Planning marine protected areas: a multiple use game

Transcription

1 1 2 3 Plannng marne protected areas: a multple use game aarten J. Punt a, *, Rolf A. Groeneveld a, Hans-Peter Wekard a, Ekko C. van Ierland a and Jan H. Stel b a Envronmental Economcs and Natural Resource Group, Wagenngen Unversty, PO Box 813, 67 EW Wagenngen. b Internatonal Centre for Integrated Assessment and Sustanable Development (ICIS), aastrcht Unversty, PO Box 616, 62 D aastrcht. * Correspondng author: Envronmental Economcs and Natural Resource Group, Wagenngen Unversty, PO Box 813, 67 EW Wagenngen. Tel: Fax: m.j.punt@gmal.com 1

2 Abstract: The EU arne Strategy Drectve has a regonal focus n ts mplementaton. The Drectve oblges countres to take multple uses and the marne strateges of neghborng countres nto account when formulatng marne strateges and when desgnatng marne protected area (PA s). We use game theoretcal analyss both to fnd the optmal sze of marne protected areas wth multple uses by multple countres, and to nvestgate the nfluences of multple use on cooperaton. To ths end we develop a model n whch two specfc uses, fsheres and nature conservaton, by multple countres are consdered n a strategc framework. The results of the paper suggest that EU marne polcy such as the arne Strategy Drectve and the comng artme Polcy may help to secure the hghest possble benefts from these PAs f these polces nduce cooperaton among countres, but only f the polces force the countres to consder all possble uses of marne protected areas. In fact cooperaton on a sngle ssue may gve a worse outcome than the non-cooperatve equlbrum. The results also ndcates that cooperaton may be hard to acheve because of defector ncentves, and therefore f the current polcy measures should be strct n enforcng cooperaton on all possble uses of PAs Keywords: arne protected areas, arne reserves, Boeconomc model, Game theory, Fsheres, Speces rchness, Speces-area curves, ultple use 2

3 Introducton The marne envronment supples several goods and ecosystem servces to socety. Yet t s also under ncreasng pressure from a varety of human actvtes, such as fsheres, ol & gas exploraton and shppng. To extract the goods and servces sustanably and to protect vulnerable ecosystems we need to manage human actvtes n the marne doman. The European Commsson s at the forefront n safeguardng and explotng ts Exclusve Economc Zones (EEZs). Ths s reflected n ts polces and new ntatves such as the Common Fsheres Polcy (EU [29a]), the Water Framework Drectve (EU [29b]), the artme Polcy (European Commsson [26]) and the arne Strategy Drectve (SD) (European Commsson [27]). All of these call for ecosystem management, a holstc vew and plannng at a regonal sea level,.e. between countres sharng a common sea, such as the North Sea. The EU SD explctly calls for the formulaton of ntegrated marne strateges by ts member states, whch should apply an ecosystem-based approach to the management of human actvtes whle enablng the sustanable use of marne goods and servces (European Commsson [27] artcle 1.1 and 1.2). Consequently we need to consder regonal seas as sngle enttes, surpassng the boundares of ndvdual countres. The necessary ntegraton of the management plans both wthn countres as well as between countres, s far from beng reached. The management plans for the ndvdual member states Exclusve Economc Zones (EEZs) are not always n concordance wth each other (Stel [23], Douvere et al. [27]) and plannng and polcymakng at the natonal level s often fragmented snce responsbltes for dfferent actvtes are dvded between dfferent organzatons, nsttutons and 3

4 mnstres (Stel [22]). The fragmentaton favors an atttude n whch the effects of human actvtes are consdered n solaton, whereas the effects are actually nterdependent and cumulatve (Ellott [22], ICES [23], de la are [25]). To manage our seas sustanably we need to develop tools and polcy nstruments that are capable of achevng goals that have been set at the EU level, and mtgate the fragmentaton at lower levels. arne Protected Areas (PAs) may be such a tool. They have been proposed for fsheres management for a long tme (Guénette et al. [1998]) and more recently also as a tool to tackle bodversty conservaton, ecosystem restoraton, regulaton of tourst actvtes and as an example of ntegrated coastal management f all of these are ncluded (Jones [22]). PAs have lmtatons, due to ther nherent statc and delneated nature versus the open and dynamc nature of the marne envronment (Allson et al. [1998]). Economsts are often skeptcal about the effects of marne reserves as fsheres management tools (e.g. Hannesson [1998], Anderson [22]). The man reasons for desgnatng PA s that has been advocated by economsts are uncertantes and shocks (e.g. Lauck et al. [1998], Sumala [22]). any such analyses, however, focus on fshng effects on a sngle stock under open access, and exclude the other effects of marne reserves such as bodversty conservaton. oreover fshng has a destructve effect on habtats (Jennngs and Kaser [1998], Jennngs et al. [21], Auster et al. [1996], Armstrong and Falk-Petersen [28]) and hence areas that are protected and where no fshng occurs may have a postve effect on the growth rate of the fsh stock outsde the reserve through habtat enhancement and the preservaton of the nursery functon of the reserve (Armstrong [27], Armstrong and Falk-Petersen [28], Schner [25a], Rodwell et al. [22]). 4

5 The multple uses of PAs and ther mpacts on the marne ecosystem requre full cooperaton among all countres that share a regonal sea such as the North Sea. On the one hand, strategc nteracton and sub-optmal polcy outcomes may occur because n general no central authorty exsts that can enforce cooperaton on these ssues. On the other hand, f the varous functons of marne protected areas (e.g. n terms of fsheres and nature conservaton) are lnked, the advantages of cooperaton may ncrease n such a way that self-enforcng agreements can be formed. In ths paper we wll analyze the problem of multple use PAs by multple countres usng game theory. Ths provdes nsghts nto the functonng of marne protected areas as a polcy nstrument for the SD and the artme Polcy Drectve. ore specfcally, we wll examne the sze of PAs that countres assgn, when these countres account for the effects of PAs on fsheres and speces conservaton separately or jontly and nvestgate the effect of playng cooperatvely versus Nash equlbrum on a sngle ssue when multple ssues are at stake. Game theory has consdered the strategc nteracton between countres n a general fsheres context (see e.g. unro [1979], Levhar and rman [198], Hämälänen et al. [1985], Vsle [1987], Hannesson [1997], Arnason et al. [2], Bjørndal and Lndroos [24], Kronbak and Lndroos [27]). The related problems of publc good provson, nternatonal envronmental agreements and enforcement has also been consdered both n terms of transboundary polluton abatement and coaltons (see e.g. äler [1989], äler and De Zeeuw [1998], Fnus [23], Wekard et al. [26], Fnus [26] and Nagashma, et al. [29]) and n terms of possble coaltons for fsheres management (Pntasslgo [23], Pntasslgo et al. [28], Pntasslgo and Lndroos [28]). Game theoretc treatments of (marne) protected areas have receved less attenton so far. Sanchrco and Wlen [21] and Beatte et al. [22] study strategc 5

6 nteracton, both between fshermen and between fshermen and polcymakers. Sumala [22] devses a computatonal model of assgnng an PA as a dfferental game between two agents. Rujs and Janmaat [27] have studed the strategc postonng of PAs. They consder two countres, the locaton of PAs and the effect of dfferent mgratory regmes n a dfferental game. Busch [28] derves some general condtons from game theory for terrestral transboundary reserves to be superor over solated reserves. Our paper contrbutes to the lterature as t consders the combnaton of multple use of protected areas wth multple agents: we examne cooperaton and defector ncentves when multple agents are present n a multple use settng,.e. by consderng mpacts on fsheres and nature conservaton. Furthermore the fsheres PA model s mproved to accommodate the habtat enhancement effects of PAs, and a conservaton game s ntroduced, that uses standard ecologcal functons to model speces rchness and consequent conservaton benefts. In secton two we present a game theoretc model that nvestgates the ssue of multple use PAs n a multple country settng by developng two separate models of PAs n a multple country settng: one model for the fsheres case and one model for the speces conservaton case. Next we lnk the separate models and nvestgate the mpacts of ths lnkage of fsheres and nature conservaton. In secton three we provde a numercal example and secton four concludes odel descrpton In our model we consder a regonal sea, such as the North sea, that s completely clamed by a number of countres. These countres have dvded the sea n Exclusve Economc Zones of equal sze. In ths sea there s only one fsh speces that s of commercal nterest for fsheres, and ths fsh speces conssts of a sngle stock. The 6

7 other fsh speces, as well as mammals and benthos have no commercal value, only exstence value. We gnore effects of tme and space, to focus exclusvely on the basc mechansms. Furthermore we are nterested n the steady state and not so much n the path towards that steady state. Our fsheres model descrbes optmal harvestng of fsh by a number of countres n a common sea wth a sngle stock. We analyze the mpact of establshng an PA n a context where each country has a fxed share n the fshng area. In the fsheres model the cost of the establshment of an PA s a reducton n harvest proportonal to the PA sze 1. We assume that assgnng an PA from a fsheres perspectve ncurs no costs other then the opportunty costs of forgone harvest. For smplcty montorng and complance costs are neglected. The gans of a country consst of an ncrease n the growth rate of the shared stock owng to an ncrease n habtat qualty n the unfshed area. Such an ncrease cannot be reached by conventonal harvest restrctons because t would only reduce the overall fshng pressure but not release one area completely, from fshng pressures, to recover habtat. Ths habtat effect of PAs s a publc good snce a sngle country bears the cost whle all countres beneft. The nature conservaton model descrbes conservaton efforts by a number of countres n the same common sea. We analyze the mpact of the total sze of an PA on the number of speces protected and the resultng costs and benefts. Each country s PA s a contrbuton to the total protected area, but the added benefts derved from the extra speces protected by ths PA accrue to all countres, makng ths another publc goods ssue. 7

8 The game on marne protected areas as fsheres management tool Our fsheres model assumes that there s only one fsh stock that s worth harvestng from a commercal pont of vew. Hence the set N of n symmetrc countres optmzes the harvest from a sngle stock, gnorng other stocks and speces. If countres cooperate they maxmze the sum of ther profts, f they defect they optmze ther own profts. Country s profts Π depend on the harvest and the ncurred costs. We use a supply sde model that assumes that the full harvest can be sold at a fxed prce p 2 : Π = ph C N (1) where H s the total harvest of country and C are the total costs of country. We assume that countres are fshng a sngle fsh stock that s unformly mxed over the fshng grounds and we model the growth of the stock wth a modfed Schaefer producton functon. We assume that the sea s completely clamed by EEZ s and that all EEZs are of equal sze. The total sze of the sea s normalzed to one, and consequently each country has an EEZ of sze 1. Each country s fshng ground s n ts EEZ. In ths model of the fsheres t s assumed that n equlbrum the harvest equals the growth of the fsh stock. The sze of the marne protected area affects both the harvest and the growth rate of the fsh stock but for smplcty we assume that the locaton of the reserve does not matter. The total growth of the fsh stock s modeled wth a modfed logstc growth functon scaled such that the carryng capacty equals one. Hence the maxmum stock sze s also one. Ths growth functon s: 8

9 (, ) ( 1 ) G X = R X X (2) wth R() the nternal growth rate of the stock, X 1, and 1 the total area protected as marne reserve. Settng asde a share of the fshng ground as a marne reserve has a postve effect on the growth rate R through enhancng the growth rate n the reserve. We model ths enhancement as a coupled producton functon: the nternal growth rate s r b n the unprotected area and (r b + r ) n the protected area. Ths assumpton s based on ncreased recrutment that s acheved by ncreasng the spawnng bomass through the absence of fshng n senstve areas. Smlar functonal forms for marne protected area modelng have been used by e.g. Schner ([25a], [25b]), who also modfes the growth rate and Armstrong [27] who modfes the carryng capacty as an effect of PAs. If we further assume that both stock and carryng capacty n the protected and unprotected area are proportonal to area we get: 189 ( 1 ) ( 1 ) = ( r + r ) X ( 1 X ) X X G X, = rb 1 X 1 + ( rb + r ) X 1 b = 1 rb + r n X X ( 1 ) (3) wth 1 s the share of the PA of an ndvdual country s EEZ. The sum of ndvdual PAs multpled wth EEZ sze equals the total PA, because each EEZ s of sze 1. Consequently each ndvdual PA s scaled by ts relatve sze n the sea n such that the sum equals the total protected area. To model the harvest we use a modfed Schaefer harvest functon: H = Q EX = nq E X (4) 9

10 wth Q() the catchablty, E the total effort level and E the effort level of player. In a standard Schaefer functon catchablty s a parameter, but n our model catchablty s a decreasng functon of. We use the followng functons for Q() and Q( ): 199 = o = ( 1 n) ( 1 n) Q q q Q q q o (5) where q o s the orgnal catchablty and q s the catchablty reducton caused by the protected area. Costs are assumed to be constant per unt of effort: C = c E (6) E Full cooperaton on PAs for fsheres When players fully cooperate they maxmze the sum of total profts: max Π = max Π (7) N wth Π the total profts. To obtan the steady state of the model we set the total growth equal to the harvest and solve for the effort level 3. Under full cooperaton the harvest s equal to the growth: H = G (, X ) R ( ) X ( 1 X ) = Q( ) EX X = 1 Q E (8) R Usng the harvest functon from (4) and substtutng the equlbrum stock gven n (8) we get the objectve functon: 214 Q E Π ( E, ) = ph cee = pq( ) E 1 cee R (9) wth the total sze of the PA. Takng the Frst Order Condton wth respect to effort we obtan 4 : 1

11 Π Q E = pq ( ) 1 2 ce = E R If we substtute the equlbrum stock gven n (8) back nto equaton (1), we get: pq 2X 1 c E = argnal beneft argnal cost (1) (11) whch dsplays the standard components of a FOC n a statc fsheres model: the frst part of the dervatve s the margnal beneft: an ncrease n effort ncreases the catch f X > 1 2, and ths addtonal catch s valued at p. The second part s the margnal cost: an extra unt of effort costs c E. Smlarly, the frst order condton wth respect to s gven by: 225 ( ) ( ) Π qo q E pr qo q E = + 2 pq E peq = rb + r rb + r (12) 226 If we substtute the equlbrum stock gven (8) n back nto equaton (12), we get: pr 1 X pq E 2X 1 = argnal beneft argnal cost (13) Ths clearly llustrates the effect of PAs. The frst part of the dervatve s the margnal beneft of an PA: an extra unt of protected area ncreases the growth and consequently the harvest by r ( 1 X ) 2 whch s n turn valued at p. The second part shows the margnal cost of such an area n the form of the forgone harvest n an extra unt of protected area, also valued at p. Solvng equaton (1) and (12) smultaneously we get a cubc equaton whch can be factorzed nto a quadratc and lnear part and solved: 11

12 235 * o E * * pq c =, E = pq = ( 2pqo + ce ) r ± cer ( r ( 8pqo + ce ) + 8pqrb ) 2 pq r cEq cer ( r ( 8pqo + ce ) + 8 pq rb ) ± ( 4 pqor + cer + 4 pqmrb ) pq * or + cer + pq rb cer r pqo + ce + pq rb ± pceqor + cer + pceq rb r E =, (14) The frst soluton s a corner soluton, and dependng on parameter values the other two may be corner solutons as well. To evaluate the nfluence of ndvdual parameters on the equlbrum protected area we use the mplct functon theorem and the frst order condtons (1) and (12) to determne the sgns of ther dervatves. The dervatves are shown n appendx I, the sgns n Table 1. Table 1: Sgns of the dervatves of equlbrum PA under full cooperaton wth respect to parameters n the fsheres game Dervatve p c E r r b q q o Sgn + - Und. - - Und Und. = Undetermned. All relatons are derved from the mplct functon theorem.assumptons used for sgns above: All parameters >, 1, Q() 1. ost sgns of the dervatves can be determned wth excepton of the dervatves wth respect to r and q o. The sgns that can be determned have the expected sgn. An ncrease n prce of fsh (p) would make harvestng more worthwhle, and thus the protected area s ncreased to ncrease the harvest. An ncrease n the cost of effort (c E ) makes a protected area more expensve because the protected area decreases the effectveness of effort through the catchablty. Therefore the protected area s decreased when cost of effort ncreases. Smlarly an ncrease n r b or q decreases protected area as they make t less effectve by ncreasng the harvest that has to be gven up to protect an area. The sgns 254 of r and qo are manly determned by the dfference between prce of fsh 12

13 and cost of effort and the parameters q and q o. If p>>c E and Q() s not too small, both sgns of the dervatves are postve. Ths s n lne wth expectatons: f the prce of fsh s large relatve to the cost of catchng t, an PA pays off: the added growth bonus and thus extra fsh to catch s more valuable than the relatve mnor costs of extra effort needed to catch that fsh. A larger growth bonus (r ) makes the PA more valuable, because t ncreases the extra avalable catch. Smlarly a larger orgnal catchablty (q o ) makes the PA even cheaper because the reducng effect on the harvest of the PA s smaller The Nash equlbrum for PAs for fsheres In a Nash equlbrum each ndvdual player wshes to maxmze hs own fsheres profts, gven that other players also maxmze ther profts. We assume that the sum of the harvest of all players s equal to the growth. The stock expressed n terms of effort and marne protected area as they can be nfluenced by an ndvdual player s: ( 1) R(, j ) H, E = G, X Q E X = R, X 1 X j = 1 Q E n Q j E j X = 1 N, j N (15) wth j all other players. For a sngle player the optmzaton problem s then: + ( 1) R (, j ) Q E n Q j E j max Π = max pe 1 cee N (16) axmzng the ndvdual proft functon n (16) s not the same as the open access regme. Even though countres only optmze ther own effort, no new entrants area allowed, thus the rent s not drven to zero. The frst order condtons wth respect to E s: 13

14 276 + ( 1) 2 Π Q E n Q j E j peq = pq ( ) 1 ce = E R, j R (, j ) (17) 277 If we substtute the equlbrum stock gven n (15) back nto equaton (17), we get: 278 ( 1) ( j ) n Q E j pq( ) 2X 1 c E = R argnal cost argnal beneft (18) whch dsplays the standard components of a FOC n a statc fsheres model: the frst part of the dervatve s the margnal beneft: an ncrease n effort ncreases the catch f X s larger than ½ plus the catch of the other players, and ths addtonal catch s valued at p. The second part s the margnal cost: an extra unt of effort costs c E. The FOC of the marne protected area n equaton (16) results n: ( n n ) ( n n ) + ( 1)( n n ) 1 n 1 b ( j ) Π qo q E n qo q j E j = 1 n pq E 1 + r + r n n q E ( + n 1 ( n n )) ( n n ) + ( 1)( n n ) 2 1 n ( b + ( j )) r q q E n q q E o o j j pe qo q + 1 n r 1 b r j n r r n n (19) If we substtute the equlbrum stock gven n equaton (15) back nto equaton (19) we get: 287 argnal beneft ( ) ( ) n 1 Q j E j n 1 Q j E j pr ( 1 X ) 1 X pq E 2X 1 = R( ) R( ) argnal cost (2) whch s smlar to the full cooperaton case but now ncludes terms for the other players. The frst part of the dervatve s the margnal beneft of an PA: an extra unt of protected area ncreases the growth and consequently the harvest by 291 r ( 1 X ) 1 X ( 1) ( j ) n Q E R j where the last term accounts for the amount harvested by the other players. The extra harvest s n turn valued at p. The second part shows the margnal cost of protected area n the form of the forgone harvest 14

15 (agan wth a modfer for the other players) n that extra unt of protected area, also valued at p. Solvng equatons (17) and (19) smultaneously for n symmetrc players we get another cubc equaton whch can be factorzed nto a quadratc and lnear part and solved: pq nc =, E = E * o E * pq * * = ( 2 pqo + cen ) r ± r ( p ( 4cEn( n + 1) )( qor + q rb ) + cen r ) 2 pq r ( 4 1 ) ( 2 ( 1) ) ( 1) (( 2 pqo + cen + cen) r + pq rb ) r ( p ( 4cEn( n + 1) )( qor + q rb ) + cen r ) 2 ( ( ) ) ( r ( p ( 3cEn + 4cEn)( qor + q rb ) + cen ( n + 1) )) 2 ( ( ) ) = ± q c n + c n r p c n n + q r + q r + c n r ± q p c n + q r + q r + r c n n E E E o b E E o b E ( 4 1 ) ( 2 ( 1) ) ( 1) q c n + c n r p c n n + q r + q r + c n r ± q p c n + q r + q r + r c n n E E E o b E E o b E (21) To evaluate the nfluence of ndvdual parameters on the equlbrum protected area we appled the same procedure as under full cooperaton, usng the mplct functon theorem and equatons (17) and (19). The results are the same as for full cooperaton (Table 2, and appendx I). Table 2: Sgns of the dervatves of equlbrum PA under the Nash equlbrum wth respect to parameters n the fsheres game Dervatve p c E r r b q q o Sgn + - Und. - - Und Und. = Undetermned. Assumptons used for sgns above: All parameters >, 1, Q( ) 1. The dfference between the protected area under full cooperaton and under the Nash equlbrum s: FC N = ( 4 ( 1 ) ) 3 8 ( 2 1 ) c r n p n + q r + q r + c n r p q r + q r + c r c r n (22) E o b E o b E E 2 pq r 15

16 where FC s the total area set asde under full cooperaton and N s the total area set asde under the Nash equlbrum. Unfortunately, gven our prevous assumptons (all parameters >, and n 2), the above dfference s unclear on the effect of ndvdual parameters on the dfference between protected area under full cooperaton and the Nash equlbrum. Two exceptons are the parameters r b and q o, both of whch ncrease the dfference. For other parameters we have to resort to smulatons. The same holds for the dfference n payoffs between full cooperaton and Nash equlbrum The game on marne protected areas for conservaton Conservaton s a man goal of marne protected areas. In ths game we measure conservaton success as the speces rchness attaned n a marne protected area. We do not model the populatons of all speces ndependently but nstead use the speces-area relatonshp to determne the number of speces that a reserve contans. The speces-area relatonshp (SPAR) has frst been put forward by Arrhenus [1921] 326 and s a curve of the general form S z = ka wth S the number of speces, A the area and k and z two parameters. It s explaned by ether the passve-samplng effect (acarthur and Wlson [1967]) or the habtat dversty hypothess (Wllams [1943]). The use of ths relatonshp has been crtczed for ts scale-ndependent applcaton and extrapolaton (Letner and Rosenzweg [1997], Rosenzweg [25]), and ts ambguous role n conservaton decsons n the Sngle Large or Several Small (SLOSS) debate (Smberloff and Abele [1976]). Despte these crtcsms SPARs can stll be used as a predctor of local speces rchness, f one accounts for the scale (Negel [23], Rosenzweg [25]). Furthermore even though SPARs may support ether several small reserves or a sngle large reserve, dependng on parameters, n our 16

17 case we assume a unform sea, mplyng that the same speces would be protected n several small reserves, and therefore we apply a SPAR on a sngle large reserve 6. For mathematcal convenence we transform the speces-area relatonshp nto a log-log relatonshp,.e. ln S = ln k + z ln A. We further assume that costs rse lnear wth the area set asde and that countres beneft from the (log) number of speces n the total area set asde. Each country has an ncentve to set some area asde, but gven that the others wll also set asde some area n a Nash equlbrum, each wll set asde less. Assumng that all areas are nterlnked and hence form one bg conservaton protected area, the total benefts of speces conservaton are: ( ln ) ( ln ln ) BD = b S c = b k + z c (23) P P P P Here b P represents the margnal global benefts from the protecton of the log number of speces, ln S, k and z are the parameters of the speces area curve and c P s the cost of protectng an area, such as montorng and enforcement costs or opportunty costs for other uses Full cooperaton on PAs for nature conservaton Under full cooperaton global welfare s maxmzed, accountng for beneft generaton n all countres. The countres maxmze: ln ln BD = b k + z c (24) P P wth agan the total protected area. Ths results n the followng frst order condton: bp z b z * P cp = = (25) cp The full cooperaton optmal PA sze s ndependent of the number of players, as they consder the protecton of the full sea, and only one optmum exsts, although the 17

18 soluton s slent on how ths should be reached. Gven symmetrc costs and benefts a far soluton would be for all countres to have an PA of sze 1 n *, resultng n a total protected area of * for the whole area The Nash equlbrum for PAs for nature conservaton In a Nash equlbrum, each country maxmzes ts prvate welfare functon, assumng all other countres also maxmze ther prvate welfare functons. We assume each country gets benefts proportonal to ther EEZ sze, 1/n. Hence the global margnal benefts of protecton accrue to the players n equal shares. Each country hence maxmzes: ln ln n 1 BD = b k + z + c N (26) n P n n j P n For an nteror soluton, the frst order condton becomes: 37 ( ) b z c n c = = N n c b * 1 Pz P P j P + ( 1) j P (27) Agan we see the strategc settng of the game from the frst order condton. The addtonal beneft of an extra unt of area s n nomnator, but t s scaled by the total area already protected. The costs of protectng one unt extra are (1/n)c P. Solvng the FOC for * for n players smultaneously, for a symmetrc soluton we get: b z = (28) * P ncp whch goes to zero for n clearly llustratng the suboptmalty of ths Nash equlbrum The dfference n PA assgned between the full cooperaton and the equlbrum s: Nash 18

19 381 b z 1 = 1 c n FC N P P (29) The dfference n PA sze between the full cooperaton case and the Nash equlbrum s ncreasng both n the number of players and n the gans of the PA. The dfference n payoff between full cooperaton and the Nash equlbrum s: b z = 1 ( ln 1+ n) (3) n FC N p W W n For the gans of an PA we see a smlar pattern: the dfference ncreases wth the gans of a protected area. If we look at the number of players however, we see that f n the dfference becomes. Ths s because the fxed payoff from a protected area under full cooperaton has to be dvded over a large number of players, leavng almost nothng for each ndvdual player, whereas each player assgns a very small protected area under the Nash equlbrum, wth both lttle revenues and costs. Consequently f we assume that cooperaton s easer f the dfferences n assgned sze are smaller and dfferences n payoff are smaller, n the nature conservaton case changes of cooperaton decrease wth ncreasng payoff of PAs (.e. an ncrease n b p or z or a decrease n c P ). If the number of players ncreases to a very large number however, cooperaton may become easer, although not necessarly so as the dfference n assgned PA becomes larger even though the payoffs become smlar Combnng the games arne protected areas affect fsheres as well as nature conservaton. In order to see how the equlbra change f we take both nto account, we now couple the games. If we combne the two games, the payoff functons of both countres change. The optmal PA sze, both from a Nash perspectve and full cooperaton s perspectve may change as well, snce we now add the two functons together. Formally, ths s 19

20 not ssue lnkage as n e.g. Folmer, van ouche and Ragland ([1993]), Barrett ([1997]) and Buchner et al. ([25]), because we are not dealng wth two separate problems that are addressed wth two dfferent strategc nstruments but wth two separate problems that are addresses wth one nstrument: the sze of the PA. When countres combne both problems the objectve functon under full cooperaton becomes: Q E W ( E, ) = pq( ) E 1 c E + b ( ln ( k ) + z ln ) c R E P P Whereas the objectve functon of an ndvdual country s: + ( 1) R (, j ) ln n 1 Q E n Q j E j W ( E, ) = pe 1 cee + ( ln ) b k + z + c N n P n n j n P The frst order condtons of both problems are respectvely: ( ) ( ) W qo q E pr qo q E bp z = + 2 pq E peq + cp = rb r rb r ( n n ) + ( 1)( n n ) pe 1 n 1 rb r ( n n j ) W o o j j = ( n n ) q q q E n q q E q q q E ( + n 1 ( n n )) n 1 n n 1 n 2 1 n b + ( j ) r q q E n q q E o o o j j pe + 1 n r 1 b r j n r r n n 1 n 1 ( n) 2 bp z + 1 n cp = N + n 1 n j q n (31) (32) (33) (34) In these two FOC we recognze the frst order condtons of the separate problems. The frst order condtons wth respect to effort are of course the same as under (1) and (18). Solvng (1) and (18) for effort and substtutng the results nto (33) and (34) yelds two quartc equatons that cannot be solved analytcally. The dervatves wth respect to parameters that can be determned wth the mplct functon theorem 2

21 are also nconclusve on the effects of parameters, unless we assume specfc values. Therefore we have to resort to smulatons. The effect of the combnaton s a combned PA that s an average between the equlbrum PAs of the separated games. As the two separate games are completely separate except for the number of players (no parameters of one game occur n the other) and both are smlar publc good games t stands to reason that the combned game suffers from the same flaws,.e. t s another publc goods game. That does not mean that ts externaltes are as bad as n the separate games. As the combned game s an average the Nash equlbrum s better then the Nash equlbrum of one of the games. We now show the effect of these combned functons n a numercal example Numercal example We now present a numercal example wth two countres, as an llustraton of the more general case. Suppose we have a sea that s fully clamed by EEZs and shared equally between two countres. Intal parameter settngs are shown n table 3. The current parameter values are selected to llustrate the functonng of the model and n a senstvty analyss we study the mpacts of devaton from these parameter values. Table 3: Arbtrary parameter values for the numercal example Parameter Value Unt p Prce 25 Euro per unt of harvest c E Cost per unt of effort 5 Euro per unt of effort r b Basc growth rate.2 - r Growth rate change PAs.8 - b p Benefts of protecton 1 Euro per log of speces number 21

22 k Speces-area curve constant z Speces-area curve exponent.2 - c P Costs of protecton.25 Euro per unt of PA Fsheres: Full cooperaton versus Nash Solvng the countres maxmzaton problem for and X, for the gven parameters usng (1) and (12) gves a value of 38.6% for the total area. How ths s dstrbuted s not relevant from the full cooperaton s perspectve, t can be 38.6% for both countres or 77.2% n one country and none n the other or any other combnaton that generates 38.6% of the total area protected. A logcal choce would be to share everythng equally and set 38.6% of the area apart n both countres. In Nash equlbrum the countres optmze ther prvate ncome. We have plotted the two response functons n Fgure 1. As can be seen from the fgure the sze of PA share each country sets apart s much smaller than under full cooperaton. Each country desgnates about 11.2% as PA, and hence 11.2% of the total area s protected. Table 4 shows the outcomes per country under full cooperaton and n a Nash equlbrum. Table 4: Full cooperaton vs. Nash Outcome for a sngle player Varable Full cooperaton Nash 38.6% 11.2% Π H X

23 .5 PA Share Country Country 1 Country PA Share Country Fgure 1: PA szes (n share of EEZ) of countres 1 and 2 as a functon of the PA sze (n share of EEZ) n the other country and the resultng Nash equlbrum of the fsheres game As can be seen from the table the outcomes of the full cooperaton are much more favorable for both countres than playng Nash, but unless a socal planner steps n or they reach some agreement by barganng the Nash outcome s gong to preval Conservaton: Full cooperaton versus Nash The conservaton game for a two player game s smlar to the fsheres game. Snce the response functons have a slope of -1 n the symmetrc case, the curves overlap completely mplyng an nfnte number of Nash equlbra. If we enforce symmetry n the outcomes as s done n (28), we are left wth one Nash equlbrum. Applyng the formulas (25) and (28) and the parameter values from table 1 we get PA szes of.4 and.8 for the Nash equlbrum and full cooperaton, respectvely. To llustrate the prsoner s dlemma of the countres, the correspondng payoffs of full cooperaton, Nash equlbrum and ther combnaton are shown n Table 5. 23

24 The table shows that the game s a basc prsoner s dlemma. If the two countres manage to coordnate ther actons they ncrease socety s total welfare, but the full cooperaton scenaro wll not be reached snce both countres can ncrease ther own payoff by defectng. The game shown here, however, does not show the full pcture, the current prsoner s dlemma may be less severe f both countres take a combned perspectve as s shown n the next secton Table 5: Payoffs n the conservaton game Country 2 Cooperate Defect Country 1 Cooperate.8,.8.1,.11 Defect.11,.1.6, Couplng both games If we couple the games we get the objectve functons as shown n secton 2.3. For the parameter values gven n Table 3 we get the response functons as shown n Fgure 2. The Nash equlbrum (as can be seen from the fgure) s at an PA of 16%, whereas the full cooperaton s optmum les at 4%. 24

25 .6 PA Share Country Country 1 Country PA Share Country Fgure 2: PA szes (n share of EEZ) of countres 1 and 2 as a functon of the PA sze (n share of EEZ) n the other country and the resultng Nash equlbrum of the combned games In table 6 some results are shown to compare the full cooperaton s outcome wth the Nash equlbrum. The results are smlar n the sense that n the combned case the Nash equlbrum s clearly not preferred from the pont of vew of socety, but n absence of a socal planner, sde payments or other mechansms ths s where socety wll end up f countres do not nternalze the postve externaltes. Table 6: Full cooperaton vs. Nash outcomes for a sngle player n the combned game Full cooperaton Nash Equlbrum 4% 16% W Π BD.6 ln S.22.4 H

26 X In fgure 3 we show the payoff to a player from assgnng an PA share, gven that the other player plays the optmum of the same strategy, and we marked the respectve Nash and Full cooperaton equlbra of the separate fsheres and conservaton games. Fgure 3 llustrates the suboptmalty of consderng the problem of PA sze n solaton. In ths partcular case the PA from a pure fsheres perspectve s to low and from a conservaton perspectve to hgh compared to the optmal PA that takes both nto account. It s therefore mperatve to combne the two, snce n specal cases only, when the optmal fsheres and conservaton PA sze concde, the actual optmum s the same n the combned functon and the separate games. Another nterestng fndng n fgure 3 s that from a socety s pont of vew cooperaton on a sngle ssue can be worse than playng Nash on that sngle ssue. Consder the payoff n the equlbrum where countres cooperate only on speces conservaton. We can see from fgure 3 that t s actually lower than the payoff of the Nash equlbrum of conservaton. If the cooperatve soluton from the conservaton game s appled the losses of the fsheres are so large that they undo all the gans from conservaton. Whether ths happens of course depends on parameter values, but the possblty s enough to demonstrate the necessty of consderng multple uses n ths multple country settng. As stated before snce the externaltes run n the same drectons n both games combnng the games does not remove defecton ncentves. What we can see from fgure 3 s that from a fsheres pont of vew combnng the games consttutes an mprovement. Furthermore the dfference between the socal optmum and the Nash equlbrum has also become smaller gvng at least some mprovement. 26

27 Payoff Fsheres equlbra Combned equlbra Full Cooperaton Nash.3.2 Conservaton equlbra PA Share Fgure 3: The total payoff of one country and the respectve equlbra of the solated games, as a functon of a) the PA sze chosen by both countres (as a share of EEZ) under full cooperaton, b) the PA sze chosen by one country (as a share of EEZ) assumng that that the other player plays Nash. For parameter values see table Senstvty analyss To llustrate the model further and provde some more nsghts nto the ncentves for cooperaton we have carred out a senstvty analyss all the parameters n the combned model. For each parameter we calculated the values of PA share and payoff for the change of a sngle parameter n steps of 1%, over a range of mnus 5% to plus 5%, whle keepng the other parameters fxed. We used the same procedure for the fsheres model to calculate the effect of parameters on the dfference between full cooperaton and Nash equlbrum because these dfferences could not be analyzed analytcally. 27

28 Dfferences between full cooperaton and Nash equlbrum n the fsheres game In Fgure 4 we show how the absolute dfference between full cooperaton and Nash equlbrum n the fsheres game both for the payoff ( Π share( FC N FC Π ) and PA ), as a result of changng the prce of fsh (p) and changng the growth bonus n the PA (r ), gven that all other parameters reman on the base level. The graphs for the other parameters are n Appendx II and the relaton between the dfference between full cooperaton and Nash equlbrum n payoff and PA share as a functon of parameters are n table 7. Table 7: The relaton between parameters and the dfference between full cooperaton and the Nash equlbrum for payoff and PA share n the fsheres game N p c E r b r q o q W A + ndcates that the parameter and dfference move n the same drecton, a - ndcates that the parameter and dfference move n opposte drectons. The effect of parameters on the dfference between full cooperaton and Nash equlbrum vares. Assumng that cooperaton becomes more lkely when ths dfference s smaller, shows that only parameter q has an unequvocal dmnshng effect on these dfferences, when t ncreases. All other parameters ncrease ether the dfference n payoff or the dfference n PA share (table 7). The same effect can be seen n Fgure 4 for two specfc parameters: f the prce of fsh (p) ncreases, the value of the harvest ncreases and ths drves up the payoff under cooperaton more than t drves up the payoff under the Nash equlbrum, hence the ncreasng dfference n payoff. The ncentve to set asde more area as 28

29 PA s also larger even under the Nash equlbrum and thus the dfference n total area set asde decreases Dfference Dfference n Payoff Dfference n Prce of fsh (p) Dfference Dfference n Payoff Dfference n Growth bonus n PA (r) Fgure 4: The dfference between full cooperaton and Nash equlbrum n payoff and PA share, as a functon of changng parameters. The same holds for the growth bonus n the PA (r ). If r ncreases the payoff of the PA becomes larger, but ths payoff s reaped to a larger extent under full cooperaton hence the ncrease n the payoff dfference. The growth bonus also ncreases the ncentves to set asde PA, so the sze assgned under the Nash 29

30 equlbrum becomes relatvely larger decreasng the dfference wth full cooperaton n PA sze assgned Senstvty analyss of the combned game In Fgure 5 we show how payoff and PA share change as a result of changng the reducton n catchablty (q ) and changng the curvature of the speces-area curve (z), gven that all other parameters reman on the base level. The graphs for the other parameters are n Appendx II and the relaton between parameters and payoff and PA share are n table 8. Table 8: The relaton between parameters, payoffs and PA share p c E r b r q o q b g z c P W A + ndcates that the parameter and varable move n the same drecton, a - ndcates that the parameter and varable move n opposte drectons. Increasng q causes both the PA share and the payoff to decrease, and ths decrease s larger for the full cooperaton case than for the Nash equlbrum (Fgure 5). If q ncreases an PA becomes more expensve, because more harvest has to be gven up per unt of protected area, hence the decrease. If we once agan assume that cooperaton s easer when there s less at stake,.e. the dfferences between payoff and PA share are smaller, then cooperaton becomes more lkely wth ncreasng q. Increasng the curvature of the speces-area curve (z), causes the PA share to ncrease but payoff falls, because we get the same beneft for more speces. In Fgure 4 we see that the effect of the z parameter, ceters parbus, has lttle effect on the dfference n equlbrum PA between the full cooperaton case and the Nash 3

31 equlbrum. Hence the dffculty of reachng a socal optmum stays just as hard n a less dverse envronment as t s n a rch envronment. Overall the dfferences n payoff and PA between the full cooperaton case and the Nash equlbrum seem to become smaller only wth decreasng gans of PAs. PA share (), Payoff per player (W) Catchablty reducton (qm) FC Payoff FC PA NP Payoff NP PA PA share (), Payoff per player (W) Curvature of the speces-area curve (z) FC Payoff FC PA NP Payoff NP PA Fgure 5: PA and payoff under full cooperaton (FC) and Nash equlbrum (NP) as a functon of parameters q and z, keepng other parameters at the base level. 31

32 Payoff per player Growth outsde PA (rb) Total payoff Fsheres payoff Conservaton payoff Fgure 6: Payoff per player n total and from the separate games as a functon of r b, keepng other parameters at the base level. Fgure 6 provdes ntuton on the mechansms n the model: ncreasng the growth outsde the PA decreases the necessty of an PA from a fsheres perspectve. oreover t ncreases the harvest and consequently the payoff of the fsheres. Therefore the optmum of the combned game moves towards the fsheres optmum and away from the conservaton optmum. As the optmum moves away from the pure conservaton optmum, the profts of conservaton fall slghtly, but ths s made up for by the fsheres profts. Increasng the growth outsde the PA consequently has the expected effect of weghng the fsheres nterests heaver then before thus movng towards the fsheres optmum. 6 32

33 61 4 Dscusson and conclusons an fndngs We now present our man fndngs and then contnue wth some lmtatons of the study and suggestons for further research. In ths paper we consdered the effects of PAs on fsheres and speces conservaton n a mult-country settng. We analyzed the resultng externaltes and the possble shfts n equlbra f two separate games of PA assgnment are combned. We focused on the man ssues and developed a model that contans some of the mportant aspects that arse n the multple country, multple use settng that PA planners, and marne polcy makers n general, face. Although the separate games do not provde major new nsghts, the combnaton of the games provdes a counterntutve result: cooperaton on a sngle ssue may be worse than Nash f we take a combned perspectve. The settng of the fsheres model s such that f PAs ncrease growth rates by more then the reduced harvest t pays to set asde some area. The results suggest that cooperaton s better than Nash, but ths s straghtforward. Furthermore the results show that the dfference n PA sze assgned between cooperaton and Nash equlbrum reduces when ether the growth bonus of PA ncreases, or when the prce of fsh ncreases. However, the payoff dfference between full cooperaton and Nash equlbrum also ncreases wth these parameters offerng lttle scope for barganng. If the catchablty reducton of PA ncreases both the dfference n assgned PA and payoff go down. Essentally f PAs become worth less, the dfferences between full cooperaton and Nash equlbrum decrease. The conservaton game offers a new approach to the conservaton problem. Although game theory has been used to analyze transboundary parks n general 33

34 (Busch [28]), to our knowledge speces-area curves were never used specfcally n a game. The results of the game are also straghtforward: for speces conservaton cooperaton s better than the Nash equlbrum, but that due to defector ncentves we have a socal dlemma. In contrast wth the fsheres game the dfference n PA assgned as well as the dfference n payoff between full cooperaton and Nash equlbrum are ncreasng n both the gans of the PA, and the number of players. Therefore t seems that achevng cooperaton wll be hard, unless there s lttle to be ganed by assgnng PAs anyway. The core result of the paper s that the combned game offers a new and counterntutve perspectve on these standard results of the separate games: f we take a combned vew, cooperaton on a sngle ssue may be worse than the Nash equlbrum of that sngle ssue. By gnorng the multple use of the PA the damage done to one of the uses s so large that t destroys all the gans from cooperaton on the other ssue. Therefore we conclude that accountng for multple uses s not just a ncety of ths model, but a necessty when plannng PAs. Furthermore the combned game has better possbltes from socety s perspectve then the sngle games do. In our numercal example the combned game ncreases the PA assgned compared to the fsheres case. Also t decreases the dfference between full cooperaton and Nash equlbrum compared to the bodversty case. Unfortunately combnng both games s not enough to overcome a prsoner s dlemma, but even a compromsed combned Nash PA may be better than cooperaton on a sngle ssue. Summarzng, PAs are a valuable tool for conservaton and fsheres management n the EEZs. Optmal use of ths tool, however, requres consderaton of ts multple 34

35 uses effects, f not then t mght be better from socety s pont of vew not to cooperate. Although not a perfect soluton The SD (and other European marne polcy) encourages countres to cooperate and more mportantly, consder the full effects of ther actons n ther marne strateges. We conclude that PAs may not be a panacea, but they wll surely help n safeguardng our marne envronment Lmtatons of the study We restrcted our analyss to symmetrc players. The ssue lnkage models of Cesar and de Zeeuw [1996] and Folmer et al. [1993] asymmetry s a requrement to break the prsoner s dlemma. Ther models consst of two prsoner s dlemmas that are completely reversed and hence they cancel each other out when lnked. If we nclude asymmetry n our model t would not play as bg a role snce our model consders only one nstrument, n contrast to the models of Cesar and de Zeeuw [1996] and Folmer et al. [1993] who consder two nstruments. Even f one country would have a hgher advantage n the fsheres and the other n speces conservaton we expect that the combned game would stll gve an average of the two. Where that average would be, depends on parameter values. Although the fsheres model suggests a postve role for PAs n fsheres management, ths result may no longer hold, f open access s present, as demonstrated n Hannesson [1998] and Anderson [22]. Even though agents may play a Nash equlbrum n our model, resultng n a suboptmal soluton, open access does not occur snce there are no potental new entrants (as the number of countres s gven). Only f a large number of agents enter the fsheres the sze of the PA s drven to zero. 35