisinitialsituationwiththeplayerk.Theabatement(x^nc_k −x^∗_k) issufficientlyhigh,theassociatedtransferishigh. Evencostly,thisoutcomeisworthy fortheplayeri.Indeed,anagreementwiththeintermediaryplayerhelpssecuringhis position whenitcomestoenterabargainingprocess withthe mostupstreamplayer. Astrengthenedposition(alternatively,ahigherimpassepoint), wouldleadto milder consequencesifadisagreementwith1hadtooccur.

Basedonthisagreement,inegotiatesinthesecondroundwithkoveralowerreduction effortandalowertransfer. Thereisnot muchlefttobebargainedsince mostofthe exchangegainshavebeengeneratedinthefirstroundbetweentheplayerskandi.

Eventually,bothkandiarebetteroffattheexpenseofj,whichisonlyinvolvedin thesecondroundofthenegotiation.Intermsofpayoffseachagentgetsashareofthe createdsurplusbutevenifjgetsahighershare(1/2inthelimitδ→ 1and1/4fork andi),hispayoffisreducedbyanamount measuredbythepositivetermintobrackets

B_i^d−B_i^nc > (B_k^nc−B_k^∗). Thislossofpayoffforjisagainforkandi. Sincethe problemissymmetric,afirstroundbetweeniandj(insteadofk)wouldimplytheterm B_i^d−B_i^nc > B^nc_j −B_j^∗. Therelativeimportanceofthesetermscanbeusedbyito decidewhethertobargainfirstwithjorwithk,showingthattheorderofthepartners inthesequentialprocessbecomesastrategicvariable. Thefirstprotocolexhibitsthat beinginvolvedinthefirstroundofnegotiationisauspicious.

SettingN ={j,k,i},Proposition6referstothedownstream-sequentialprotocolunder ATS.Inthisprotocol;i=3suffersfromscarcityandbargainsw ith2and1overanabate-mentin x1x2inexchangefor monetarytransfers. WereferthisgametoΓ^down_ATS(j,k,i).

SettingN ={i,k,j}Proposition6referstotheupstream-sequentialprotocolunder UTI.Inthisprotocol; i=1suffersfromscarcityandbargains with2and3overan abatementinx2x3inexchangeformonetarytransfers. WereferthisgametoΓ^up_UTI(i,k,j).

Thesetwogamesareidenticalforapermutationofplayersifplayersarehomogeneous whileΠ differsinthesetwogamesotherwise.

Askingforatransfer Alternatively,weproposetheobverseofthepreviousprotocol. Inthesecondprotocol,theplayerwhoownspropertyrightsovertheestate(1underATS and3underUTI)bargainstwiceasaproposer.Inroundtwojbargainswithiknowing thatanagreementhasbeenreachedinthefirstroundwithk.

Proposition7 TheRubinsteinbargainingsolutionshowsthatwhenjbargainswithkin round1andwithiinround2

1. Waterextractions(x^∗_i,x^∗_j)aresolutionof

B_j(xj) = B_i(xi) xk = x^nc_k

2. Equilibriumpayoffsaftertransfersare

π_j^∗ = B_j^nc+µjΠ π_k^∗ = B_k^nc+µkΠ

π_i^∗ = B_i^nc+(1−µj−µk)Π

withΠ =B_i^∗−B_i^nc+B_j^∗−B_j^nc>0istheadditionalsurp lusfromtheupstreambar-gainingprotocolcomparedtothenoncooperativeoutcome.µi=_(1−δ^(1−δ_kδ^k_i^)(1−δ_)(1−δ^j_iδ⁾_j)and µk=_(1−δ^δ2(1−δ_kδiⁱ_)(1−δ^)(1−δ_i^j_δ⁾_j)thesharingparameters.

Proof. seeproof1.4.7.

Thesecondprotocolalsoimpliesthatwaterextractionsarenotsocialwelfare max i-mizing. Theplayerkbehavesconsonantlytothenon-cooperativecase,indeed:x^∗_k=x^nc_k.

Itturnsoutthatx^∗_i<x^c_iandx^∗_j<x^c_j. Analogouslytothecaseofaskingforwater,each agentgetsitsimpassepointandanadditionalshareofthenetsurplus.Incontrasttothe previousprotocol,itturnsoutthatbeinginvolvedinthesecondroundisnowopportune. Thereasonisthattheimpassepointofjinthesecondroundweakenshisrelativepos i-tioninregardstoplayeri.Intheeventofaperpetualdisagreementwithi,agentjwill consumex^nc_j,butwillbearthecostofthetransfertok(τjk>0). Forthatreason,jis urgedtosignanagreementwithioveralowlevelofwaterextraction(x^c_j>x^∗_j)orahigh reduction(x^nc_j −x^∗_j),implyingahighvalue B^nc_j −B_j^∗ ,whichincreasesthetransferτij

paidbyjanddecreasesthetransferτjk.

Inthelimitδ→ 1,agentsjandkgetthesameshareofthecreatedsurplusπj−B_j^nc= πk−B_k^nc=¹₄Π ,andiisbetteroffπi=B_i^nc+¹₂Π . Askingforatransferfavorstheagent whosuffersfromscarcityandensuresanequaltreatmentfortheremainingagentssince jandkgetanidenticalshareofthecreatedsurplus.

SettingN ={j,k,i},Proposition7referstotheupstream-sequentialprotocolunder ATS.Inthisprotocol; j=1bargainswith2and3overanabatementin x1andx2in exchangefor monetarytransfers. WereferthisgametoΓ^up_ATS(j,k,i).

SettingN ={i,k,j}Proposition7referstothedownstream-sequentialprotocolunder UTI.Inthisprotocol; i=1suffersfromscarcityand j=3bargains with1and2 overanabatementinx2x3inexchangefor monetarytransfers. Wereferthisgameto Γ^down_UTI (i,k,j).Onceagain,ifplayersarehomogeneous,Π isidenticalinbothgamesand Γ^up_ATS(j,k,i)=Γ^down_UTI (i,k,j).

Inordertocomparetheupstream-sequentialandthedownstream-sequentialprotocol, weprovidedanexamplehereafter.

1 .3 .4 Examp le

Letusillustratetheoutcomesofthedifferentbargainingprotocolsdiscussedabovewith anexample. AssumethefollowingbenefitfunctionasinAmbecandSprumont(2000)

Bi=axi−bi

2x²_i

Inthenoncooperativecase,theoptimalityconditionB_i=0impliesx^nc_i =a/bi.

Playersarerankedsuchthatx2= λx1andx3= λ²x1with λ >0. Forλ <1, x1>x2>x3andforλ>1,x1<x2<x3.Itimpliesλb2=b1andλ²b3=b1.Playersare homogeneousforλ=1.

Toensurenegotiations weset(1+λ+λ²)x1> E (scarcityconstraint)andE >

(1+λ)x1underATS.Thefirstplayercanextractx^nc₁ =x1.Theresultsaregiveninthe Table1.2withrespectto: Π=b1((1+λ+λ²)x^nc₁ −E)²>0and

SIM^(sim−ats) Downstream^(D−ats) Upstream^(U−ats)

x1 1

Table1.2:SolutionsunderATS

Itcanbeshowthatintermsofwaterextractionthatx^D−ats₁ >x^sim−ats₁ >x^U−ats₁ , x^U−ats₂ >x^sim−ats₂ >x^D−ats₂ andx^D−ats₃ >x^sim−ats₃ >x^U−ats₃ andintermsofpayoffs π₁^sim−ats>π₁^U−ats>π^D−ats₁ ,π₂^D−ats>π^sim−ats₂ >π^U−ats₂ andπ₃^U−ats>π₃^D−ats>π^sim−ats₃ . Thisexampleshowsthatthe mostupstreamagentprefersthesimultaneousnegotiation whilethe mostdownstreamagentpreferstheupstream-sequentialnegotiationandthe

Protocol

SIM^(sim−uti) Downstream^(D−uti) Upstream^(U−uti)

x1 1 Table1.3:SolutionsunderUTI

middleagentthedownstream-sequentialnegotiation.

UnderUTI,weset _λ¹2+_λ¹+1 x3>E (scarcityconstraint)andE> ¹_λ+1 x3,the lastplayercanextractx^nc₃ =x3. TheresultsaregiveninTable1.3.

Wecannotethatthereisarelationbetweentheadditionalgeneratedsurplusunder ATSand UTI,indeed, λ³S^sim−ats= S^sim−utiarerespectivelythegeneratedsurplusin bothcases. Onceagain,theadditionalsurplusisidenticalinthepresenceofhomogeneous players.Ifnegotiationsaresimultaneous,theATSproceduregeneratesagreatersurplus forλ>1.Inotherwords,enforcingtheATSprinciplebringsagreatersurplusifextraction costsincreasesastherivergoesdownstream.ThismakessenseastheATSprincipletends tofavortheplayersupstream. Conversely,ifextractioncostsdecreaseastherivergoes downstream,thenΠ^sim_UTI>Π^sim_ATSsinceλ<1.

Itcanbeshowthatintermsof waterextractionthatx^U−uti₁ >x^D−uti₁ >x^sim−uti₁ , x^D−uti₂ >x^sim−uti₂ >x^U−uti₂ ,x^U−uti₃ >x^D−uti₃ >x^sim−uti₃ .Intermsofpayoffs,thereisan identicalorderingforthethethreeplayersπ^sim−uti>π^D−uti>π^U−uti.