Fondazone En Enrco Matte Endogenous Coalton Formaton n Global Polluton Control Mchael Fnus and Banca Rundshagen NOTA DI LAVORO 43.001 JUNE 001 Coalton Theory Network Unversty of Hagen, Germany Ths paper can be downloaded wthout charge at: The Fondazone En Enrco Matte Note d Lavoro Seres Index: http://www.feem.t/web/actv/_actv.html Socal Scence Research Network Electronc Paper Collecton: http://papers.ssrn.com/paper.taf?abstract_d Fondazone En Enrco Matte Corso Magenta, 63, 013 Mlano, tel. +39/0/5036934 fax +39/0/5036946 E-mal: letter@feem.t C.F. 97080600154
1. Introducton Concern about transboundary and global polluton problems ranks promnently on the agenda of nternatonal poltcs and has led to the sgnature of several nternatonal envronmental agreements (IEAs), as for nstance the Oslo Protocol on sulfur reducton n Europe n 1994, the Montreal Protocol on the depleton of the ozone layer n 1987 and the Kyoto-Protocol on the reducton of greenhouse gases n 1997. Ths concern s also reflected n numerous recent papers on the formaton of coaltons n nternatonal polluton control snce the appearance of Barrett (1991), Bauer (199) Black/Lev/de Meza (199), Carraro/Snscalco (1991), Chander/Tulkens (1991), Hoel (199) and Tulkens (1979). The fundamental assumpton of all models s that IEAs must be self-enforcngly desgned snce there s no nternatonal agency that can establsh bndng agreements (Endres 1996). The man problem analyzed by these models s free-rdng n nternatonal polluton control. There are two types of free-rdng whch negatvely affect the success of an IEA (Fnus 000, ch. ). The frst type of free-rdng s the ncentve of a country remanng a non-sgnatory (or to choose a low abatement level), beneftng from the (hgher) abatement efforts of (other) sgnatores. The second type of freerdng relates to the ncentve of a sgnatory to volate the sprt of an agreement. Through free-rdng a country can reduce ts abatement effort substantally, though envronmental qualty wll only be affected margnally. Thus a country can (temporarly) net a free-rder gan. Though models dffer wth respect to the specfcaton of the utlty functon of governments and wth respect to the stablty concept they employ, they can be classfed n two groups. Dynamc Game Models The frst group may be called dynamc game models (DG-models). These models assume an nfntely repeated game where governments agree on some contract n the frst stage that has to be enforced n subsequent stages by usng credble threats (Barrett 1994a, b, Fnus/Rundshagen 1998 and Stähler 1996). Credblty s defned n these papers n terms of renegotaton-proof equlbrum. Whereas Barrett and Fnus/Rundshagen consder only sngle devatons from the oblgatons of an IEA, Stähler also consders multlateral devatons, however, he restrcts the number of countres to three whch allows to draw only lmted conclusons for transboundary and global envronmental problems. Barrett analyzes the condton for stablty of a grand coalton targetng at a globally optmal soluton. He fnds that stablty may be jeopardzed even n a supergame framework and even f agents are almost perfectly patent. Therefore, Fnus/Rundshagen extend hs work consderng the formaton of subcoaltons and of less ambtous abatement targets. They show that the allocaton of the abatement
burdens crucally affects the success of IEAs, a grand coalton s unlkely to form and that a subcoalton may acheve more than the grand coalton. Reduced Stage Game Models The second group of models may be termed reduced stage game models (RSG-models). Two varants may be dstngushed: a) models applyng the concept of nternal&external stablty and b) those applyng the concept of the core to determne the equlbrum coalton structure. Both varants model coalton formaton as a two-stage game where countres decde n the frst stage on coalton formaton and n the second stage countres choose ther emsson (abatement) levels and how they dstrbute the gans from cooperaton. Varant a): Internal&External Stablty Up to now, all models of varant a have assumed that n the frst stage countres have only the choce between accedng to an IEA or to reman a non-sgnatory where non-sgnatores play as sngletons. In the second stage, t has been assumed that sgnatores jontly maxmze the coalton s welfare and that non-sgnatores maxmze ther ndvdual welfare. Sgnatores and non-sgnatores play non-cooperatvely aganst each other. Ether sgnatores and nonsgnatores choose smultaneously (Nash-Cournot assumpton, e.g., Carraro/Snscalco 1991 and 1993, Bauer 199 and Hoel 199) or sequentally (Stackelberg assumpton, e.g., Barrett 1994b). For the dstrbuton of benefts, three typcal assumptons have been made. ) no dstrbuton (Bauer 199), ) dstrbuton accordng to the Nash-barganng soluton (Botteon/Carraro 1997 and 1998) and ) dstrbuton accordng to the Shapley value (Barrett 1997b and Botteon/Carraro 1997 and 1998). If countres are assumed to be ex-ante symmetrc 1, all sgnatores receve the same payoff and a redstrbuton s obvously not necessary. In the case of asymmetrc countres, assumptons and are popular allocaton rules, though other rules (as for nstance the Kala-Smorodnsk-soluton, Kala/Smorodnsk 1975, or the 1 Ex-ante symmetrc refers to the assumpton that all countres have the same payoff functon though they may receve dfferent payoffs dependng whether they become a sgnatory or reman a non-sgnatory (or n the general context whch coalton they jon). In the case of symmetrc countres the Nash-barganng, the Shapley value and probably any other barganng rule of cooperatve game theory mples no redstrbuton of the ntal payoff allocaton. However, strctly speakng, despte the assumpton of symmetrc countres any asymmetrc allocaton of payoffs cannot be ruled out a pror, though t would probably requre some addtonal motvaton that should be endogenous to the model.
proportonal soluton, Kala 1977) are also concevable. 3 Snce payoffs are receved at the end of the second stage and snce the rules of the game (choce of emsson levels, dstrbuton of payoffs and procedure of coalton formaton) are exogenously fxed, the two stages can be reduced (va backwards nducton) to one stage that smplfes computatons a lot. Coalton formaton can be studed wth the help of a partton functon (see secton ), whch contans all nformaton about the payoffs a country receves n dfferent coalton structures. In equlbrum, nternal stablty mples that no sgnatory has an ncentve to leave the coalton; external stablty mples that no non-sgnatory wants to accede to the agreement (see secton 3.. for a formal defnton). A key result of these models s that, generally, the number of sgnatores falls short of the grand coalton and the equlbrum coalton s rather small. Moreover, whenever cooperaton would be needed most from a global pont of vew, the coalton acheves only lttle. 4 Ths result s also remnscent to the dynamc game models of Barrett (1994a, b) and Fnus/Rundshagen (1998). Varant b): Core In models of varant b t s analyzed n the frst stage whether there s an ncentve for one country or a group of countres to devate from some coalton structure. In the second stage, t s assumed that coaltons choose an emsson vector, whch maxmzes the aggregate payoff to the coalton and that n the case of asymmetrc countres a transfer scheme s establshed. If a country or some countres devate, t s assumed that the remanng players splt up nto sngletons and ) mnmax the devatng player(s) (α-core), ) maxmn the devatng player(s) (β-core) or ) play a Nash strategy (γ-core). Smply speakng, a coalton structure s an equlbrum f no member can acheve a hgher payoff n any other coalton structure. The key result of these models (Chander/Tulkens 199, 1995, 1997) s that by choosng a cleverly desgned transfer scheme (whch resembles that of Kaneko s rato equlbrum), the grand coalton establshng the socally optmal emsson vector can be sustaned as an equlbrum. 3 4 Botteon/Carraro (1997 and 1998) ncely demonstrated the mportance of the allocaton rule for the number of sgnatores, for the composton of sgnatores and for the success of an IEA through a smulaton exercse. That s, n equlbrum, global welfare generated by the coalton s close to that n the Nash equlbrum and the gap between the coalton equlbrum and the socal optmum s large. Ths result s demonstrated n Fnus (000b, ch. 13) for a large number of models belongng to varant a. 3
Evaluaton and Comparson The advantage of DG-models s that they capture both dmensons of free-rdng. In contrast to RSG-models, these models also consder the possblty that countres may jon an agreement but do not fully comply wth the terms of the agreement. These models have been crtczed for ther assumpton of an nfnte tme horzon and that they specfy threat strateges n terms of a temporary expanson of emssons. It has been argued that poltcal agents act n a fnte tme horzon and that emssons are not used n realty to punsh free-rders. 5 Moreover, though these models would allow to account for smultaneous coaltons, gven the nature of these models, the study of smultaneous coaltons would be a complex undertakng and would have to rely on smulatons. The advantage of RSG-models s related to ther smplcty, though all share the dsadvantage of capturng only one dmenson of the free-rder problem, namely, that a country free-rdes on the abatement efforts of other countres. 6 The second dmenson of a country accedng to a coalton but volatng the agreement s not captured. By assumpton countres nstantaneously reoptmze ther strategy n the second stage and therefore a temporary free-rder gan s ruled out. 7 A partcular drawback of those RSG-models, whch apply the stablty concept of nternal&external stablty, s that they assume exogenously that there wll be only one non-trval 8 coalton (sgnatores). Of course, ths assumpton smplfes computatons tremendously, however, a pror t s not clear whether the co-exstence of several coaltons could not also be an equlbrum. Ths concern s taken up by those RSG-models that apply the stablty concept of the core. Ths varant of models allows for the co-exstence of several coaltons. An other advantage s that results do not have to rely on smulatons (as ths s the case for almost all prevously mentoned models). Moreover, the core concept allows studyng stablty of an 5 6 7 8 See Fnus (000a, b) for a dscusson of these ssues and possble arguments of defense. In the models applyng the concept of nternal&external stablty free-rdng mples to reman a non-sgnatory. In the reduced stage models applyng other concepts (and whch allow for the possblty of the co-exstence of several coaltons), free-rdng mples that a country belongs to a coalton that contrbutes less to jont abatement than other coaltons. See sectons and 3. Ths allows for two nterpretatons of an equlbrum n RSG-models: t has to be assumed that ether a) countres comply wth the terms of an IEA once they have acceded or that b) the volaton of an IEA s mmedately dscovered. The frst nterpretaton s not n accordance wth the emprcal evdence on the complance of numerous IEAs (see the lterature cted n Fnus (000a, ch. ); the second nterpretaton requres an optmstc vew as to the possbltes of montorng and the flexblty of alterng strateges. A non-trval coalton means a coalton of at least two countres. 4
IEA where the members gradually adjust to the targets of the treaty (e.g., Chander/Tulkens 1991 and 199, German/Tont/Tulkens 1996 and German/Tont/Tulkens/de Zeeuw 1998). It thus allows capturng an mportant feature of actual IEAs where sgnatores acheve ther long-term targets not n one step but n a step-by-step fashon. The dsadvantage of ths varant of models s that from a postve pont of vew they do not contrbute very much to explan the formaton of IEAs n realty: none of the exstng IEAs consttutes a grand coalton and transfers played a neglectable role f at all n the past. Due to the assumpton that a) there are no problems to admnster and to enforce transfers and b) f a devaton occurred the remanng players would resolve the coalton, ths optmstc result s obtaned, renderng the co-exstence of coaltons only a theoretcal possblty whch never materalzes. Our model framework belongs to the group of RSG-models and reles on work of Bloch (1997) and Y (1997). Lke the core concept, our framework allows for the possblty of several coaltons. However, equlbrum coalton structures are determned by applyng "new concepts" to study coalton formaton. Some of these concepts, whch we call coalton formaton games, have been appled to problems of ndustral economcs and nternatonal trade whereas some others have not been appled yet (see the lterature cted n Fnus 001, ch. 15). Our man concern s to compare the equlbrum coalton structures n global polluton control under dfferent coalton formaton games and to evaluate these games wth respect to ther theoretcal propertes and how they capture features of the formaton of exstng IEAs. Snce these concepts mply dfferent rules accordng to whch coaltons form, they also have a normatve dmenson. For nstance, one key ssue s whether membershp n an IEA should be based on unanmous agreement by all sgnatores (exclusve membershp) or whether all countres should be allowed to accede to an IEA f they wsh do so (open membershp). It should be ponted out rght at begnnng that for the complexty of allowng for the possblty of the co-exstence of several coaltons we have to pay a prce n terms of makng the smplfyng assumpton of ex-ante symmetrc countres. Thus, the ssue of redstrbutng the benefts of a coalton and the choce of the abatement target s trvally solved. 9 Moreover, for some concepts t s necessary to assume a specfc payoff functon (though smulatons are not necessary) to derve nterestng conclusons. However, t should be kept n mnd, that all models mentoned above share ths dsadvantage, except the reduced stage game models of varant b. 9 There s no redstrbuton and countres choose that emsson level whch maxmzes a coalton s aggregate payoff (whch also maxmzes a country s payoff). 5
In what follows, we lay out the model framework, present defntons and derve some general results n secton, whch form the bases of determnng the equlbrum coalton structures under varous coalton formaton games n secton 3. In secton 4 we compare the equlbrum coalton structures, draw conclusons and evaluate the concepts theoretcal and practcal potental to explan the formaton of IEAs and pont to ssues of future research.. Model.1 Defntons We consder a RSG comprsng two stages where n the frst stage countres form coaltons and n the second stage coaltons choose ther optmal strategy vector. The detals of the frst stage are summarzed n the rules of a coalton formaton game. We postpone the detals untl secton 3. In the second stage, we assume that the members of a coalton maxmze aggregate payoffs to the coalton and play a Nash-Cournot strategy aganst other coaltons. Snce we assume exante symmetrc countres, ths mples that those countres that belong to the same coalton choose the same emsson level and receve the same payoff. We therefore do not have to consder transfers among coalton members n the second stage. Transfers between coaltons are also ruled out. Assumpton 1: Rules of the Second Stage Countres are assumed to be ex-ante symmetrc, and to maxmze a coalton s aggregate payoff. There are no transfers. If the rules of the second stage are ex-ante specfed, the two stages can be reduced to one stage. All relevant nformaton on whch the decson n the frst stage s based can then be compactly summarzed n the per-membershp partton functon (Bloch 1997). Defnton 1: Equlbrum Valuaton or Per-Membershp Partton Functon Let c = { c 1,..., c M } denote a coalton structure wth M coaltons, c C where C denotes the set of coalton structures, c j c k = j k, c1...... c M = I, then an equlbrum valuaton s a mappng whch assocates to each coalton structure c C a vector of ndvdual payoffs π( c ) = { π1( c j, c ),..., πn( c k, c )} where π( c ) Π (C) s the set of payoffs whch results from the maxmzaton of players accordng to a partcular rule, a gven sharng rule of the gans from cooperaton and a gven coalton structure. The frst argument n π (c,c) refers to the coalton to whch country belongs, the second to the partcular coalton structure. 6
For Assumpton 1 and strctly concave payoff functons (see secton.), there s a unque optmal strategy vector for each possble coalton structure n the second stage. Thus, the set of equlbrum valuatons s unquely defned. The coalton formaton may be summarzed as follows: Defnton : Coalton Formaton Game In the coalton formaton game a country decdes, based on the per-membershp partton functon, on the membershp accordng to the rules of a coalton formaton game. The partton functon s determned by the rules of the second stage n the game. Note that due to the assumpton of symmetry, notaton smplfes. Thus, nstead of wrtng for example C={ c 1 3, c, c } where c 1 3 ={{1, }, {3}}, c ={{1}, {, 3}}, c ={{1, 3}, {}} (mplyng all coalton structures that two countres can form and where one country remans a sngleton are equlbrum coalton structures), we wrte C ={(, 1)} or c =(, 1) where the entres ndcate the coalton szes and the astersks equlbrum coalton structures. 10 For the subsequent notaton t wll prove helpful to order coaltons accordng to ther sze. That s, c=(c 1,..., c M ) where c 1... c M. An mportant defnton to compare dfferent coalton structures s that of coarsenng and concentraton (e.g., Y (1997), p. 05 and Bloch (1997), p. 334). Defnton 3: Coarsenng of a Coalton Structure c = ( c 1,c,...,c M ) s a coarsenng of = c ( c 1, c,..., c M ), M < M f and only f there s a 1 1 1 1 sequence of coalton structures c = ( c 1, c,...,c M(1) ), c = ( c 1, c,...,c M() ),..., R R R R c = ( c,c,...,c ) wth M( 1) = M() 1 for all =,..., R such that 1) c = c 1 and 1 M(R) c = c R and )..., R. r 1 r c = c \ {c,c } { c r + c r } for some, j {1,...,M ( r )} and for all r =, r r j j A coalton structure c s coarser than a coalton structure c f c can be obtaned by mergng coaltons n c. For example coalton structure (6, 5) s coarser than coalton structure (5, 5, 1) snce (6, 5) can be obtaned by mergng coaltons 5 and 1 n coalton structure (5, 5, 1). However, many coalton structures cannot be compared under coarsenng as for nstance (5, 5) and (6, 4). Then a comparson may be possble under the crteron of concentraton. 10 Thus c 1 denotes a coalton wthn a coalton structure and 1 c a partcular coalton structure. 7
Defnton 4: Concentraton of a Coalton Structure c = ( c 1,c,...,c M ) s a concentraton of c = ( c 1, c,..., c M ), M M f and only f there s a 1 1 1 1 sequence of coalton structures c = ( c 1, c,...,c M(1) ), c = ( c 1, c,...,c M() ),..., R R R R c = ( c,c,...,c ) such that 1) c = c 1 and c = c R r and ) 1 r r r c = c \{c,c } 1 M(R) r r r j(r) ( r ) j( r ) r {c + 1,c 1}, c c for some (r), j(r) = 1,..., M(r) and for all r=,..., R. (r) That s, c s a concentraton of c f one can obtan c by movng one member at a tme from a coalton n c to another coalton of equal or larger sze. Through ths process, coaltons n c may sequentally be dssolved. For nstance (6, 5) s a concentraton of (5, 5, 1) snce the sngleton coalton s dssolved and ths player jons a larger coalton of sze 5. However, (6, 4) s also a concentraton of (5, 5), though no coaltons are dssolved. Unfortunately, however, also concentraton does not allow for a complete orderng of coalton structures. For nstance, (4, 3) and (5, 1, 1) cannot be ranked under concentraton. From the defntons t follows that every coalton c whch s coarser than a coalton c mples that c s a concentraton of c, however, the opposte s not true. A formal proof of ths relaton s provded n Y (1997).. Propertes of the Global Polluton Game In ts most general form, the payoff functon of the global emsson game may be wrtten as follows: (r) j(r) [1] N π =β(e ) φ ( e ) j j1 = where we assume β > 0 for all e max < e, β < 0 for all e > 0, φ > 0 and φ 0 for all ej > 0. That s, benefts from emsson (n the form of consumpton and producton of goods) ncrease n emssons at a decreasng rate. Damages ncrease n global emssons at a constant or ncreasng rate. Unfortunately, as ponted out n the Introducton, the amount of conclusons, whch can be derved for the general functon [1], s very lmted. Therefore, we consder two addtonal examples that have wdely been used n the lterature on coalton formaton. [] [3] 1 π = b(de e ) c( e ) N j j1 = 1 c π = = N b(de e ) ( e j) j1 8
Whereas payoff functon [] assumes constant margnal damages, payoff functon [3] assumes lnear margnal damages. Ths mples orthogonal reacton functons for payoff functon []. That s, the slope of the reacton functon s zero. In contrast, for payoff functon [3] reacton functons are downward slopng n emsson space wth a slope greater than 1 and less than zero. 11 Thus payoff functon [3] exhbts a more nterestng pattern than [] wth respect to the nteracton of agents. However, also for payoff functon [3] only a lmted amount of general propertes can be establshed whch allow makng precse predctons about coalton formaton. Therefore, n most parts of the paper we wll llustrate results based on payoff functon []. Followng Y (1997), the global emsson game maybe vewed as a postve externalty game f a strategy s seen as emsson reducton from some status quo. 1 That s, f a country or a coalton reduces emssons, all other countres or coaltons beneft from abatement efforts as well. In the standard framework wthout consderng coalton formaton ths s an mmedate mplcaton of π / ej < 0. In the context of coalton formaton ths fact also appears but also other facets of t. To demonstrate those, t s helpful to look frst at emssons of countres belongng to dfferent coaltons and at global emssons resultng from dfferent coalton structures. Proposton 1: Emssons a) Let emssons of a member of a coalton c be denoted by e and of a coalton c j by e, j then for any coalton structure c we have e > e ff c < c. b) Let coalton structure c be coarser than coalton structure c and denote total emssons T by e, then T T e(c ) < e(c). c) Let coalton structure c be more concentrated than coalton structure c, then T T e ( c ) < e ( c ) for payoff functons [] and [3]. j j 11 1 Note that the followng results do not depend on the exact specfcaton of functons [] and [3]. For the results t s only mportant that payoff functon [] mples constant margnal damages and payoff functon [3] lnear margnal damages. In the context of coalton formaton payoff functon of type [] has been used for nstance by Barrett (1994b, 1997a), Bauer (199), Botteon/Carraro (1997) and Hoel (199), payoff functon of type [3] by Barrett (1994b), Carraro/Snscalco (1991), Fnus/Rundshagen (1998) and Stähler (1996). Only for the concept of the core Chander/Tulkens (1995, 1997) were able to produce results based on a general payoff functon of type [1]. In our context t seems obvous to defne the status quo as the Nash equlbrum wth only sngleton coaltons. 9
Proof: a) follows from the frst order condton of a member of a coalton c k whch s gven T by β (e k) = c kφ (e ). Snce for any e T T c kφ (e ) ncreases n c k and β < 0, e k ncreases n the sze of the coalton c k. b) s demonstrated by contradcton. Assume e T(1) < e T() would be true where e T(1) are global emssons before and e T() after coaltons c and c j have merged. Moreover, assume that there s a thrd coalton c k that s not nvolved n the merger. Then for members of coaltons (1) T(1) T() () (1) () k: β (e k ) = c kφ (e ) < c kφ (e ) =β (e k ) ek > ek (1) T(1) T(1) T() () (1) () (and j): β (e ) = c φ (e ) < (c + c j) φ (e ) < (c + c j) φ (e ) =β (e ) e > e hold whch obvously volates the ntal assumpton of e T(1) < e T(). c) follows from routne computatons whch shows that global emssons are gven by payoff functon []: c = and payoff functon [3]: M T e Nd c b = 1 e T Nd = M c 1+ c b = 1 respectvely and where M c ncreases through concentraton snce, assumng c c j, = 1 j k k j k,j k holds (Q.E.D.). [(c 1) + (c + 1) + c ] c = (1 + c c ) > 0 Proposton 1a s an mmedate mplcaton of Assumpton 1: coalton members maxmze the aggregate payoff to ther coalton. The larger a coalton, the more do ther members care about the negatve mpact ther emssons exhbt on other countres. Consequently, global emssons decrease f coaltons merge and form larger coaltons as stated n Proposton 1b. Only for coalton structures, whch cannot be ranked under coarsenng, such a general concluson s not possble. However, as Proposton 1c demonstrates, global emsson decrease for payoff functons [] and [3] f coalton structures become more concentrated. An mmedate mplcaton of Proposton 1 s the followng corollary. Corollary 1: Global Emssons The grand coalton produces the lowest global emssons, the degenerated coalton structure consstng of sngletons produces the hghest global polluton. Proof: Follows from Proposton 1b and the facts that a) N s coarser than c f c (N) and b) c s coarser than (1, 1,..., 1) (f c (1, 1,..., 1)) for each coalton structure c (Q.E.D.). We can now turn to the facets of a postve externalty game n the context of coalton formaton. One facet s that members of small coaltons enjoy a hgher payoff than members of large coaltons for any gven coalton structure. 10
C 1 : π (c,c) < π (c,c) ff c < c j j j Snce all countres suffer equally from emssons but members of smaller coaltons choose hgher emssons n equlbrum (and therefore have hgher benefts), members of smaller coaltons are better off than members of larger coaltons. A somewhat more specfc though stll very general feature of the postve externalty game s the followng. C : a) π (c,c) < π (c,c ) where c c, c and c s coarser than c. b) π (c,c) < π (c,c ) where c c, c and c s more concentrated than c. Whereas condton a) clams that coaltons whch are not nvolved n a merger are better off after the merger, condton b) clams that coaltons whch are not nvolved n a concentraton are better off f other coaltons form a more concentrated coalton structure. Obvously, condton b) s stronger than a). C a) and b) are mplcatons of Proposton 1 b) and c), respectvely: global emssons decrease through coarsenng (concentraton) whch has a postve effect on outsders. 13 Condtons 1 and reveal a typcal feature of a postve externalty game, namely that of freerdng. Smaller coaltons and those coalton members, whch are not nvolved n formng larger coaltons amng at reducng global damages, beneft from the abatement efforts of the more actve players. Other features, though more specfc, are also related to ths problem. The next two condtons deal wth the effect on members of an "old" and "new" coalton f a member leaves hs old coalton to jon a new coalton j. C 3 : π(c,c) < π (c \{k},c ) where c = c \ { c, c j } {c j {k},c \{k}}, c c j. If a member of the coalton leaves hs coalton to jon a larger or equal-szed coalton j, the members of the old coalton are better off. The old members beneft from the ncreased abatement efforts of the new and larger coalton (payoff functon [] and [3]) whle they keep ther efforts constant (payoff functon []) or reduce ther effort (payoff functon [3]). For payoff functons [] and [3] t turns out that the free-rder ncentve can be compactly summarzed as follows. 13 Outsders ncrease ther emssons after the concentraton (see the proof of Proposton 1) and hence receve hgher benefts and lower damages. 11
C 4 : πk(c,c) > π k(cj { k }, c ) where c = c \ { c, c j } {c j { k}, c \{ k}} f cj c. If a member k of the coalton leaves hs coalton to jon a larger or equal-szed coalton j, then the devator becomes worse off. From C 4 t also follows that jonng a coalton that s smaller by at least two members s proftable. Condton C 4 stresses that - once there s some (mnmal) amount of concentraton - sngle members have only a margnal effect on reducng global damages by jonng larger coaltons but ther benefts decrease substantally through ncreased abatement efforts. Wth respect to a merger the followng condton can be establshed. C 5 : π j(c j,c,c) < π j(cj c,c ) and π(c,c) < ( ) π (cj c,c ) f cj c < ( ) 0 where c = c \ { c, c j } {c j c } and cj c. The smaller coalton must be at least half the sze of the bgger coalton to be not worse off after the merger of coalton and j. For c j = c = 1 coalton c gans from a merger. The larger or equal-sze coalton j always gans. In order to compactly summarze our results n sectons 3 and 4, we make the followng assumpton related to C 4 and C 5, whch we assume to hold n the remander of ths paper. Assumpton : Indfference of Payoffs If players are ndfferent between beng a member of a smaller or larger coalton, they jon the larger coalton. In the followng, Assumpton mples for nstance that f a sngleton s ndfferent between remanng a sngleton and jonng a coalton of sze two and the larger coalton lkes hm to jon, we assume he wll do so (see condtons C 4 and C 5 ). 14 Our dscusson of condtons C 1 to C 5 may be summarzed as follows. Proposton : Incentve Structure of the Global Emsson Game The general payoff functon [1] satsfes condtons 1 and a, payoff functon [] satsfes condton C 1 to C 5 and payoff functon [3] condtons C 1 to C 4. Proof: The statement wth respect to the general payoff functon has already been proved above. The statement wth respect to payoff functons [] and [3] s proved n Appendx 1 (Q.E.D.). 14 We take ths assumpton from Ray/Vohra (1999). It reduces the amount of knfe-edge equlbra but does not affect the fundamental results. 1
In secton 3 t turns out that condtons C 1 to C 5 are essental for characterzng equlbrum coalton structures for most coalton games. For only a few games C 1, C (strong verson) and C 3 are suffcent to draw some conclusons, for some other games also C 4 s needed and for some games wthout C 5 almost nothng can be sad. Snce we fnd t nterestng to compare equlbrum coalton structures between dfferent coalton games, t s evdent that we have to work wth payoff functon [] to make progress. Ths s also evdent when consderng the subsequent propertes that we wll use n secton 3. The subsequent Proposton 3 wll be helpful n that t allows drawng mmedate conclusons wth respect to global welfare f equlbrum coalton structures under dfferent coalton games can be ranked accordng to concentraton. Proposton 3: Concentraton and Global Welfare Assume payoff functon []. If c s a concentraton of c, global welfare s hgher under c. Proof: See Appendx (Q.E.D.). At a more general level, we fnd: Proposton 4. Welfare n the Grand Coalton The grand coalton produces the hghest global welfare. Proof: Follows obvously from (Q.E.D.). max π max π+ max π+... + max π j k (N) c1 j c k cm For the cartel formaton, open-membershp and the exclusve membershp -game the followng defnton and proposton wll turn out to be useful. Defnton 5: Stand-alone Stablty of a Coalton Structure c={ c 1,..., c M } s stand-alone stable ff π (c,c) π ({ },c ) where c = c \ c { c \{ }, {}} I. A coalton structure c s stand-alone stable f and only f no player fnds t proftable to leave her coalton to be a sngleton, holdng the rest of the coalton structure constant. From ths t follows mmedately that the degenerated coalton structure consstng only of sngletons s stand-alone stable by defnton. 13
Proposton 5: Stand-alone Stablty for Payoff Functons [] and [3] For payoff functon [] all coalton structures where no coalton comprses more than three coalton members s stand-alone stable. For payoff functon [3] a coalton structure may only be stand-alone stable f no coalton comprses more than two coalton members. Proof: See Appendx 3 (Q.E.D.). Interestngly, assumng payoff functon [3], the more concentrated a coalton structure s, the more lkely t s that the stand-alone stablty fals. That s, the more coaltons of sze two have formed, the hgher s the ncentve to take a free-rde and to become a sngleton (see Appendx 3). To derve the equlbrum coalton structure under the exclusve membershp -game and the sequental move unanmty game but also to evaluate equlbrum coalton structures n generally, the followng defnton and proposton wll turn out to be useful. 15 Defnton 6: Pareto-optmal Coalton Structures (POs) A coalton structure c s Pareto-optmal f there s no other coalton structure c where at least one player s better off and no player s worse off,.e., c c wth π (c,c ) > π (c,c) for some j I : π j(c j,c ) < π j(c j,c). Ths mples that there s no other coalton structure c whch weakly Pareto-domnates c. Proposton 6: Pareto-optmal Coalton Structures and Grand Coalton The grand coalton s a Pareto-optmal coalton structure. Proof: Let c = (c 1,...,c M) where c 1... c M. Then π 1 (c 1, c) π j (c j, c) cj c1 by condton C a. Snce the grand coalton s a PO (Q.E.D.). 1 1 1 j j j j= M N π (N) c π (c, c) + c π (c, c), π1(n) π 1(c 1, c) holds. Hence Whereas for general payoff functon [1] and also for payoff functon [3] a more specfc characterzaton beyond Proposton 6 s not possble, for payoff functon [] the entre set of POs can exactly be determned. 15 It may be worthwhle to pont out that our defnton of a Pareto-optmal coalton structure assumes a fxed behavor of the coalton s members. That s, a coalton maxmzes the coaltons welfare and plays a Nash strategy aganst outsders. 14
Proposton 7: Pareto-optmal Coalton Structures for Payoff Functon [] For payoff functon [] the set of Pareto-optmal coalton structures, PO C (N ) = {N } { c = { c 1,..., c M } M, c 1... c M,{ c 1,..., c M }\cj j {1,..., M}, π(c M,c) > π(n )}. PO C (N ), s gven by PO C (N c ) PO PO Proof: Follows from the followng facts: a) {N} C (N) by Proposton 6. b) c C (N) PO => c C ( ~ c ) c c. c) If c {N} {c = {c 1,..., c M} M, c 1... c M, {c 1,..., PO c M}\cj C (N c j) j {1,..., M}, π (c M, c) >π (N)}, then the smallest coalton has no ncentve to form the grand coalton 16 and no other coalton has an ncentve to partcpate n formng any other coalton structure snce each sub-coalton structure s a PO tself (Q.E.D.). j Table 1: Pareto-optmal Coalton Structures for Payoff Functon [] N Pareto-Optma N Pareto-Optma 1 (1) 7 (7), (6, 1), (5, ) () 8 (8), (7, 1), (6, ) 3 (3) 9 (9), (8, 1), (7, ) 4 (4), (3, 1) 10 (10), (9, 1), (8,), (7, 3) 5 (5), (4, 1) 11 (11), (10, 1), (9, ), (8, 3) 6 (6), (5, 1) 1 (1), (11, 1), (10, ), (9, 3), (8, 3, 1) Assumpton s assumed to hold. PO Proposton 6 mples that C (N) are computed recursvely. For N=1 to N=1, the set of PO Pareto-Optma are lsted n Table 1. For nstance for N=6, C (N) = {(6), (5,1)}. (6) s the grand coalton. (5, 1) s a PO snce (5, 1)\{5}=(1) s a PO (see N=1), (5, 1)\{1}=(5) (see N=5) s a PO and π (1, (5, 1)) >π (6, (6)). For practcal purposes of determnng C PO (N), t s helpful to note that a necessary condton for a coalton structure c=(c 1, c,..., c M ), c 1 c... c M-1 c M, to qualfy as a PO s c >c +1 for any =1...M-1 due to C 5. (Suppose the opposte s true, then there s an ncentve for at least two coaltons to merge.) 16 Recall that due to C 1 the members of the smallest coalton derve the hghest payoff n a gven coalton structure. Due to Assumpton, n a PO t s assumed that the smallest coalton wll partcpate n the grand coalton f ths leaves ts members ndfferent. 15
3. Coalton Structures 3.1 Introducton We consder sx dfferent versons (coalton games) how the frst stage of the coalton formaton process can be modeled: 1) cartel formaton game, ) open membershp game, 3) exclusve membershp -game, 4) exclusve membershp Γ-game, 5) sequental move unanmty game and 6) equlbrum bndng agreement game. These games can be structured wth respect to two dstngushng features. The frst feature s the tme dmenson of the coalton formaton process. Ether the formaton process s modeled as a one-shot game or as a dynamc process. We choose ths feature to group the games n ths secton. Coalton games 1 to 4 assume smultaneous choce of membershp (secton 3.), whereas games 5 and 6 assume a sequental choce of membershp (secton 3.3). The second feature concerns the membershp. In open membershp type of games all players can freely accede to a coalton f they want. In exclusve membershp type of games external players need the consent of the members of a coalton before they can jon. Games 1 and assume open membershp, games 3 and 4 exclusve membershp, and games 5 and 6 mply de facto exclusve membershp, though ths s not explctly spelled out n the defnton of these games. In what follows we ntroduce the coalton games n subsectons 3. and 3.3 and derve the equlbrum coalton structures. In secton 4 we dscuss the equlbrum coalton structures, evaluate the coalton games wth respect to ther theoretcal consstency, ther practcal applcaton and compare the equlbrum coalton structures among the dfferent games. 3. Smultaneous Choce of Membershp 3..1 Prelmnares In order to select the equlbrum coalton structure(s) from equlbrum valuatons, we need an equlbrum concept. For coalton games 1 to 4, we use the concepts Nash equlbrum (Nash 1950), strong Nash equlbrum (Aumann 1959) and coalton-proof Nash equlbrum (Bernhem/Whnston/Peleg 1987) that n our context may be defned as follows (see, e.g., Bloch 1997). Defnton 7: Nash and Strong Nash Equlbrum Coalton Structure (NE and SNE) Let G = {I, Σ = { Σ }, π(c( σ )) = { π (c,c)} } be the frst stage of the coalton formaton I I game wth players I, strategy vectors σ Σ (proposals for coaltons), resultng coalton structures c and vectors of payoff functons π. Further, let C(c, σ ) be the set of S coalton structures that a subgroup of countres S c can nduce f the remanng countres j I\ S c 16
play σ j. For a fxed strategy vector σ defne the reduced game for subgroup c S as G s = {c S,{ },{ (c,c(c S, )} } σ Σ S π S c σ. Then σ s called a Nash equlbrum (strong Nash c equlbrum) wth the resultng Nash equlbrum (strong Nash equlbrum) coalton S structure (NE, SNE) c f no sngleton c = { } (no subgroup c S ) can ncrease hs (at least one members ) payoff (wthout reducng the payoff of any other member) by nducng another coalton structure. That s, c( σ ) s a NE f I and c C({ }, σ ): π ( c,c ) π ( c,c ), c( σ ) s a SNE f no subcoalton c S I can nduce a coalton structure c S C(c, σ ) S S wth π (c,c ) π (c,c) c and π (c,c ) < π (c,c) for at least one c. 17 Defnton 8: Coalton-Proof Nash Equlbrum Coalton Structure (CPNE) For I = {1} σ s a coalton-proof Nash equlbrum f and only f t s a Nash equlbrum. Assume that I = n > 1 and that coalton-proof Nash equlbrum coalton structures have been defned for all m< n. Then - σ s self-enforcng f and only f for all c S I, equlbrum of s G σ. S c I, σ S s a coalton-proof Nash c - σ s a coalton-proof Nash equlbrum of G wth the coalton-proof coalton structure (CPNE) c f and only f t s self-enforcng and there does not exst another selfenforcng strategy σ such that π(c( σ ),c( σ )) π(c,c ) I and π (c( σ ),c( σ )) > π (c,c ) for at least one. 18 Whereas a NE requres that a coalton structure s mmune to devatons by sngle countres, a SNE also requres that devatons of any subgroup of countres are not benefcal to the devators. From ths t follows that any SNE s a NE too. That s, C SNE C NE. Moreover, a necessary condton for a SNE s that t s a Pareto optmal coalton structure,.e., C SNE C PO. That s, a NE should not be weakly Pareto-domnated by an other NE. However, one has to be aware that, generally, not every weakly Pareto-undomnated NE s a SNE,.e., C PO SNE C. 17 18 Orgnally, a SNE s defned as an equlbrum where no subcoalton can devate such that the welfare of each member ncreases. We use a stronger verson here to be consstent wth Assumpton. For example for payoff functon [] and N=3, accordng to our defnton c = (3) s a SNE, but not c = (,1), whch could also be a SNE accordng to the orgnal defnton. The prevous footnote (wth approprate changes) apples to CPNE as well. 17
In contrast to a SNE, a CPNE must only be mmune to devatons whch are self-enforcng. SNE CPNE Consequently, C C. For nstance, π (1, (3,1)) >π (4, (4)) >π (, (, )) s true for payoff functon []. Hence, c=(, ) cannot be a SNE snce ths coalton structure s Paretodomnated by grand-coalton. However, a country has an ncentve to leave the grand coalton f the other countres reman n the coalton whch may be the case dependng on the assumptons of the coalton formaton rules. Hence, c=(, ) mght be a CPNE and n fact s one n the open-membershp game and the exclusve membershp -game. Snce coalton-proofness consders self-enforcng devatons of subgroups of countres, the CPNE NE specal case of sngle devatons s entaled n the defnton and C C. Thus, we have SNE CPNE NE C C C. Of course, one may wonder whether requrng a coalton structure to be Pareto-effcent wth respect to the entre set of coalton structures s not an unduly restrctve condton. In partcular snce t turns out below that n some coalton games, as n many other games of economc nterest, no SNE exsts. However, the man weakness of the SNE-concept s that t does not mpose any consstency requrement on devatons. A devaton s deemed feasble even though ths devaton may be subject to further devatons. In contrast, coalton-proofness rules out such non-credble devatons. A coalton structure s only allowed to be challenged by self-enforcng devatons. The weakness of the CPNE-concept s that self-enforcng devatons are defned n a narrow sense: t only allows subsequent devatons by those players who devated ntally. Thus, only nternal consstency but not external consstency of devatons s ensured by ths concept. Though for future research one may want to develop a concept takng up ths concern, we are not aware of any better concept presently. However, there s no doubt that such an extenson would ntroduce a great complexty that may be dffcult to handle. 3.. Cartel Formaton Game The cartel formaton game 19 has been wdely appled n the envronmental economcs lterature (e.g., Barrett 1994b, 1997a, b, Bauer 199, Carraro/Snscalco 1991, 1993 and Hoel 199) because of ts smplcty. Its roots go back to d Aspermont et al. (1983) who used ths settng to study cartel formaton n an olgopoly. We study ths game for reference reason, though t restrcts the number of non-trval coaltons to one. That s, t s exogenously assumed that there s one group of countres (sgnatores) whch form a coalton and that all other countres (non-sgnatores) play as sngletons. The equlbrum coalton sze s found by applyng the concept of nternal&external stablty. 19 The term s taken from Bloch (1997). 18
Defnton 9: Stablty n the Cartel Formaton Game Denote the non-trval equlbrum coalton of sgnatores by c S, c S, non-sgnatores by j c S and let the coalton structure be gven by c = ( c S,1,...,1). 1) Internal Stablty: There s no ncentve for a sgnatory to leave the coalton. That s, π(c S,c) π(1,c ) 0 c S where c = c \ cs {c S \{}} {} ) External Stablty: There s no ncentve for a non-sgnatory to jon the coalton. That s, π j(cs {j},c ) π j(1,c) < 0 j cs where c = c\{{ j },{ c S }} {cs {j}}. 0 From Defnton 9 t s evdent that nternal stablty corresponds to the defnton of standalone stablty (Defnton 5). The defnton of external stablty mples de facto an open membershp rule. That s, non-sgnatores may accede to the coalton f ths s benefcal for them. From Defnton 9 t s also apparent that only sngle devatons are consdered when determnng an equlbrum. That s, an nternally and externally stable coalton structure s de facto a NE. Due to the smple structure of the game, t s straghtforward to state the followng result. Proposton 8: Equlbrum Coalton Structure n the Cartel Formaton Game Let c S be the largest coalton for whch c = ( c S,1,...,1) s stand-alone stable. Then c s the most concentrated equlbrum of the cartel formaton game. Under condton C 5 the equlbrum s unque f c = ( c S,1,...,1) s stand-alone stable for all c < S c S. For payoff functon [] c = 3 and for payoff functon [3] c {1,}. S Proof: c satsfes Defnton 9.1 because t s stand-alone stable and also Defnton 9. because (cs + 1,1,...,1) s not stand-alone stable. Assume that there s an other coalton structure c = (c S,1,...,1), cs cs that satsfes Defnton 9. From Defnton. 9.1 t follows that cs < cs. Hence c s the most concentrated equlbrum. If c = (c 1,1,...,1) s stand-alone stable for all c1 < cs, then c cannot be an equlbrum snce sngletons would jon the coalton due to Assumpton and condton C 5 untl the coalton s of sze c S. The fnal statement s an mplcaton of the results above and Proposton 5 (Q.E.D.). The result above s not terrbly nterestng snce the equlbrum number of sgnatores s ndependent of the number of countres. Thus, one may wonder whether the result changes f some of the assumptons are modfed. One possblty could be to assume an exclusve nstead of an open membershp rule. Exclusvty would mply that only f sgnatores are S 0 We use "<" nstead " " to be consstent wth Assumpton. 19
wllng to accept a non-sgnatory, an "external" player s allowed to jon ther club. However, t s easly checked that ths modfcaton has no effect n our context (under the condtons of Proposton 8). A second possblty could be to assume a Stackelberg nstead of a Nash-Cournot strategy of sgnatores n the second stage. Such an assumpton has been made by Barrett (1994b, 1997b): sgnatores maxmze the jont payoff of the coalton, takng the reacton of non-sgnatores nto consderaton. For payoff functon [] ths change has no effect snce reacton functons are orthogonal. For payoff functon [3] one fnds c s [, N]. 1 That s, the exact number of sgnatores depends on the parameter values N (total number of countres), b and c (beneft and cost parameter). The advantage of Barrett s model verson s that t can also explan IEAs whch comprse more than 3 countres and that the coalton sze can be related to the parameters of the model. It therefore allows drawng some conclusons of poltcal relevance. For example a major fndng derved from ths verson s that whenever cooperaton would be needed most from a global perspectve, a coalton acheves only lttle. The dsadvantages of ths verson are: 1) For ex-ante symmetrc countres t s dffcult to justfy why some countres (sgnatores) have more nformaton than others (non-sgnatores). ) The Stackelberg assumpton mples rratonal behavor of countres. For nstance, consder the condton of nternal stablty. As a sgnatory, a country has an nformatonal advantage as a Stackelberg leader but assumes that, provded t would become a non-sgnatory, t looses ths nformaton. A thrd possblty of a dfferent assumpton s derved by notng, as ponted out above, that Defnton 9 mples a myopc behavor of players when decdng on ther membershp. Players only consder the mmedate reacton to ther change of membershp but gnore possble chan reactons that may be trggered by ther decson. For example assume N=5 and payoff functon []. Accordng to Proposton 8 c S = 3. cs 5 snce (5, (5)) π <π (1, (4,1)) and π j(4,(4,1)) <π j(1,(3,1,1)). That s, vewng coalton formaton as a sequental process startng from the grand coalton, players leave the coalton as long as a coalton s not standalone stable. However, a player of the grand coalton who does not only look one step ahead should realze that π (5, (5)) >π (1, (3,1,1)) and therefore may refran from takng a free-rde. 1 For a comparson of equlbrum coalton structures for dfferent payoff functons under the Nash-Cournot and Stackelberg assumpton see Barrett (1997a) and Fnus (000b, ch. 13). For an extensve dscusson of ths and related ssues see Fnus (000b, ch. 13). 0
Ths feature of farsghtedness has been proposed by Carraro/Morcon (1997). We skp to gve a formal defnton of ths modfcaton snce such a coalton formaton game consttutes a specal case of the more general case of an equlbrum bndng agreement game (EBAG), whch we dscuss n subsecton 3.3.. 3 The only dfference s that here the equlbrum number of non-trval coaltons s restrcted to one, whereas n the EBAG multple coaltons are possble. In the present context, t may only be worthwhle to pont out that the equlbrum coalton structures are determned recursvely. That s, one starts by checkng whether cs = 1 s stable, whch t s by defnton. Thus, the nterm largest stable coalton s defned as c S = 1 Then one checks for c S = by computng F : =π (c S, c = (c S,1,...,1)) π (1, c = (c S,1,...,1)). If F 0, then c S: = cs =, otherwse we stll have c S = 1. For payoff functon [], F 0, and therefore cs = cs =. Also for cs = 3, F 0, and hence cs = cs = 3. For S c = 4, F<0 and thus c S = 3. However, as argued above, for c S = 5 and c S = 3, F 0 and therefore cs = 5. More generally, we have: Proposton 9: Equlbrum Bndng Agreement n the Cartel Formaton Game (Payoff Functon [] a) In the cartel formaton game an equlbrum bndng agreement s gven by c = ( c,1,...,1) where c N s determned as follows: S S( ) 1) :=0, c : = 1, c : = 1. ) c : = c + 1 S S S S 3) Let F : = π = = ( c S, c ( c S,1,...,1)) π (1,c ( c S(),1,...,1)). If F 0, then : = + 1 and c = c. As long as c < N, go to step. If c = N stop. S( ) S S Then the most concentrated equlbrum coalton structure s gven by c = ( c S,1,...,1) where c = maxc N. S S() S c S b) For payoff functon [], F (<)0 f nteger (e.g., [3.5]=3 and [3]=3). c S ( < ) 1+ cs() c S() wth [ ] the next lower c) The number of coaltons n c s I(3.5)=I(4)=4 and I(3)=3)). M I(N/3) wth I( ) the next hgher nteger (e.g., Proof: a) Obvous and therefore omtted. b) Follows from the evaluaton of F. c) s proved n Appendx 4 (Q.E.D.). 3 Though ths modfcaton should be grouped under sequental games, we dscuss t here snce the forces are best understood n connecton wth the ordnary cartel formaton game. 1
For payoff functon [] we have c S() ={1,, 3, 5, 8, 1, 18, 6, 38, 55, }. Thus for nstance for N=5, c {(1, 1, 1, 1, 1), (, 1, 1, 1), (3, 1, 1), (5)} of whch the most concentrated coalton structure s the grand coalton. 3..3 Open Membershp Game In the open membershp game of Y/Shn (1995) players can freely form coaltons as long as no outsder s excluded from jonng a coalton. Players choose ther membershp by smultaneously announcng a message m (or n the dcton of Y/Shn they "announce an address"). Players that have announced the same message form a coalton. That s, f and only f m =m j, then {} {j} ck. For nstance, f N=4 and m 1 =m =m 3 =1 and m 4 =, c={{1,, 3}, {4}} forms. If country 3 changes ts message to m 3 =, then c={{1, }, {3, 4}}. Snce a country can always leave a coalton by announcng a sngleton address, a basc prerequste for a coalton structure to qualfy as a NE n the open membershp game s that a coalton structure s stand-alone stable. For payoff functon [] ths mples that only coaltons structures c=(c 1,..., c M ) wth 3 c 1... c M qualfy as NEs. For nstance, suppose N=4, then c 1 =(3, 1), c =(, ), c 3 =(, 1, 1) and c 4 =(1, 1, 1, 1) are potental NEs. However, coalton structure c 1 cannot be a NE snce a country belongng to the coalton comprsng three countres has an ncentve to announce the same address as (to jon) the sngleton country that follows from condton C 4. Though the sngleton prefers to reman a sngleton (whch follows from condtons C 3 ), t cannot deny the accesson under the open membershp rule. Coalton structures c 3 and c 4 cannot be NEs snce, gven the announcements of the other countres, a sngleton has an ncentve to announce the same address as an other sngleton due to C 4. Coalton structure c s a NE snce no country n a coalton comprsng two countres has an ncentve to become a sngleton or to be a member of a coalton comprsng three countres CPNE NE due to C 4. Due to C C, coalton structure c s the only canddate to qualfy as a potental CPNE. It s easly checked that the only coalton structure that could challenge c s the grand coalton whch requres that two or four countres devate. However, snce the grand coalton s subject to further devatons, whch are not n the nterest of any ntal CPNE PO devator, we have C = c = (,). Snce c C (4), there s no SNE. Due to the smple structure of the game, one can derve qute general results (Y 1997 and Y/Shn 000).