endogenous minimum participation in international environmental treaties

Environ Resource Econ (2009) 42:411–425DOI 10.1007/s10640-009-9264-x

Endogenous Minimum Participation in InternationalEnvironmental Treaties

Carlo Carraro · Carmen Marchiori · Sonia Oreffice

Received: 7 August 2008 / Accepted: 16 January 2009 / Published online: 12 February 2009© Springer Science+Business Media B.V. 2009

Abstract Many international treaties come into force only after a minimum number ofcountries have signed and ratified the treaty. Minimum participation constraints are particu-larly frequent in the case of environmental treaties dealing with global commons, where free-riding incentives are strong. Why do countries that know they have an incentive to free-rideaccept to “tie their hands” through the introduction of a minimum participation constraint?This article addresses the above issues by modeling the formation of an international treatyas a three-stage non-cooperative coalition formation game. Both the equilibrium minimumparticipation constraint and the number of signatories—the coalition size—are determined.This article, by showing that a non-trivial partial coalition, sustained by a binding mini-mum participation constraint, forms at the equilibrium, explains the occurrence of minimumparticipation clauses in most international environmental agreements. It also analyses theendogenous equilibrium size of the minimum participation constraint.

Keywords Agreements · Climate · Negotiations · Policy · Participation rule

JEL Classification H0 · H4 · O3

C. Carraro (B)Department of Economics, University of Venice, CEPR, CEPS, CESifo and Fondazione ENI E. Mattei,S. Giobbe 873, 30121 Venice, Italye-mail: [email protected]; [email protected]

C. MarchioriLondon School of Economics and Fondazione ENI E. Mattei, Venice, Italy

S. OrefficeDepartamento de Fundamentos del Análisis Económico,Universidad de Alicante, Ctra. San Vicente del Raspeig, 03080 Alicante, Spaine-mail: [email protected]

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412 C. Carraro et al.

1 Introduction

Several articles have provided a thorough economic analysis of a country’s incentives to signan international treaty.1 Particular attention has been given to international environmentaltreaties, because they are often aimed at providing a global public good and are thereforecharacterized by strong free-riding incentives. This literature, by using a non-cooperativegame-theoretic approach, has also explored which policy strategies and negotiation rules canbe designed to increase the number of signatories of an international environmental treaty(see, for example, Barrett 2003, 2007 and Finus 2003, 2008).

A feature of many international treaties that has been largely neglected by the theoreticalliterature is the clause that a treaty will only come into force if a minimum number of signato-ries or a minimum degree of effectiveness is achieved. For example, in the case of the KyotoProtocol, the clause is twofold: the Protocol comes into force only if at least 55 countries (outof the Annex 1 list of the 1992 UNFCCC) sign and ratify it and if these countries represent atleast 55% of reference year emissions (Article 25). More generally, almost all internationalenvironmental treaties contain a minimum participation clause. According to Rutz (2001),only two out of the 122 multilateral environmental agreements provided by the Center forInternational Earth Science Information Network do not contain any minimum participationrule. In 81 cases, the participation rule asks for a minimum number of signatories. In 22cases, unanimity is required for the treaty to come into force, namely all negotiating coun-tries must sign and ratify the agreement for it to be effective. In the remaining 17 cases, theminimum participation rule is coupled to other requirements, i.e., for these agreements it isnot sufficient that a certain number of countries ratify the treaty, but these countries have tosatisfy additional criteria.2

Despite the importance of the minimum participation rule (and of the unanimity rule asa particular case), very few theoretical studies have analyzed this issue. A previous studyby Black et al. (1992) shows that the introduction of a minimum participation constraintincreases the number of signatories of an international environmental agreement. However,it does not prove that countries actually have an incentive to introduce a minimum partici-pation constraint among the rules of a given treaty.3 Similarly, Rutz (2001) shows, in a veryspecific case, that a minimum participation rule, if it exists, can result in an increase in thenumber of signatories. However, the rule is imposed exogenously and is not shown to be anequilibrium outcome of the game.

This article endogenizes the minimum participation rule as the equilibrium outcome of athree-stage coalition formation game. Our goal is to address the following questions: Why docountries agree to have a minimum participation constraint among the clauses characterizingan international treaty? Is a minimum participation rule a way to offset free-riding incentives?Why do countries that know they have an incentive to free-ride, accept to “tie their hands”through the introduction of a minimum participation constraint? What is the endogenousequilibrium level of the minimum participation constraint?

This article answers the above questions by analyzing a non-cooperative game of coalitionformation. In the theoretical literature on international agreements, the decision of countries

1 Refer to the surveys in Barrett (2003, 2007); Carraro and Marchiori (2003); Finus (2003, 2008); Rubio andUlph (2006).2 For example, the Montreal Protocol on the ozone layer asks for 11 instruments of ratification representingat least two-thirds of 1986 estimated global consumption of the controlled substances.3 Moreover, Black et al. (1992) design the minimum participation constraint to solve a profitability problem(cooperation becomes profitable only if a minimum number of countries sign the agreement), whereas in thisarticle we also solve a stability problem (cooperation is undermined by the presence of free-riders).

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Endogenous Minimum Participation in International Environmental Treaties 413

to sign a treaty is often modeled as a two-stage game (see Barrett 1997b, 2003; Carraroand Siniscalco 1998; Tulkens 1998; Carraro and Marchiori 2003; Finus 2003, 2008; Rubioand Ulph 2006; for a few surveys). In the first stage (the coalition stage), countries non-cooperatively decide whether or not to sign the treaty by anticipating the consequences oftheir decision on the economic variables under their control. Then, in the second stage ofthe game (the policy stage), the equilibrium value of these variables is set. In this article,we introduce an additional stage preceding the coalition and policy stages, to endogenizethe choice of the minimum participation level. In this first stage, called the minimum par-ticipation stage, countries select the number of countries that must sign the treaty for it tocome into force.4 This decision is taken non-cooperatively and unanimously by anticipatingits implications on the second and third stage of the game.

Using the above theoretical framework, this article shows that, at the equilibrium, a non-trivial coalition will form, and will be sustained by a binding minimum participation con-straint. It also shows that the number of signatories is increased by the presence of a minimumparticipation rule, even when this rule is endogenous. This explains the occurrence of mini-mum participation clauses in most international environmental agreements. In addition, thearticle explores under which conditions it is optimal for countries to adopt the unanimity rule(in many treaties, as discussed in Rutz 2001, the minimum participation constraint coincideswith unanimous participation).

The idea that a non-trivial partial coalition can belong to the equilibrium structure of acoalition formation environmental game is not new. The aforementioned literature on inter-national environmental agreements (see, for example, Carraro and Marchiori 2003 or Finus2008 for a summary of the main results) shows that: (i) even in cases where there are positivespillovers (e.g., in the case of public good provision) and even without any commitment tocooperation, countries may form a coalition, i.e., may decide to sign a treaty in order to coop-erate to achieve a common target (Carraro and Siniscalco 1992; Barrett 1994 among others);(ii) the coalition is usually formed by a subgroup of the negotiating countries (Barrett 1994),but the small initial coalition can be expanded by means of transfers or through “issue link-age,” under some restrictive conditions (Carraro and Siniscalco 1993; Carraro and Siniscalco1995; Botteon and Carraro 1997); (iii) the outcome of the game is found to strictly dependon the membership rules (open membership, exclusive membership, coalition unanimity, …)that are adopted by the negotiating countries (Carraro and Marchiori 2003; Finus 2003);(iv) finally, when multiple coalitions form, the equilibrium is characterized by several, oftensmall, coalitions (Bloch 1997, 1996; Ray and Vohra 1997, 1999; Yi 1997, 2003).

The novelty of this article is the endogenization of the minimum participation constraintand the analysis of a country’s reasons to adopt it despite the presence of incentives to free-rideon the other countries’ emission abatement.

The structure of the article is as follows. Section 2 provides the main assumptions anddefinitions. Section 3 analyses the coalition game for a given value of the participation con-straint. Section 4 endogenizes the participation constraint, discusses the incentives to adopta participation rule and analyses its equilibrium value. A concluding section summarizes themain results and describes the scope for further research.

4 Results are robust to considering a given proportion of the externality globally produced (i.e., percentageof total emissions) as minimum participation constraint that countries signing the treaty must represent for thetreaty to come into force.

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2 Assumptions and Definitions

The three-stage game proposed in this article involves n countries, i = 1, . . . , n with n ≥ 3.Countries are assumed to be ex-ante identical. This assumption is necessary for each countryto have the same strategy space and payoff function in the third stage of the game (policystage).5 The structure of the game can be formally described as follows. In the first stage—theminimum participation stage—countries non-cooperatively choose the minimum participa-tion rule α by unanimity voting. They unanimously select the α representing the share of then countries that must sign the treaty for it to come into force, α ∈ [0, 1] being the strategy set.Let c ∈ {1, . . . , n} denote the number of cooperating countries; that is, the number of coun-tries that decide to sign the treaty.6 At the equilibrium and for a given α, a coalition c can formonly if c ≥ αn. In the second stage—the coalition stage—countries decide non-cooperativelywhether or not to sign the environmental treaty. The strategy set is thus binary: whether or notto join the coalition. The third stage—the policy stage—is as follows. Those countries thathave signed the agreement play as a single player and divide the resulting payoff accordingto a given burden-sharing rule (any of the rules derived from cooperative game theory). Theremaining countries play non-cooperatively against the coalition and against one another.7

In this last stage, the strategies are represented by the specific policy values that each countrycan set. In many IEAs such as the Kyoto Protocol, these are the emission reduction targetsthat each country decides to achieve.

The third stage of the game can be reduced to a partition function under suitable assump-tions (Bloch 1997; Ray and Vohra 1997, 1999; Yi 1997, 2003 , corresponding to assumptionsA.4 and A.5 below). The gains from cooperation can thus be described by a per-memberpartition function p : � → �n , a mapping which associates each coalition structure π witha vector of individual payoffs in �n . In particular p j (Ci ;π) represents the payoff of playerj belonging to the coalition Ci in the coalition structure π . Therefore, our study of coalitionformation consists of the analysis of the first and second stage of the game.

As to the first stage game, if α = 0, no restriction is introduced and a treaty comes intoforce whatever the number of its signatories. This is the so-called “open membership” rule,one of the most commonly assumed membership rules (Yi 1997, 2003). If α = 1, a treatycomes into force only if all countries decide to sign the treaty. In this case, we have what istermed the “coalition unanimity” rule (Tulkens 1998; Carraro and Marchiori 2003) or fullparticipation constraint. If α = 0.55, and countries are symmetric, we have the 55% ruleintroduced in the Kyoto Protocol. Therefore, by endogenizing α, we can determine whetherit is optimal for countries to opt for the open membership rule, the coalition unanimity ruleor any other minimum participation rule defined by α ∈ [0, 1].

In what follows, we formally state the assumptions characterizing the three-stage game.

5 The assumption of symmetry is standard and broadly used in the literature of international environmentalagreements (see Barrett 2007; Finus 2008; Rubio and Ulph 2006). The advantage of this assumption is toreasonably simplify the policy stage of the game, which allows us to focus on the coalition formation process.6 When c = 1, the coalition structure is trivial. All players are singleton and play non-cooperatively againstone another.7 These behavioral assumptions coincide with those defining the PANE equilibrium of Chander and Tulkens(1995). They can be contrasted with the traditional cooperative game approach (e.g., Chander and Tulkens1997) and with the repeated game approach (Barrett 1994, 1997). Moreover, note that the regulatory approachoften proposed in public economics is not applicable given the absence of a supranational authority.

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A.1. All players decide simultaneously at all stages.8

A.2. All players are ex-ante identical.9

Developments of the theory of endogenous coalition formation (Bloch 1997; Yi 2003) havestressed the implications of allowing players to form different coalitions. However, in manyenvironmental treaties (e.g., the case of climate change such as the Kyoto protocol, or ofthe Montreal protocol), the negotiating agenda focuses on a single agreement. Therefore,we assume that:

A.3. Players are invited to sign a single agreement. Hence, those that do not sign cannotpropose a different agreement.

From a game-theoretic viewpoint, A.3 implies that only one coalition can form, the remain-ing defecting players playing as singletons.

A.4. The third stage emission game has a unique Nash equilibrium for any coalition struc-ture.

A.4 is a necessary assumption for the existence of a well-defined partition function andallows us to focus on the second stage of the game; that is, on the process of coalition for-mation. In order to convert the strategic form game into a partition function, the competitionamong the coalition and the singletons must also be specified. The standard and perhapsmost natural assumption is that:

A.5. Inside a coalition, players make their policy decisions cooperatively in order to max-imize the coalitional surplus, whereas the coalition and the singletons compete with oneanother in a non-cooperative way.

Given A.5 and the symmetry among players, a coalition Ci can be identified just by its size,ci . Therefore, a coalition structure with only one coalition π = {c; 1n−c} can be denotedby π = {c}, the size of the only coalition that can possibly form. The partition functionis then obtained as a Nash equilibrium payoff of the game played by the coalition and thesingletons.Following Bloch (1996), we assume that the payoff inside a coalition is divided equallyamong the coalition members. In fact, in the case of identical players, equal sharing israther a result than an assumption. As shown by Ray and Vohra (1999), in a symmetriccontext any of the sharing rules derived from cooperative game theory (e.g., the Nash bar-gaining solution or the Shapley value) approximates the equal division rule. Therefore, inthe context of the present model each player inside a coalition receives the same payoffas the other members. This also implies that players’ payoff depends only on the coalitionsize and not on the identities of the coalition members. The per-member partition function(partition function hereafter) can thus be denoted by p(c; 1n−c), which represents the payoffof a player belonging to the size-c coalition.We also introduce an assumption which is quite natural in the case of international environ-mental agreements and simplifies the analysis of the coalition formation game:

A.6. Each player’s payoff function increases monotonically with respect to the coalitionsize (the number of signatories).10

8 By contrast, Barrett (1994) assumes that the group of signatories is Stackelberg leader with respect tonon-signatories. In Bloch (1997), it is assumed that players play sequentially.9 As previously explained, this assumption means that each country has the same strategy space and payofffunction in the third stage of the game.10 Note that the monotonically increasing payoff function is a weaker requirement than superadditivity, sinceit does not impose any structure on the rate of increase. Monotonicity of the payoff function excludes the

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Since the global environment is a public good, a country that finds it convenient to reduceits own emissions provides a positive contribution to the welfare of all countries, both insideand outside the coalition. Formally, A.6 implies that both members and non-members’ par-tition functions are increasing in c for all values of c ∈ {1, . . ., n}. As for those outside acoalition, the assumption that the effort undertaken by cooperators makes them better off istypically referred to as ‘positive externality’. The presence of positive externalities, in turn,gives room to the well known ‘free-riding problem’, which is a key feature of the gameanalyzed in this article. Players’ payoffs will be fully characterized in Sect. 3, where thecoalition stage is formally described.Finally, the first stage of our game is a sort of constitutional stage in which the rules of thegame are set. Therefore:

A.7. In the minimum participation stage, decisions are taken under unanimity.

Players decide the minimum participation rule by unanimity voting, as no player can beforced to accept a minimum participation rule he does not agree upon. The decision is takennon-cooperatively and simultaneously. Each player’s strategy consists of choosing a ruleα ∈ [0, 1] . [0, 1] represents the strategy set, where α = 0 entails the absence of participationconstraint, α = 1 implies that all players are required to join the coalition for it to comeinto force, and any fraction α ∈ (0, 1) represents a minimum participation constraint of αncountries. A player’s utility associated to these strategies is represented by the expected valuebetween a cooperator’s and a free rider’s partition functions of the coalition formed underα. The exact payoffs will be defined in Sect. 4 , which specifically analyses the minimumparticipation stage. Finally, the decisions made in the first stage of the game are irreversible.Given that this is the constitutional stage, it is assumed that once a minimum participation ruleis set, it cannot be renegotiated in the later stages of the game. This is a standard assumptionin sequential games.

In the next sections, we will analyze the equilibrium of the three-stage game with endoge-nous minimum participation. Given the partition function described above, we can directly fo-cus on the coalition stage of the game and then move, by backward induction, to the minimumparticipation stage, in which the optimal minimum participation constraint is determined.

3 The Coalition Stage

The second stage of the game is a binary choice game (joining the coalition or behaving asa lone free-rider) whose outcome is a single coalition structure π = {c}, where n – c playersremain singletons. Given the above assumptions, a coalition member’s partition function isincreasing in c for all values of c ∈ {1, . . .n} and the presence of positive spillovers impliesthat the partition function of any player outside the coalition is also increasing in c for allvalues of c ∈ {1, . . .n}.

Let us denote by Q(c) the relative non-member partition function p(1; c)−p(1; 1n), wherep(1; 1n) is a player’s partition function when no coalition is formed. Similarly, the relativeper-member payoff function P(c) is equal to p(c; 1n−c) − p(1; 1n). From A.6, we focuson the case in which P(c) and Q(c) are monotonic. As said, this case is relevant because itcharacterizes environmental negotiations and more generally the provision of public goods.

Footnote 10 continuedpossibility of “exclusive membership equilibria”, where the group of cooperating players can refuse entry to aplayer who wants to join the coalition (Cf. Yi 1997). Our main results can also be extended to non-monotonicpayoff functions.

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Let us start the analysis by considering the case in which no minimum participation ruleis introduced. In this case, α = 0 (open membership rule) and the equilibrium of the coa-lition stage is completely characterized by the following two properties, first derived in thecartel literature (d’Aspremont et al. 1983) and then often used in the environmental literature(Carraro and Siniscalco 1993; Barrett 1994; and many others).

3.1 Profitability

A coalition c is profitable if each cooperating player gets a payoff larger than the one hewould obtain when no coalition forms. Formally:

P (c) ≥ P (1) = 0, (1)

where P(c) is the relative payoff of a cooperating player member of the coalition of size c,and P(1) is the relative payoff associated to the singleton structure.

3.2 Stability

A coalition is stable if it is both internally and externally stable. It is internally stable if nocooperating player is better off by defecting in order to form a singleton.11 Formally12:

P (c) ≥ Q (c − 1) . (2a)

It is externally stable if no singleton is better off by joining the coalition c. Formally:

Q (c) > P (c + 1) (2b)

To determine the stability of a coalition we use the stability function L(c) = P(c) −Q(c − 1) proposed in Carraro and Siniscalco (1992). If L(c) is positive, there is no incentiveto free-ride on the coalition of size c.

The size of the stable coalition is determined by conditions (2a)(2b),13 i.e., the largest inte-ger c* smaller than c, with c defined by L(c) = 0 and L′(c) < 0 (assumption A.6 guaranteesthe uniqueness of c).14 The size c* identifies the minimal externally stable coalition and themaximal internally stable one. Hence, the equilibrium coalition structure is π∗ = {c∗}, withn − c∗ singletons. Note that c* is not necessarily greater than one. There exists an extensiveliterature (e.g., Barrett 1994) showing that an equilibrium stable coalition generally existsunder A.1–A.6.12 Let us denote by c∗

0 the coalition formed under the open membership ruleα = 0. Then:

Proposition 1 Let A.1–A.7 hold and α∗ = c∗0/n. The equilibrium of the coalition game for

any value of α ∈ [0, 1] is the coalition structure:π∗(α∗) = {c∗

0, 1n−c∗0} (4a)

for any minimum participation rule such that 0 ≤ α ≤ α∗, and

11 Yi (1997) denotes this condition by the term “stand alone stable”.12 We assume that if a player is indifferent as to joining the coalition or defecting, then he joins the coalition.13 (1) is only a necessary condition implied by internal stability under A.6.14 This result, as well as many others on the size of stable coalitions, is shown in Carraro and Marchiori(2003).

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π(α) = {cα = αn, 1n−cα } (4b)

for any minimum participation rule such that α∗ < α ≤ 1.

Proof If α ≤ α∗, the equilibrium coalition is not modified by the presence of a minimumparticipation constraint α. Indeed, if in the first stage players set a minimum participationrule α ≤ c∗

0/n, the number of signatories necessary for the treaty to come into force issmaller than or equal to the number of countries that would sign the treaty anyway. Hence,the equilibrium coalition remains c∗

0. On the other hand, if in the first stage players agree ona minimum participation rule α > α∗ = c∗

0/n, then in the second stage they have the choiceeither to form a coalition of size cα = αn, or not to form any coalition at all. Indeed, allcoalitions larger than cα are unstable, because all coalitions larger than c0* are unstable andcα > c∗

0. Moreover, no coalition smaller than cα = αn can form, by definition of minimumparticipation. Given that, by assumption A.6, Q(c − 1) > P(c) > P(1) = 0, players preferto form the coalition cα = αn. �

Proposition 1 identifies the relationship between the minimum participation rule decidedin the first stage of the game and the equilibrium of the coalition stage. This relationshipbetween the value of α and the equilibrium of the coalition stage, jointly with the incentivestructure that supports the equilibrium coalition c∗

0, is crucial to determine the equilibriumvalue of α in the minimum participation stage.

4 The Minimum Participation Stage

When countries deal with the choice of the membership rule, they consider two aspects:

(1) If it is convenient to introduce a minimum participation constraint rather than to nego-tiate under the open membership regime;

(2) Which minimum participation constraint is optimal.

As previously discussed, the decision in the first stage of the game is taken under theunanimity rule, as no player (country) can be forced to accept a participation constraint hedoes not agree upon. In order to achieve unanimity on a value of α, all players must (weakly)prefer the equilibrium supported by the agreement on a minimum participation constraint tothe one that would emerge without a minimum participation rule. This is the case if, whateverthe choice of a player in the second stage, a player’s payoff when the minimum participationconstraint is introduced is larger than (or equal to) the payoff he/she would receive in theabsence of any minimum participation constraint. Specifically, we can state the following:

Proposition 2 Let A.1–A.7 hold. In the first stage of the game, all players agree to introducea minimum participation constraint α = c/n iff :

P(c) ≥ Q(c∗0) (5)

Proof For all players to agree on a minimum participation constraint, we must have:

– P(c) ≥ P(c∗0), i.e., all players who cooperate in c and in c∗

0 are better off in c;– Q(c) ≥ P(c∗

0), i.e., all players who are free-riders with respect to c but cooperators in c∗0

are better off free riding with c;– P(c) ≥ Q(c∗

0), i.e., all players who cooperate in c but free-ride with respect to c∗0 are better

off in c;

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– Q(c) ≥ Q(c∗0), i.e., all players who free-ride both with respect to c and to c∗

0 are better offfree riding with c.

If these four conditions are satisfied, all players find it convenient to agree on the minimumparticipation constraint α that sustains the coalition of size c. Notice that c > c∗

0 (otherwisethe coalition is of size c∗

0) by Proposition 1. Hence, P(c) > P(c∗0) by Assumption A.6. More-

over, Q(c) is also monotonically increasing, which implies Q(c) > Q(c∗0) when c > c∗

0. Thesetwo results, the definition of c∗

0, and c > c∗0, imply Q(c) > Q(c − 1) > P(c) > P(c∗

0). As aconsequence, Eq. 5, i.e., P(c) ≥ Q(c∗

0), is a necessary and sufficient condition for all playersto agree on the selection of a minimum participation rule such that the equilibrium coalitionstructure becomes {c, 1n−c}. �

Can inequality (5) be met for c ≤ n? The answer is positive. Given the monotonicityof the functions P(c) and Q(c), it is possible—if n is sufficiently large—to find a coalitionc large enough to satisfy P(c) ≥ Q(c∗

0). In particular, assume that n is such that there is ac+ < n that satisfies P(c+) = Q(c∗

0). The monotonicity of P(c) implies that (5) is met for allc ∈ {c+, . . ., n}.

Let us consider a simple example, as an illustration of Proposition 2 and of the scope ofcondition (5). The following basic economic-environmental model with linear cost and qua-dratic benefit functions is also used in Barrett (1994, 1997); Carraro and Siniscalco (1992);Chander and Tulkens (1997) and many others.

Let n be the number of identical countries which emit a pollutant that damages a commonenvironmental resource. The utility u of each country depends on the benefits arising fromthe use of the environment for production and consumption activities and on the damageresulting from polluting emissions:

u = B(xi ) − D(X) i = 1, 2, . . . , n

where xi is country i’s emission level and X is the total amount of polluting emissions(X = �n

i=1xi ). Let the benefit function B(xi ) be quadratic and the cost function (or damagefunction) be linear:

B(xi ) = k

(dxi − x2

i

2

)

where d is the maximum level of emissions and k is a technological parameter which param-eterizes total benefits from emissions (the higher k, the larger the benefit);

D(X) = m

2[xi + (X − xi )]

where m parameterizes the level of perceived damage from pollution.A country’s payoff can, therefore, be written as follows:

u = k

(dxi − x2

i

2

)− m

2[xi + (X − xi )]

In a coalition structure π = {c, 1n−c}, countries inside the coalition act cooperatively andchoose their emission level by maximizing the sum of the payoffs of all members. Hence,for a coalition structure π = {c, 1n−c}, the payoff of a cooperating player can be written as:

uc = k

(dxc − x2

c

2

)− m

2

[xc + (

(c − 1) xc + (n − c) x f)]

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where xc and x f are the emission levels of cooperators and free-riders, respectively. Differ-entiating uc with respect to xc yields x∗

c = d − mc/2k, which is the optimal emission levelfor a cooperating country.

A free-rider’s payoff function is given by the following expression:

u f = k

(dx f − x2

f

2

)− m

2

[x f + (n − c − 1) x f + cxc

].

Differentiating u f with respect to x f yields x∗f = d − m/2k. By replacing the values of

x∗c and x∗

f into the payoff functions uc and u f , we obtain the payoffs for cooperators andfree-riders as a function of the coalition size c:

P(c) = k

[d

(d − mc

2k

)− 1

2

(d − mc

2k

)2]

+ m

2

[(d − mc

2k

)+ (c − 1)

(d − mc

2k

)+ (n − c)

(d − m

2k

)]Q(c) = k

[d

(d − m

2k

)− 1

2

(d − m

2k

)2]

+ m

2

[(d − m

2k

)+ (n − c − 1)

(d − m

2k

)+ c

(d − mc

2k

)]Let m = 2k and k = d = 1. By computing the stability function L(c) = P(c)− Q(c−1),

equalizing it to zero and solving with respect to c, the following result can be derived: c∗0 = 3.

It can also be shown that condition (5) is verified for c ≥ 5. Therefore, Proposition 2 holdsin this setting.15

However, (5) is only a necessary condition to identify the minimum participation constraintwhich is chosen by the n countries. The inequality (5) defines the set of minimum partici-pation rules that would be voted with the unanimous consensus of all countries. However,which rule is actually going to be adopted?

To answer this question, we must write the optimization problem that each country solveswhen deciding the optimal minimum participation rule. Given symmetry, in the first stage ofthe game a country does not know whether in the coalition stage it will be a cooperator or afree rider. Therefore, it determines its optimal choice in the first stage (the value of α in thestrategy set [0,1]) by solving:

maxc

E X (c) = c

nP(c) + n − c

nQ(c)

s.t. P(c) ≥ Q[c∗0], with 1 ≤ c ≤ n (6)

where E X (c) is a country’s expected payoff in the first stage of the game, c/n is the proba-bility of being a cooperator in the second stage (given that the rule α = c/n is chosen in thefirst stage), whereas (n − c)/n is the probability of being a free-rider in the second stage ofthe game.

The equilibrium minimum participation rule α◦ = c◦/n is the solution to (6), where c◦is the maximizer of E X (c) subject to (5). The optimization problem in (6) identifies thetrade-off that countries face. On the one hand, a high value of c, namely of α, increases thepayoff of both cooperators and free-riders. On the other hand, a high value of α reduces

15 Condition (5) may hold also in a quadratic benefit-quadratic damage model. Using for example the modelproposed in Diamontoudi and Sartzetakis (2006), where the cooperators’ payoff function ωs(.) correspondsto our payoff function P(.), and the free riders’ payoff function ωns(.) corresponds to Q(c), condition (5) holdsfor c ≥ 4.

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the probability of being a free-rider in the second stage, thus reducing the probability ofreaping the highest payoff. Note that the equilibrium in the first stage always exists and isunique. In particular, α = 0 represents the equilibrium solution when countries do not unan-imously agree on any α > 0, i.e., no country finds any α > 0 more profitable than α = 0. Ifat least one country finds a α◦ > α∗ optimal, then by symmetry all countries will: α◦ will beunanimously voted and will represent the unique equilibrium (α◦ dominating strategy for allplayers).

Under what conditions is E X (c) increasing in c for all values of c satisfying (5)? Thisquestion is relevant because it identifies the case in which c◦ = n, or α◦ = 1, i.e., in which theminimum participation rule requires the ratification by all countries and therefore coalitionunanimity. A sufficient condition is provided by the following:

Proposition 3 Let A1–A7 hold. The equilibrium minimum participation rule is α◦ = 1, i.e.,the treaty comes into force only when all countries sign and ratify it, if:

P(n) ≥ Q[c∗0] (7a)

(1 + γP )P(c + 1)/Q(c + 1) ≥ 1 (7b)

∀c ≥ c+, where c+ is the solution of P(c+) = Q(c∗0) and γP = (c/P(c + 1)) ∗ (�P(c)/�c)

is the quasi-elasticity of the function P(c).

Proof The first condition, (7a), is the necessary condition already established by Proposi-tion 2. It is met for n sufficiently large. The second condition, (7b), can be proved as follows.A discrete change of n*EX(c) with respect to c can be written as follows:

� [cP(c) + (n − c)Q(c)] /�c

= [�c ∗ P(c + 1) + c ∗ �P(c) + �(n − c) ∗ Q(c + 1) + (n − c) ∗ �Q(c)] /�c

= P(c+1)+c ∗ �P(c)/�c−(�c/�c) ∗ Q(c+1)+n ∗ �Q(c)/�c−c ∗ �Q(c)/�c

= P(c + 1) ∗ [1 + (c/P(c + 1)) ∗ �P(c)/�c] −−Q(c + 1) [1 − ((c/c) ∗ n/Q(c + 1))∗(�Q(c)/�c) +(c/Q(c + 1)) ∗(�Q(c)/�c)]

Then, let γP = (c/P(c + 1)) ∗ (�P(c)/�c) and γQ = (c/Q(c + 1)) ∗ (�Q(c)/�c) bethe quasi-elasticity of P(c) and Q(c), respectively. Substituting above, we get:

� [cP(c) + (n − c)Q(c)] /�c = P(c + 1) ∗ [1 + γP

] + Q(c + 1) ∗ [γQ ∗ (n/c − 1) − 1

]The equilibrium minimum participation constraint is α = 1 if � [cP(c) + (n − c)Q(c)] /�cis positive for all c ∈ [

c+, n], i.e.,

P(c + 1) ∗ [1 + γP

] + Q(c + 1) ∗ [γQ ∗ (n/c − 1) − 1

] ≥ 0 (8)

Dividing by Q(c+1) both sides (Q(c+1) is a positive number), we get:

(P(c + 1)/Q(c + 1)) ∗ [1 + γP ] + γQ ∗ (n − c)/c ≥ 1 (9)

Note that both terms on the left hand side are positive.Hence, if (1 + γP ) ∗ (P(c + 1)/Q(c + 1)) ≥ 1, then (9) holds. �Note that P(c+1)/Q(c+1) < 1 for c > c∗

0 by definition of stability, and (n − c)/c < 1 if c isrelatively high with respect to n. Both (n − c)/c and P(c+1)/Q(c+1) decrease with c whenP and Q are monotonic and c∗

0 ≥ 2. Therefore, unless the quasi-elasticities strongly increase

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with c, (9) is less likely to be met for high values of c. The strong condition to be met forEX(c) to be increasing for all c ≥ c+, including c = n, is (7b), i.e.,

(1 + γP )P(c + 1)/Q(c + 1) ≥ 1

which coincides with (9) for c=n. This condition is satisfied only if γP is sufficiently large,i.e., the profitability function increases sufficiently rapidly with c. The curvature of the coa-lition’s payoff function is therefore crucial. The intuition is that high and increasing valuesof P(c) when c is close or equal to n offset the loss induced by the impossibility of free-ridingin the second stage.

However, in situations in which there are increasing free-riding incentives for larger coa-lition sizes, condition (7b) is not likely to be satisfied. Suppose, for instance, that P(c) isconcave. In such a case, the quasi-elasticity of P(c) becomes smaller and smaller as c movestoward n. At the same time, by definition of stability, the difference between Q(c) and P(c)increases with c. Therefore, P(c + 1)/Q(c + 1) < 1 also becomes smaller and smaller. Asa consequence, (1 + γP )P(c + 1)/Q(c + 1) may become smaller than one and (7b) maynot be met. If benefits from cooperation do not increase with the number of cooperators, orincrease too slowly, then it is not optimal to set a minimum participation constraint such thatall countries must sign and ratify the treaty for it to come into force (that is, α = 1). Instead,the optimal minimum participation rule will be smaller than one, which in turn implies that,at the equilibrium, some countries free-ride on the cooperative behavior of the others. Bycontrast, if γP is relatively large—e.g., because P(c) is convex—the ratio P(c +1)/Q(c +1)

is closer to one and (7b) is more likely to be satisfied for c = n. If the benefits from coopera-tion are rapidly increasing with the number of cooperators, no player has an incentive to takethe risk of being a free-rider. Note that this conclusion crucially depends on the assumptionof symmetric countries.

The evidence actually shows that several IEAs met the condition of high and increasingprofitability for high levels of participation, leading to the selection of the unanimity par-ticipation rule. Rutz (2001) reports that more than one fifth of multilateral environmentalagreements with a minimum participation provision indeed require full cooperation for thetreaty to come into force.16

In the case identified by (7b), all countries not only prefer to “tie their hands” in the firststage, but also decide to “tie all hands”, thus giving up the possibility of free-riding in thesecond stage in order to reap the benefits arising from full cooperation. As said, 22 out of122 multilateral environmental agreements share this feature.

Using the simple economic model previously introduced, we can compute the expectedutility function EX(c) and find that it is proportional to the following expression:

[c(

d − mc

2k

)(kd

2+ mc

4− m N

2

)+ (N − c)

(d − m

2k

) (kd

2+ m

4− m N

2

)]

16 In some international environmental agreements, the equilibrium number of signatories is above the min-imum participation clause introduced in these treaties. Indeed, there may be circumstances related to thespecific shape of players’ payoff functions in which the minimum participation rule is not binding. Consider,for instance, a case discussed in Carraro and Marchiori (2003). In that case, beyond a certain participationthreshold k, all players have the incentive to join the coalition. As such, a minimum participation rule equalto k/n would lead to the grand coalition. In this case, the minimum participation rule simply induces a band-wagon effect. This is an example of why the number of signatories can be larger than those identified by theminimum participation rule.

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Moreover, this function increases monotonically with c within the interval[c+, n

]if and

only if the following inequality holds:

E X (c) − E X (c − 1) > 0 f orc ∈ [c+, n]That is:

c(

d − mc

2k

)(kd

2+ mc

4− mn

2

)− (c − 1)

(d − m(c − 1)

2k

) (kd

2+ m(c − 1)

4− mn

2

)

−(

d − m

2k

)(kd

2+ m

4− mn

2

)> 0

It can be shown that the above inequality holds for 1 < c < 43 n. Therefore, the expected

utility function E X (c) increases monotonically with c within the interval(1, 4

3 n). As a con-

sequence, the equilibrium minimum participation constraint is α◦ = 1 (coalition unanimity)and the equilibrium coalition size is c◦ = n (full cooperation).

It is worth noting that in the presence of structural asymmetries, the minimum participationconstraint α = 1 does not necessarily guarantee full cooperation because some countries maybalk from signing the agreement (the profitability condition may not be met for all countriesif they are different from one another). In such a situation, the requirement of full cooperationmay even be counterproductive because it may lead to no cooperation, unless coupled with atransfer scheme that guarantees the profitability of the agreement for all countries (e.g., theone proposed in Chander and Tulkens 1995, 1997). Allowing for players’ heterogeneity in thecontext of endogenous minimum participation rules would require a more complicated gamestructure with strategy sets also including compensation schemes across different players.We leave this extension to future research.

5 Conclusions

In this article, we endogenised the choice of a minimum participation rule in international(environmental) treaties. A minimum participation rule was modeled as the equilibrium out-come of a three-stage coalition formation game. In the first stage, countries unanimously setthe minimum coalition size that is necessary for the treaty to come into force. In the secondstage, they decide whether to sign the treaty. In the last stage, they set the equilibrium valuesof the policy variables.

When choosing a minimum participation rule, countries face the following trade-off: onthe one hand, by setting a high participation constraint, they can increase the payoff of bothcooperators and free-riders. On the other hand, a high minimum participation rule reduces theprobability of being a free-rider in the second stage, thus reducing the probability of reapingthe highest payoff.

In the first part of the analysis, we identified the condition under which all countries find itstrategically convenient to limit their freedom to free-ride and introduce a minimum partici-pation constraint. This condition is represented by (5). We also showed that the endogenouslychosen minimum participation rule achieves the goal of enlarging the equilibrium numberof signatories. This prediction is consistent with the results of traditional models with exog-enous minimum participation rules and explains why most environmental treaties include aminimum participation rule. In the second part, we focused on the stringency of the partic-ipation constraint. We characterized the optimal minimum participation rule and analyzedunder which conditions it coincides with coalition unanimity. We showed that the choice offull participation as a minimum participation rule crucially depends on the quasi-elasticity

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of the payoff function of cooperating countries. When this is sufficiently large for c close ton, then all countries prefer a minimum participation constraint that induces full cooperation.

These results are meant to be useful for the design of actual policy agreements. A min-imum participation rule is generally helpful to enhance the environmental effectiveness ofan international agreement. The optimal rule can even be coalition unanimity, in particularwhen the number of negotiating countries is not too large. However, the unanimity constraintis effective only if the profitability condition is met for all countries, otherwise it may becounterproductive. Indeed, a country for which the agreement is not profitable would notsign it. As a consequence, unanimity would prevent the treaty to come into force, even ifall other countries would sign it. This case, however, can be a feature of games with asym-metric players, and does not arise in the symmetric player framework considered in thisarticle. Extensions of our results to asymmetric players would also be useful to identify thecharacteristics of countries that belong to the equilibrium coalition.

Acknowledgments The Authors are grateful to Scott Barrett, Maria Paz Espinosa, Michael Finus, Jeff Fren-kel, Jim Hammitt, Marita Laukkanen, Cynthia Lin, Rob Stavins, Martin Weitzman, two anonymous reviewersand participants at the Coalition Theory Network Workshop and at seminars in Paris and Harvard. The usualdisclaimer applies.

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