Cooperation in international environmental negotiations due to a preference for equity

Journal of Public Economics 87 (2003) 2049–2067www.elsevier.com/ locate/econbase

C ooperation in international environmental negotiationsdue to a preference for equity

a,b , a*Andreas Lange , Carsten VogtaCentre for European Economic Research (ZEW), Postfach 103443, 68034Mannheim, GermanybInterdisciplinary Institute for Environmental Economics, University of Heidelberg, Heidelberg,

Germany

Received 27 February 2001; received in revised form 28 August 2001; accepted 27 November 2001

Abstract

This paper demonstrates that cooperation in international environmental negotiations canbe explained by preferences for equity. Within aN-country prisoner’s dilemma in whichagents can either cooperate or defect, in addition to the standard non-cooperativeequilibrium, cooperation of a large fraction or even of all countries can establish a Nashequilibrium. In an emission game, however, where countries can choose their abatementlevel continuously, equity preferences cannot improve upon the standard inefficient Nashequilibrium. Finally, in a two stage game on coalition formation, the presence of equity-interested countries increases the coalition size and leads to efficiency gains. Here, even astable agreement with full cooperation can be reached. 2002 Elsevier B.V. All rights reserved.

Keywords: International environmental negotiations; Cooperation; Equity preference; Coalition forma-tion

JEL classification: C7; D63; H41; Q00

1 . Introduction

It is well known that international cooperation is needed in order to dealeffectively with environmental problems, such as global warming and the

*Corresponding author. Tel.:149-621-1235-208; fax:149-621-1235-226.E-mail addresses: [email protected](A. Lange),[email protected](C. Vogt).

0047-2727/02/$ – see front matter 2002 Elsevier B.V. All rights reserved.doi:10.1016/S0047-2727(02)00044-0

mailto:[email protected]

mailto:[email protected]

2050 A. Lange, C. Vogt / Journal of Public Economics 87 (2003) 2049–2067

depletion of the ozone layer. Since pollution crosses borders and causes negativeeffects on a global scale, any country that reduces emissions supplies a publicgood, e.g. the reduction of climate change or the protection of the ozone layer.

The impact of a single country on the global pollution level, however, is rathersmall. Hence, when reducing emissions, a country must bear abatement costs butbenefits only little, i.e. the incentives for a country to reduce emissions are small.Rather, the country has an incentive to free-ride on (possible) emission reductionsmade by the rest of the world. Consequently, although it pays if all partiescooperate, each agent unilaterally has an incentive to defect from cooperation.Thus, the incentive structure can be described as a prisoner’s dilemma, clearlyleading to an undesired outcome. This incentive problem is rather severe,because—contrary to locally bounded pollution—there is no supranational authori-ty that can enforce cooperative behavior.

However, the last two decades have shown an increasing number of negotiationsand protocols on subjects of (global) environmental concern. Perhaps mostprominent are the Montreal Protocol on Substances that Deplete the Ozone Layer(1987), the Basel Convention on the Control of Transboundary Movements ofHazardous Waste (1989), several agreements under the 1979 Convention onLong-Range Transboundary Air Pollution, and the UN Framework Convention onClimate Change (1992), which established a process leading to the Kyoto Protocol(1997) which specifies targets for the reduction of greenhouse gases. However, itis not absolutely clear whether these agreements go substantially beyond actionsthat countries would have done in their self-interest anyway. For example,Murdoch and Sandler (1997) argue that the Montreal Protocol does not provecooperative behaviour, instead it might be a symbolic policy agreement. TheKyoto Protocol has not yet been ratified by most countries, although it specifiessubstantial emission reduction targets. However, despite the drop out of the US,the political agreement of the remaining countries that was reached at 6thConference of the Parties (COP6) in Bonn in July 2001, seems to stabilize theprotocol. Clearly, whatever the result of the ongoing negotiations will be, anyadopted policy must be in the interest of each party involved.

In attempting to explain cooperation and coalition formation, most theoreticalmodels use a two-period structure as introduced by Barrett (1992, 1994). Here,countries must first decide whether or not to join a coalition. In a second step, boththe coalition and the remaining countries choose their emission levels non-cooperatively. A coalition is stable if no country wants to join the coalition and nocountry has an incentive to leave. Simulations by Barrett (1992, 1994, 1997),Carraro and Siniscalco (1993), and Hoel (1992) have shown that—although thereis cooperation—the coalition size is rather small.

The question remains how the cooperation that is observed in prisoner’sdilemma situations as well as in some international agreements can be explained.Here, besides the symbolic policy arguments mentioned above, the literature refersto repeated games: For prisoners dilemma situations cooperation can prevail due to

A. Lange, C. Vogt / Journal of Public Economics 87 (2003) 2049–2067 2051

1an infinite repetition of the one stage game. In this paper, however, we follow adifferent approach and study the effect of equity (fairness) preferences for the

2formation of cooperation. Fair behavior was observed in experimental economics,arguments related to fairness and equity, however, can also be found in

3international negotiations as on the abatement of greenhouse gases.Often it is called for an equalization of per capita emission levels in the long

run. On the one hand, specifically developing countries claim that developedcountries with high per capita emission levels of greenhouse gases are responsiblefor global warming; for example the leader of the Indian delegation at thenegotiations on the Rio Convention 1991 demanded that, within the negotiations,‘the principle of equity should be the touchstone for judging any proposal’(Dasgupta, 1994, p. 133). On the other hand, environmental interest groups indeveloped countries also demand emissions based on equal per capita levels. Forexample, a petition called ‘The Hague Mandate’, circulated by the Climate ActionNetwork (CAN), was signed by more than 85 international organizations. In thisthey call for immediate action on climate change and argue that ‘no citizen has aright to pollute more than any other;. . . every country has a duty to ensure its

4emissions do not exceed its global per capita share.’ A government, facing voterswith such preferences, clearly has to take those into account. Thus, using fairnesspreferences for international negotiations can be motivated by political economyarguments. The weight that a government attaches to the relativity argument willthen depend on the impact these interest groups have on the national policy.

In this paper, we look at the implications of equity preferences for internationalcooperation in reduction of some global pollutant like the greenhouse gas CO . We2

concentrate on the case of symmetric countries, i.e. countries which are identicalwith respect to both their abatement costs and their benefits from abatement. Forthose, clearly, all the fairness arguments mentioned above coincide: equal payoffsare obtained if (per capita) emissions as well as abatement efforts coincide. Werely on a preference structure given by the ERC theory (Bolton and Ockenfels,

1By the different Folk theorems, basically any payoff vector can be sustained as a Nash equilibriumunder certain circumstances.

2For a survey of bargaining experiments, see Roth (1995).3An overview of alternative equity criteria for climate change policy is given by Cazorla and Toman

(2001). The principle of equity is even fixed in Article 3 of the Convention on Climate Change as wellas in the decision approved by the COP 6 in Bonn which states that measures shall be implemented‘ . . . with a view to reducing emissions in a manner conducive to narrowing per capita differencesbetween developed and developing country Parties’.Other equity arguments often used are with respect to equal burden shares or equal percentagereduction of some pollutants. For example, equal measures for the signing countries are used in theLRTAP protocol on reduction of sulphur emissions (minus 30% based on 1980 level) or in the LRTAPprotocol concerning the control of volatile organic compounds (minus 30% based on 1999 level).

4The text can be found atwww.foeeurope.org/dike/news/1001-01en.htm,the list of signers at:www.foeeurope.org/dike/news/1001-01.htm.

www.foeeurope.org/dike/news/1001-01en.htm,







www.foeeurope.org/dike/news/1001-01.htm.








52000, 2001). This theory explains most of the behaviour of agents observed in6diverse experiments, but deviates little from the traditional utility concept. The

utility of an agent is not solely based on the absolute payoff but also on therelative payoff compared to the overall payoff to all agents. Given a certainrelative payoff share, the utility is strictly increasing in the own absolute payoff ofthe agent. Given a fixed absolute payoff, the agent is best off when receiving justthe equal (fair) share. To both sides of this equal share, i.e. when receiving less ormore than the fair amount, utility is lower, even if the absolute payoff does not

7change. By trading off equity and absolute payoff, this preference structure is inline with the observation by Cazorla and Toman (2001, p. 238) that ‘Equity mightbe one motivation for countries to pursue GHG emissions policies. However,equity principles will not override other elements of national self-interest.’

We first study a symmetricN-country prisoner’s dilemma game. We show thatthere may be Nash equilibria in which countries cooperate: If, for example, the restof the world plays cooperatively, a country can get the equal share by choosing tocooperate as well. Hence, if it values its relative payoff being close to the equalshare more than its absolute payoff, it will choose to complete the grand coalition.Clearly, partial cooperation can also occur. For such equilibria, we show that thefraction of cooperating countries is rather large (.50 percent).

Note, however, that in the prisoner’s dilemma, the countries have only thediscrete choice of cooperating or defecting, while with respect to the globalwarming problem, countries can choose their emission level continuously. Wetherefore analyse a symmetric continuous emission game. Here, ERC alone cannotimprove upon the non-cooperative Nash equilibrium with standard preferences.The reason is that as long as a country receives less than the equal share and hasthe possibility to increase its absolute payoff, it will do so. Since this holds for allcountries, the unique Nash equilibrium therefore is identical to the symmetricNash equilibrium with standard preferences.

This result changes, however, if a stage of coalition formation is introduced.Here, in contrast to the traditional models by Barrett (1994), Carraro andSiniscalco (1993), etc., coalitions that involve a rather large fraction of countriescan be stabilised. If each country puts enough weight on getting close to the equalpayoff share, even the grand coalition can be obtained in equilibrium. In general,

5ERC hereby stands for Equity, Reciprocity, and Competition.6As already noted by Bolton and Ockenfels, this theory can generate cooperation in the standard

prisoner’s dilemma.7Note that such a preference for equity is self-centred only and is distinct from altruism (Bolton and

Ockenfels, 2001). A country’s utility is determined solely by its own absolute and relative payoff. Forour application to the abatement of greenhouse gases, countries (voters) clearly might also care aboutother aspects of equity as for example inequalities within the rest of the world. However, each countrycan only influence its own emission levels, and thereby has an impact on its relative position to the restof the world. Inequalities among other countries it cannot directly influence. Therefore, it seems to beappropriate to include the average of other countries only.


the presence of (some) equity-interested countries increases the incentive to join acoalition and, therefore, leads to a larger coalition size in equilibrium. Theefficiency gains are maximal if countries that are interested in equity stay outsidethe coalition and reward a higher abatement effort of the coalition with anexpansion of their own abatement level.

The paper is organised in the following way: In Section 2, the basic features ofthe ERC theory are introduced. We then study the discrete version of a symmetricN-country prisoner’s dilemma in Section 3, whereas the emission game is analysedin Section 4. Here, we look at the Nash equilibria of the one shot game first andthen study stable coalitions in the two-stage coalition formation game. The finalsection—as always—concludes.

2 . The preference structure

The analysis in this paper relies on a preference structure in which players—along with their own absolute payoff—are motivated (non-monotonously) by therelative payoff share they receive, i.e. how their standing compares to that ofothers. With this, we rely on the ERC model by Bolton and Ockenfels (2000) butuse a full information framework. This theory explains the behaviour of peopleobserved in a rather large group of experiments better than the standard theory. Asalready mentioned, it can also be rationalised for countries (governments) whosecitizens have some preference for fair sharing abatement efforts among countries.

Let the (non-negative) payoff to agenti be denoted byy , i 5 1, . . . , N, thei

relative share bys 5 y /(o y ). The utility function is given by:i i j j

a u y 1 b r ss d s di i i i

wherea , b $ 0, u( ? ) is differentiable, strictly increasing and concave, andr( ? ) isi i

differentiable, concave and has its maximum ats 5 1/N. Throughout this paperi

we assume that agent’s disutility from disadvantageous inequality is larger than ifthe agent is better off than average, i.e.r(1 /N 2 x)# r(1 /N 1 x) for all x [ [0,1 /N]. The types of countries are characterised by the relative weightsa /b .i i

3 . The prisoner’s dilemma

In this section we study a simple symmetricN-country prisoner’s dilemma (PDgame) where each country can cooperate, ‘c’, or defect, ‘d’. In terms of the climatechange problem: The country either reduces emissions by a (pre-) specifiedamount or it doesn’t. This situation may occur when countries have to decidewhether or not to ratify an international agreement which already specifies theemission levels that must be abated.


3 .1. The payoff structure

The total number of cooperating countries is denoted byk. For any givenk, thepayoff to a country is given byB(k) if the country defects (tries to free-ride). If acountry plays cooperatively, it must bear some additional costsC(k). Its payoff istherefore given byB(k)2C(k). We assume decreasing marginal benefits for acountry if the number of contributers rises, i.e.B( ? ) is increasing and concave.Furthermore, the total cost of cooperation,kC(k), increases ink.

In order to generate the standard incentive structure of a PD game, we assumethat B(k 1 1)2B(k),C(k 1 1), i.e. playing cooperatively reduces the absolutepayoff, given an arbitrary number of ‘c’-countries. To make more cooperationattractive from both the social and the individual point of view, we make thefollowing assumption:

(i) NB(k 1 1)2 (k 11)C(k 1 1)$NB(k)2 kC(k) ‘socially desirable’

(ii) B(k 1 1)2C(k 11)$B(k)2C(k) ‘individually desirable’

Furthermore, we assume that payoffs for both cooperating and defecting countriesare non-negative for allk.

3 .2. The Nash equilibria

In the following, we analyse Nash equilibria in the one shot PD game wherecountries choose simultaneously. Assume thatk countries, aside from countryi,play cooperatively. Then countryi chooses to play ‘c’ if and only if:

B(k 11)2C(k 1 1)]]]]]]]]a u(B(k 1 1)2C(k 11))1 b rS Di i NB(k 1 1)2 (k 11) C(k 1 1)

B(k)]]]]$ a u(B(k))1 b rS Di i NB(k)2 kC(k)

This is equivalent to countryi playing ‘c’ if:

B(k 1 1)2C(k 1 1) B(k)]]]]]]]] ]]]]r 2 rS D S DNB(k 1 1)2 (k 1 1) C(k 1 1) NB(k)2 kC(k)]]]]]]]]]]]]]]]]a /b #d(k)[ (1)i i u(B(k))2 u(B(k 11)2C(k 1 1))

In other words, in order to choose ‘c’ the country must be overcompensated for theloss in absolute payment by moving closer to the average payment.

The general conditions for a Nash equilibrium of this ERC-PD game are thengiven by:

* *a /b #d(k 21) for k countries (playing ‘c’) (2)i i


* *a /b $d(k ) for the remainingN 2 k countries (playing ‘d’) (3)i i

*We now have a closer look at the number of countriesk that may possibly*cooperate in a Nash equilibrium. On the one hand, as long asd(k 2 1), 0, there

* *is no chance of having a coalition of sizek . Here,a /b .d(k 2 1) for all typesi i

and condition (2) cannot hold for any country. On the other hand, the conditions*for a Nash equilibrium given by (2) and (3) immediately imply that ifd(k 2 1).

*0 then there are types ((a /b ) ) of countries such thatk countriesi i i51, . . . ,N

*cooperate andN 2 k countries free-ride. These types—for example—could be* * * *given bya /b 5d(k 21) for i 51, . . .k , anda /b 5minhd(k 2 1), d(k )j fori i i i

8*i 5 k 1 1, . . . , N.

2Example 1. Let B(k)5 km, C(k)5 c, wherec .m, r(s)5 2 (s 2 1/N) /2, and1 km 2 c] ]]u(y)5 y. Then, as easily shown:d(k 2 1).0 if and only if 2 ,N Nkm 2 kc

(k 2 1) m 1]]]]] ]2 , or—equivalently—[22N /k] . 0. Therefore, if in equilib-NN(k 2 1) m 2 (k 2 1) c

rium some countries cooperate, then at leastN /2. Fig. 1 illustrates the function*d(k) for m 5 1, c 51.5, and N 5 10. An equilibrium in which k countries

* *cooperate results if the ERC-type ofk countries is below the line atk 5 k 2 1* * *(a /b #d(k 21)), and above the line atk 5 k for the remaining N 2 ki i

*countries (a /b .d(k 2 1)). In particular, if for all i: a /b #d(N 2 1)5 3.1?i i i i2410 , the grand coalition would be stable.

Fig. 1. Graph ofd(k) for B(k)5 k, C(k)5 1.5k, N 510.

8 *In other words, for a given distribution of ERC-types,d(k 2 1). 0 is necessary but not sufficient* *to get a coalition size ofk . For a given payoff structure withd(k 21).0, however, there exist

*ERC-types such thatk is an equilibrium coalition size.


In order to find feasible coalition sizes, we must therefore study conditions inwhich d( ? ) is positive. Note that in (1) the denominator ofd(k) is positive, sinceplaying ‘d’ always maximises the absolute payoff. The sign of the numerator,however, depends on the numberk of cooperating countries. It is negative fork 5 0 and positive fork 5N 21, since both, defection and cooperation of allcountries equalise countries’ payoffs and thereby maximiser( ? ). Therefore,d(0),0,d(N 2 1) and no country unilaterally cooperates whereas all countriesplaying ‘c’ can establish an equilibrium, provided that all countries’ types (a /b )i i

9are smaller thand(N 21).In general, however, there may also be equilibria where only a certain number

*k of countries cooperate. Remember that we assumed that countries suffer morefrom disadvantageous inequality than if they are better off than average, i.e.r(1 /N 2 x)# r(1 /N 1 x) for all x [ [0, 1 /N]. Therefore, in order to obtaind(k).0, it is necessary that by choosing ‘d’, a country further deviates from the equalshare (1/N) than by playing ‘c’, i.e.:

B(k) 1 1 B(k 1 1)2C(k 11)]]]] ] ] ]]]]]]]]2 . 2N NNB(k)2 kC(k) NB(k 1 1)2 (k 11) C(k 11)

Straightforward calculus shows that this is equivalent to:

0,B(k 11) C(k) Nk 1B(k) C(k 1 1) N[k 112N]

1C(k) C(k 1 1)[Nk 2 2k(k 11)]

or

N k 1 12N]] ]]]0,B(k 11) C(k) 1B(k) C(k 1 1) Nk 11 k(k 11)

N]]F G1C(k) C(k 1 1) 22k 1 1

N N]] ]F G5 B(k 1 1) C(k) 2B(k) C(k 1 1)k 1 1 k

NB(k) N]] ]]F G F G1 2C(k) C(k 1 1) 22 (4)k k 1 1

*We can use this inequality to study the numberk of agents that playcooperatively in equilibrium. First, note that we assumed payoffs to be non-negative and thereforeNB(k)2 kC(k). 0. Thus, the second summand is negativefor k ,N /22 1. For payoff functions that satisfy the requirement that total costsproportionally increase more than the total benefits, i.e.:

9Thus, if we look at the question of whether or not to ratify an already negotiated internationalagreement, we see that no country would unilaterally ratify if it expects the others not to do so. If,however, all except of one country have already ratified, then this country may decide to ratify as well.


(k 1 1) C(k 1 1) NB(k 1 1)]]]]] ]]]. (5)

kC(k) NB(k)

10the first bracket in (4) is negative as well. As a consequence, inequality (4)cannot hold andd(k),0 for k ,N /22 1. Thus, for any given vector of types, if acountry plays ‘c’ in equilibrium, then, in total, at least half of the countriescooperate. For PD games where inequality (5) does not hold, we obtaind(N /221). 0, suggesting that the minimal coalition size can be smaller thanN /2. As long

11as the number of countries exceeds 8, however, this is not the case. We cansummarise our results in the following proposition:

Proposition 1. (Prisoner’s Dilemma)For any given payoff structure of the PDgame with ERC preferences, there is always an equilibrium in which all countriesdefect. Additionally, depending on the distribution of ERC-types, there can beequilibria in which some or all countries cooperate. Then—at least—N /2countries cooperate if N . 8 or the total costs of cooperation proportionallyincrease more than the benefits, i.e. inequality (5) holds.

This proposition shows that if there is a coalition of cooperating countries, then itis rather large. The intuition is that in order to make cooperation attractive for acountry, it has to be closer to the equal payoff share than when playing ‘d’. Thiscan only be the case if the number of countries that receive the (smaller)cooperative payoff is already large.

4 . The emission game

In the last section, we assumed that countries have only the discrete choicewhether or not to cooperate. Now, we look at an emission game where countriescan choose their emission levels continuously. Again we concentrate on the case ofcountries that are homogeneous with respect to their payoff functions. It willbecome obvious that, here, ERC alone cannot improve upon the non-cooperativeNash equilibrium with standard preferences where only the absolute payoffmatters. However, introducing more structure to the game, i.e. if countries play acoalition game as in Barrett (1994), ERC may yield a rather large coalition size oreven support the grand coalition.

Let the number of countries again be denoted byN. Each country must choose

10 B(k 1 1) NB(k 1 1)2 (k 1 1) C(k 1 1)]] ]]]]]Note that (5) is equivalent to . , i.e. a situation where private benefits

B(k) NB(k) 2 kC(k)

of a defecting country increase faster than social benefits from cooperation.11The proof is given in Appendix A. In it, we derive an upper bound ford(k) which is negative for

* *k ,N /22 1. Therefore,d(k 2 1). 0 can only be the case ifk $N /2.


maxits abatement levelq [ [0, e ] (i 51, . . . , N). Reduction of emissions from ai imaxstatus quo levele induces some costsC(q ) that are assumed to be increasingi i

and convex in the abatement level,C9( ? )$0, C0( ? )$0 Abatement creates somebenefitsB(Q) in terms of abated damage from climate change, whereQ 5o qi i

denotes the aggregate abatement level. Benefits from abatement are increasing and(strictly) concave,B9( ? )$ 0, B0( ? ), 0. The payoff to a country is therefore

12determined byB(Q)2C(q ). In order to guarantee interior solutions we assumei

further C9(0)50 and lim C9(q )5`, and B9(0). 0.maxq →e ii i

4 .1. Nash equilibria in the one-shot emission game

We again analyse Nash equilibria when countries act simultaneously. Takingqj

( j ± i) as given, countryi choosesq to maximise:i

a u y 1 b r ss d s di i i i

wherey 5B(o q 1 q )2C(q ) denotes the absolute, ands 5 y /o y denotesi j±i j i i i i j j

the relative payoff to countryi. By choosingq , a country directly influences itsi

own abatement costs and the benefits from abatement. It thereby also has animpact on the payoff to the rest of the world, which enters its own utility throughthe relative payoff. The first order condition is therefore given by:

O yjj±i]]]05 a u9( ? )1 b r9( ? ) B9(Q)2C9(q )f g2i i i3 4O yjF G

j

yi]]]2 b r9( ? ) (N 2 1) B9(Q)2i O yjF G

j

5 a u9( ? ) B9(Q)2C9(q )f gi i

O y 2 y O ys dj i jj j±i]]] ]]]1 b r9( ? ) B9(Q)2 C9(q ) (6)2 2i i3 4O y O yj jF G F G

j j

O sj12Ns j±ii]]] ]]5 a u9( ? ) B9(Q)2C9(q ) 1 b r9( ? ) B9(Q)2 C9(q )f gi i i i3 4O y O yj j

j j

(7)

12Note that although using the same notation as in the previous section, the domain of the cost andbenefit functions now is continuous abatement instead of discrete coalition sizesk.


The reaction of countryi to a given abatement policy for the rest of the world canbe calculated from this first order condition. Let us first study the two extremecases,a 5 0 and b 5 0, respectively. Forb 50, i.e. an absolute payoff maxi-i i i

miser, the first order condition reduces toB9(Q)2C9(q )50. For a 50, thei i

country is solely interested getting the equal payoff share. Hence it would chooseq to satisfy NC(q )5o C(q ). For a , b ± 0, the chosen abatement level isi i j j i i

between the levels for those extreme cases.In a Nash equilibrium, the first order condition must be satisfied for all countries

simultaneously. Sincer9(1 /N)50, it follows immediately that for all typesa /b ,i i

i 5 1, . . . ,N there is a symmetric equilibrium where all countries choose the same*abatement level, i.e.s 51/N for all i. Here, the resulting abatement levelq isi

* *given by the first order conditionB9(Nq )2C9(q )5 0. It corresponds to theNash equilibrium when agents are only interested in their absolute payoff andtherefore resembles the Nash equilibrium in the PD game where all countriesdefect.

This equilibrium is the only one assuming that for at least one countrya greateri

than 0. To see this, assume that there is an asymmetric equilibrium, i.e. somecountry i receives less, another countryj more than the equal share. In this case,on the one hand,s , 1/N implies thatr9(s ). 0, and hence from Eq. (6), wei i

obtain B9(Q)2C9(q ).0. On the other hand, fors . 1/N we haver9(s ), 0,i j j

and therefore Eq. (7) impliesB9(Q)2C9(q ), 0. These two inequalities, how-j

ever, imply that countryj has larger (marginal) abatement costs than countryi,which clearly contradicts the assumed payoff distribution. Hence, only symmetricequilibria exist. Here, ifa .0 for at least one country, we getB9(Nq)2C9(q)5 0i

from Eq. (6). Only in the extreme (unlikely) case that all countries are solelyinterested in equal payoff could any arbitrary abatement level be implemented as asymmetric equilibrium.

We can summarise the result in the following proposition:

Proposition 2. (Emission game)In the emission game for ERC preferences, the* *equilibrium is given by B9(Nq )5C9(q ). It is unique as long as at least one

country draws utility from its absolute payoff (a .0).i

Thus, in contrast to the prisoner’s dilemma, ERC does not add any equilibria inwhich there is more abatement effort (cooperation). The existence of equilibria inthe PD game that mimick cooperative behaviour, therefore, only arises in thepresence of discrete action sets where countries have the binary choice of eithergetting more or less than the equal share and therefore might choose to cooperateif this brings them closer to the equal share. Having a continuous decisionvariable, small changes of abatement policy and—thereby—small changes ofrelative payoff are possible. Sincer9(1 /N)5 0, a country will always deviate fromthe equal share in order to increase its absolute payoff. The only situation where


this is not possible, resembles the standard Nash equilibrium in which the absolutepayoff is already maximised.

4 .2. Coalition formation in the emission game

We now study a two-stage game of international negotiations as introduced byBarrett (1994), Carraro and Siniscalco (1993), and Hoel (1992). Let us againassume that all countries are identical with respect to their payoff function. In afirst stage, countries decide whether or not to join the coalition. Here, each countrytakes the decisions of the other countries as given. Each country anticipates,however, that the abatement levels, which are chosen in the second stage, dependon whether it does or does not enter the coalition. In stage 2, countries inside andoutside the coalition simultaneously select their abatement levels. The coalitionthereby maximises its collective benefits but plays Nash against the non-signatory

13countries which simultaneously maximise their individual utility. As shown byCarraro and Siniscalco and Hoel, the coalition size for payoff-maximisingcountries is 2.

We first look at the case of countries that have identical ERC-types. In a secondstep, we then study the case of heterogeneous ERC-types in which countries aresolely interested either in their absolute payoff or in equity.

4 .2.1. Cooperation of identical ERC-typesWe solve the coalition formation game backwards. That is, for any coalition size

k, we first study the first order conditions for the choice of the abatement levelinside and outside the coalition. Then, in the second step, the equilibrium coalitionsize is determined by a stability condition such that that there is an incentive toneither leave nor join the coalition. In contrast to standard preferences (herecorresponding tob 5 0), for ERC preferences, the coalition size can be muchlarger. Instead of solving the game in general, we will show that if countries onlyvalue the relative payoff high enough, i.e.a /b is below a certain bound, then eventhe grand coalition can be stable.

For countries outside the coalition, the first order condition is again given by(6), whereas the coalitionS maximises the utility of a representative member bychoosing the abatement policy for all its members,i [ S. It is plain that all

Scountries within the coalitionS must have the same abatement level,q , since allScountries are assumed to be of the same type. Hence, the absolute (y 5B(Q)2

S S S SC(q )) and the relative payoffs (s 5 y /(ky 1o y )) are equal for allj[⁄ S j

members of the coalition. This leads to the following condition:

13Such an simultaneous move game for standard preferences was analyzed by Carraro and Siniscalco(1993) and Hoel (1992). This Cournot–Nash behavior at the second stage also resembles thepartial-agreement Nash equilibrium concept used by Chander and Tulkens (1997). In contrast to thisassumption, Barrett (1994) assumes that the coalition behaves as a Stackelberg leader.


O sjj[⁄ S S]]05 au9( ? )1 br9( ? ) kB9(Q)2C9sq df gf g O yj

j

Ss]]2 br9( ? ) (N 2 k) kB9(Q) (8)O yj

j

S5 au9( ? ) kB9(Q)2C9sq df g

O s Sj 12Nsj[⁄ S S]] ]]]1 br9( ? ) 2 C9sq d1 kB9(Q) (9)3 4O y O yj jj j

From the last section we already know that ifs , ( . )1 /N for j [⁄ S, thenj

B9(Q). ( , )C9(q ). Analogously, for the coalition, we obtain from (8) and (9)jS S 14that if s , ( . )1 /N thenkB9(Q). ( , )C9(q ). SinceB9(Q), kB9(Q) (k $ 2),

the first order conditions therefore imply that for countries within a coalitionS 15

s # 1/N, and thus:

SkB9(Q)$C9sq d

This inequality can now be used to show the following proposition:

Proposition 3. (Coalition game)In the symmetric coalition game for identicalERC preferences (type a /b), the grand coalition is stable if a /b is sufficientlysmall, i.e. countries are interested enough in being close to the equal share.

Note first, that within the grand coalition, the emission level satisfies the condition* *NB9(Nq )5C9(q ), independently of the ERC-types. Here, countries clearly

receive the equal share. If countryi leaves the coalition (k 5N 2 1), then from theinequalities above we obtain:

S S S(N 2 1) B9 (N 2 1) q 1 q $C9sq d$C9 q .B9 (N 2 1) q 1 q (10)s d s ds di i i

for all ERC typesa /b. Let us now look at the abatement levels that would result ifthe ERC-typea /b goes to zero. In this case, all countries get more and moreinterested in getting their equal share, and their abatement levels will converge to a

˜˜level q. However, inequality (10) still must hold in the limit, i.e. (N 2 1) B9(Q )$

14 S SAgain assuming thats . 1/N then r9( ? ), 0, and (9) implieskB9(Q),C9(q ).15 SAssuming to the contrary thats . 1/N and therebys ,1/N for some countryj outside thej

Scoalition, the inequalities implyC9(q ),B9(Q), kB9(Q),C9(q ), which contradicts the assumptionj

of increasing and convex abatement costs.


˜˜ ˜C9(q ). This, however, implies, thatNB9(Q ).C9(q ), and therefore, in the limit,the absolute payoff of a country leaving the coalition (and of all the othercountries) is smaller than within the grand coalition. Therefore, as long asa /b issmall enough, the absolute payoff remains lower, and—due to the asymmetricpayoff share—the utility derived from the relative payoff is also smaller than inthe grand coalition. Thus, no country has an incentive to leave the grand coalitionif a /b is small enough.

4 .2.2. Coalition of heterogeneous ERC-typesApparently, if one allows for heterogeneous ERC-types, starting in the grand

coalition, countries that have the largesta /b will have the greatest interest toi i

leave the coalition in order to obtain a larger absolute payoff. In the following weconcentrate on the extreme case in which countries are either interested in theirabsolute payoff (b 5 0) or in equity (a 50). The former are referred to asi i

A-countries, the latter as B-countries. In total, there areN A-countries andNa b

B-countries;k of these A-countries andk B-countries form the coalition. Thea b

abatement levels are denoted byq , q for signatory countries,q and q foras bs an bn

countries outside the coalition.Let us first look at the behaviour of B-countries. Outside the coalition, any

B-country can arrive at the equal share by choosing the average abatement costlevel. Thus:

1]]C q 5 k C q 1 k C q 1 N 2 k C qs d f s d s d s d s dgbn a as b bs a a anN 1 ka b

Therefore, for a B-country inside a coalition to have no incentive to leave, it alsomust receive the equal share:

1]]C q 5 k C q 1 N 2 k C q 1 N 2 k C q (11)s d f s d s d s d s d s dgbs a as b b bn a a anN 2 kb

Thus, in any equilibrium, all B-countries choose the same abatement level,q [q 5 q , and receive the equal share:b bn bs

1]C q 5 k C q 1 N 2 k C qs d f s d s d s dgb a as a a anNa

A-countries outside the coalition again maximise their absolute payoff,B(Q)2C(q ). The first order condition is given by:an

B9(Q)5C9(q ) (12)an

whereas the coalition maximises the utility of a representative A-type-member byguaranteeing its B-members the equal share, i.e. Eq. (11). The first order conditionfor choosingq is therefore given by:as


≠qbs]]05B9(Q) k 1 k 2C9(q ) (13)F Ga b as≠qas

k C9(q )b as]]]]]5B9(Q)k 11 2C9(q ) (14)F Ga asN 2 k C9(q )b bs

By construction, for any givenk and k , every B-country is indifferent to beinga b

either inside or outside the coalition. For a coalition to be stable, an A-countrymust not have an incentive to join or leave the coalition. In general, for anykb

there will be a certain number of A-countries,k , that will join the coalition. Wea

have multiple equilibria.The properties of the equilibria are best understood by considering an example.

2Consider the following quadratic payoff functions:B(Q)5 10(Q 2Q /2) and2C(q)55000q , and N 5N 5 6.a b

Table 1 presents both the total abatement level and the payoffs to cooperatingand non-cooperating A-countries as a function of the number of A- and B-countries inside the coalition. Here, if no B-country enters the coalition, allA-countries cooperate. If all B-countries are within the coalition, then only twoA-countries enter. In general, when B-countries join the coalition, they successive-ly drive out A-countries. This can be illustrated by the equilibria in Table 1: Thefirst, second, fourth, and sixth entering B-country reduces the number of A-countries in the coalition and thereby lowers the payoffs to both, signatory andnon-signatory countries. The payoffs to all countries increase, however, when athird or a fifth B-country enters the coalition and the number of cooperatingA-countries stays at 4 and 3, respectively. Therefore, in this example, the presenceof equity-interested countries leads to efficiency gains which are largest if allB-countries stay outside the coalition.

The results of the simulations—which appear to be robust to variations of thepayoff functions and the total number of countries—can be summarised asfollows:

Result 4. The larger the total number of equity-oriented countries (N ) is, theb

higher the incentives are for A-countries to join the coalition. In other words, for agiven k , the number of cooperating A-countriesk increases inN .b a b

Result 5. The more B-countries join the coalition, the smaller the incentives arefor A-countries to do so. In other words, in equilibrium,k andk are negativelyb a

correlated.

Result 6. The total abatement level increases with the number of B-types outsidethe coalition. A joining B-country improves the payoffs only if it does not driveout an A-country.


Table 1Payoffs to non-cooperating, cooperating countries and aggregate abatement as a function ofk and ka b

a(N 5 6, N 5 6)a b

k kb a

0 1 2 3 4 5 6an0 p 0.113 0.113 0.156 0.240 0.354 0.490 –as

p – 0.113 0.142 0.202 0.284 0.382 0.492*Q 0.012 0.012 0.016 0.025 0.037 0.051 0.067*

an1 p 0.113 0.115 0.168 0.264 0.388 0.543* –as

p – 0.114 0.148 0.215 0.302 0.404* 0.518Q 0.012 0.012 0.017 0.027 0.040 0.055* 0.073

an2 p 0.113 0.118 0.185 0.293 0.430* 0.587 –as

p – 0.115 0.157 0.230 0.321* 0.427 0.544Q 0.012 0.012 0.019 0.030 0.044* 0.061 0.080

an3 p 0.113 0.122 0.206 0.330 0.480* 0.650 –as

p – 0.117 0.166 0.245 0.341* 0.451 0.571Q 0.012 0.013 0.021 0.034 0.050* 0.068 0.088

an4 p 0.113 0.130 0.233 0.375* 0.540 0.726 –as

p – 0.120 0.176 0.260* 0.361 0.475 0.597Q 0.012 0.014 0.024 0.039* 0.056 0.076 0.097

an5 p 0.113 0.142 0.269 0.432* 0.618 0.821 –as

p – 0.122 0.184 0.273* 0.378 0.494 0.619Q 0.012 0.015 0.028 0.045* 0.064 0.086 0.110

an6 p 0.113 0.161 0.318* 0.507 0.718 0.943 –as

p – 0.124 0.188* 0.279 0.385 0.503 0.629Q 0.012 0.017 0.033* 0.053 0.075 0.100 0.126

a The equilibria are marked with *.

The rationale of Results 4 and 5 is the following: If an A-country enters thecoalition and the coalition increases its abatement efforts, B-countries outside thecoalition increase their abatement activities as well and thereby additionallyreward the entering country. If the number of such equity-oriented B-countriesoutside the coalition gets larger, this external reward for joining a coalitionincreases and, therefore, the equilibrium coalition size increases. Analogously, ifB-countries join the coalition, less countries outside the coalition reward theentering A-country by an increase of their abatement activities. Hence, theincentives for A-countries to enter the coalition decrease and the number ofA-countries that are inside the coalition in equilibrium gets smaller. If allB-countries cooperate, the external reward has vanished and the incentives forA-countries to join the coalition are qualitatively the same as with standard


preferences for all countries: Only two countries enter. Result 6 reflects the factthat, in general, the more countries cooperate, the higher the efficiency gains areand the closer the aggregate abatement level is to the efficient one. The impact ofA- and B-countries on the decision of the coalition, however, differs in thefollowing way: A joining A-country is interested in the absolute payoff and,consequently, the re-optimising coalition increases its abatement effort because thepositive effect on one more country is now taken into account. A joiningB-country, however, is not primarily interested in the absolute payoff, but in theequal share. Therefore, the coalition will not increase the total abatement level thatmuch because the B-country refrains from deviating from the abatement level ofnon-cooperating countries. Consequently, the efficiency gains are larger if anA-country enters the coalition than if a B-country joins. Therefore, B-countries arewelcome inside a coalition only if their entering does not drive out an A-country.

5 . Conclusion

In this paper, we studied three different games that model basic features ofinternational cooperation to provide a public good such as environmental quality.Instead of countries that are solely interested in their absolute payoff, we studiedequilibria based on ERC-preferences. This theory explains the behaviour of peopleobserved in experiments by assuming that a part of utility depends (non-monoton-ously) on the received relative payoff share.

We demonstrated that for a discrete prisoner’s dilemma, cooperation can result.If it does, then the fraction of cooperating countries is rather large (above 50%).For the one-shot emission game, which better describes the action set with respectto environmental problems like global change, however, ERC does not change theequilibrium at all: Countries (in equilibrium) still behave as if they wereexclusively maximising their own payoff.

We then studied the process of international environmental negotiations in thestandard two stage coalition formation game. Here, even the grand coalition can bestable if all countries put enough weight on getting close to the equal payoff share.In general, the presence of countries that are highly motivated through obtainingthe equal share leads to efficiency gains. In particular, for countries that areinterested exclusively in their absolute payoff, the incentive to join a coalitionincreases with the number of equity-oriented countries that stay outside thecoalition, but it becomes smaller when those countries enter the coalition. Thus, byentering a coalition, equity-oriented countries may drive out countries that aremotivated by their own absolute payoff and thereby worsen the payoffs for allcountries.

In conclusion, observing international environmental agreements that are signedby a large number of countries may indicate that (at least some) countries are notsolely interested in their absolute payoff, and their behaviour could be explained


by the ERC-theory. Crucial for this result, however, is the structure of the game:Countries must have a discrete choice at one stage of the game.

Note, however, that, in this paper, we exclusively analysed the case of countrieswhich are symmetric with respect to their payoffs. In general, countries clearly areheterogeneous with respect to their benefits and costs of abatement of CO , and a2

preference for equity would probably not regard an equal payoff structure. Instead,equity might be preferred with respect to the abatement targets as percentages ofthe business-as-usual-emissions or in terms of CO emissions per capita. The2

implications of such a preference for equity in the case of heterogeneous countriesremain subject to further research.

A cknowledgements

We would like to thank Michael Finus, Axel Ockenfels, three anonymousreferees, and Lans Bovenberg as the co-editor of this journal for helpful commentsand suggestions to improve the paper. Funding of the research group ‘In-stitutionalization of International Negotiation Systems’ by the Deutsche For-schungsgemeinschaft (DFG) is gratefully acknowledged.

A ppendix A. (Proof of Proposition 2)

We have to show thatd(k),0 for k ,N /22 1. Using (4), this is equivalent to:

B(k 1 1) (k 1 1) C(k 1 1)]]] ]]]]]2F GB(k) kC(k)

1 1 N]] ]] ]]F G1 (k 11) C(k 1 1) 2 22 , 0F G k 1 1kC(k) NB(k)

From the monotonicity and concavity ofB( ? ) it follows that B(k 11) /(k 1 1),B(k) /k. Furthermore, total cost of cooperationkC(k) increase ink. Therefore:

B(k 1 1) (k 1 1) C(k 1 1) k 1 1 1]]] ]]]]] ]] ]2 # 2 15k kB(k) kC(k)

Since we assumed payoffs to be non-negative,B(k)$C(k). Thus:

1 1 (k 1 1) C(k 1 1) N 2 k N 2 k]] ]] ]]]]]]] ]](k 1 1) C(k 1 1) 2 $ $F G N NkC(k) NB(k) kC(k)

For k ,N /221 we therefore obtain:


B(k 1 1) (k 1 1) C(k 1 1)]]] ]]]]]2F GB(k) kC(k)

1 1 N]] ]] ]]F G1 (k 11) C(k 1 1) 2 22F G k 1 1kC(k) NB(k)

1 N 2 k N] ]] ]]F G# 1 22k N k 11N(k 11)1 2(N 2 k) k(k 1 1)2 (N 2 k) Nk]]]]]]]]]]]]5

Nk(k 1 1)

3 2The numerator equals22k 1 (3N 2 2) k 2N(N 23) k 1N which can easily beshown to be negative for 1# k ,N /221 as long asN . 8. Hence, forN . 8 wehaved(k 21), 0 for k ,N /2.

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Cooperation in international environmental negotiations due to a preference for equity