Raouf Boucekkine
email: raouf.boucekkine@rennes-sb.com

Carmen Camacho
email: carmen.camacho@psemail.eu

Weihua Ruan
email: wruan@pnw.edu

Benteng Zou
email: benteng.zou@uni.lu.

may come from teaching and research institutions in France or abroad, or from public or private research centers

Keywords: Coalition splitting, environmental agreements, constitutional vs technological heterogeneity, differential games, multistage optimal control. JEL classification: C61, C73, D71

We propose an alternative dynamic theory of coalition breakdown. Motivated by recent coalition splitting events through unilateral countries' withdrawals, we assume that: i) the payoff sharing rule within coalitions is not necessarily set according to any optimality and/or stability criterion, and, ii) players initially behave as if the coalition will last forever. If the sharing rule is non-negotiable or if renegotiation is very costly, compliance to these rules may become unbearable for a given member because the rule, being too rigid, would make exit preferable as time passes. We examine this endogenous exit problem in the case of time-invariant sharing rules. Assuming a Nash non-cooperative game after (potential) splitting where players play Markovian, we characterize the solutions of the endogenous exit problem in a linear-quadratic frame with endogenous splitting time. We find that splitting countries are precisely those which use to benefit the most from the coalition. Suitable sharing rules should be used to prevent coalition splitting. When initial pollution is high, all shares should be low enough and none of the players should detain a payoff share larger than 1/2. If initial pollution is small, we provide with an explicit interval for the sharing rule values preventing the collapse of the coalition. Finally, we demonstrate that the latter properties are qualitatively consistent with the optimal behaviors and equilibrium outcomes resulting from players anticipating the end of the coalition and acting accordingly.

Introduction

While the classical literature on coalitions had essentially addressed the question of coalition formation and stability (with a few exceptions though, see [START_REF] Bolton | Economic theories of the break-up and integration of nations[END_REF], numerous papers on coalition break-ups have been written in the last few years. This abundant literature is essentially motivated by the recent numerous withdrawals of countries from international organizations and agreements, some highly impactful. Beside some of the decisions taken by the Trump administration, which may seem "idiosyncratic" 1 , the United Kingdom withdrawal from the European Union on January 31, 2020, or Canada withdrawal from Kyoto Protocol on December 13, 2011, 10 years after the US, are two of these striking break-up events which have attracted the attention of economists and political scientists. Just as a way of illustration let us mention papers investigating the impact of Brexit [START_REF] Sampson | Brexit: The Economics of International Disintegration[END_REF]; La Torre et al., 2020; the special issue of the Oxford Review of Economic Policy, vol 33, 2017; etc.), or the economic consequences of the U.S. withdrawal from the Kyoto Protocol and the Paris agreement [START_REF] Bucher | Economic consequences of the US withdrawal from the Kyoto/Bonn Protocol[END_REF][START_REF] Zhang | US withdrawal from the Paris Agreement: Reasons, impacts, and China's response[END_REF]; Nong and Siriwardana, 2018; ...). 2 This paper is a methodological contribution to this rising literature. The traditional game-theoretical settings proposed to study the design of international agreements and the stability of coalitions are quite diverse: they range from cooperative to non-cooperative, from static to dynamic through repeated games or fully dynamic set-ups, and they often include interesting procedural ingredients, typically on enforceability of the agreements. Regarding the particular problem of coalition splitting, one finds two main conceptual settings. The first one is based on the theory of coalition stability, which is anchored in the cooperative games literature. 3 A second type of setting uses the traditional Nash non-cooperative theory with individual optimizing strategies (plus a Pareto-like criterion to evaluate efficiency). An essential part of this literature uses dynamic games. Ngo Van Long (see, for example, Van Long, 2010, for a survey) is one of the principal contributors to this line of research.

Our paper departs from the latter dynamic games literature in two major ways:

• First of all, the sharing rule within coalitions is not necessarily set according to any optimality and/or stability criterion.

• Second, players initially behave as if the coalition will last for ever. 1 For example: on July 7, 2020, the Trump administration formally notified the United Nations that the U.S. was pulling out of the World Health Organization, which became effective as of July 6th, 2021.

2 See also the empirical study of [START_REF] Mayer | The cost of non-Europe, revisited[END_REF] on the cost of being non-EU, and the general theoretical investigation of [START_REF] Gancia | A theory of economic unions[END_REF] on the gain of being in some economic unions and partnerships. 3 Two different frames are considered in this literature. The first looks at static or repeated games as explained by [START_REF] Tulkens | Cooperation vs. free riding in international environmental affairs: two ap-proaches[END_REF] and surveyed by [START_REF] Bréchet | Efficiency vs. stability in climate coalitions: A conceptual and computational appraisal[END_REF]. The second deals instead with dynamic games and, after [START_REF] Petrosjan | Stable Solutions of Differential Games with Many Participants[END_REF], searches for sharing mechanisms that will ensure the coalition stability (see Zaccour, 2007, for more details).

At first glance, the two departures seem rather realistic. The coalitions are typically based on a number of, say, constitutional rules specifying the duties and benefits corresponding to each member of the coalition. Usually, the coalition may obviously entail large heterogeneities across members, in particular in multi-country coalitions, technological, demographic or geographic notably. It's unlikely that the constitutional rules at the dawn of the coalition can cope with all these discrepancies, and meet any kind of optimality in the sense of the criteria given above (for example, Nash bargaining or Shapley value). It's also quite reasonable that when engaging in essential (and somehow existential) collective initiative like a strategic political alliance or a long-term environmental agreement, no member will start playing against it, they will rather act optimally in accordance with the constitutional rules, as if the coalition will last till the end time agreed upon. If the constitutional rules are non-negotiable or if renegotiation is very costly, compliance to these rules may become unbearable as time passes, either because the rules, being too rigid, would make exit preferable (endogenous exit) or because an exogenous (symmetric or asymmetric) shock occurs undermining the political, economic or historical rational behind the coalition creation. We shall consider the endogenous exit problem in this paper, specializing in the case of time-invariant and non-negotiable constitutional rules. As we shall see, the problem remains largely nontrivial in this benchmark case.

While realistic and fitting a variety of situations in very different contexts, our assumptions entail two theoretically unpleasant features: predetermined (non-optimal) shares under the coalition setting and time inconsistency. As we will see clearly when solving the dynamic games posed, our alternative frame involves the absence of forward-looking behavior in the coalition stage, which enables a forward induction solution method. We study this case till its ultimate consequences, including policy implications (which does make perfect sense as this case is based on reasonably realistic assumptions). Nonetheless, we also provide with the solution to the standard forwardlooking counterpart where players do anticipate the coalition breakdown from t = 0 and behave accordingly. Despite that the induced solution scheme is opposite to the counterpart in our alternative case and consists in backward induction, we show that the ultimate optimal splitting problems are analogous. Moreover the main policy implication (related to the sharing rule) is qualitatively the same. That's to say the equilibrium outcomes generated by our alternative frame is quite far from irrationality (if rationality corresponds to the pure forward-looking case).

In our theory, since we focus on coalition splitting, we specialize in the simple case where one single country can potentially break down its tie to the coalition. After the split, if any, a two-players non-cooperative game sets in: the splitting country and the remaining coalition treated as a single player. This mimics many of the recent coalition withdrawals occurrences mentioned above, and we believe the theoretical approach taken is deep enough to highlight some of the key determinants of coalition breakdowns as we will argue throughout the text. Another critical aspect of our theory is (non)-negotiability. Non-negotiable here means that the core principles establishing the initial coalition (say, the sharing rules) cannot be redefined to strictly cope with the preferences of any single country. For instance, we have in mind the case of the U.S. splitting from the 2015 Paris agreement or the case of UK splitting from the EU. We do precisely formalize this problem by designing a tractable dynamic game-theoretic frame assuming time-invariance of the sharing rules under coalition.

In this paper, we specialize in the theoretical literature of environmental agreements to fix the ideas. Fundamental work has already been accomplished on these issues in a variety of frameworks ranging from multistage games à la [START_REF] Carraro | Strategies for the international protection of the environment[END_REF] to dynamic games [START_REF] Hoel | Intertemporal properties of an international carbon tax[END_REF][START_REF] Xepapadeas | Induced technical change and international agreements under greenhouse warming[END_REF][START_REF] Dutta | A strategic analysis of global warming: Theory and some numbers[END_REF]. Here, we take a different approach to better suit the question of coalition splitting as we have posed it just above. We consider a set of countries which are initially bound in a coalition, and whose aim it to maximize a given joint payoff subject to a public bad. The coalition is based on time-invariant (typically suboptimal) constitutional rules, namely the sharing quotas of the benefits and the costs of the coalition. Given all these ingredients, under which conditions a country initially belonging to this coalition may eventually optimally decide to split at a finite date, and when? What are the determinants (Constitution, technology,...etc) of splitting and of the duration of the coalition? Is it possible to identify time-invariant consititional rules which prevent the coalition breakdown?

In our framework, the splitting time is an explicit optimal control in the hands of any member of initial coalitions. The recent contribution of [START_REF] Colombo | A dynamic analysis of international environmental agreements under partial cooperation[END_REF] is close to our setting in investigating partial cooperation in international environmental agreements. There, all players are identical and so is the share of each player in joint welfare (regardless of whether there is full or partial cooperation). In their setting, the coalition that is optimally set at the initial time will last forever. Our work is also related to the seminal contribution of [START_REF] Benchekroun | Temporary natural resource cartels[END_REF], who study the temporary natural resource cartels where the cartels' ending time is known for all players at the beginning of the game, while in our setting it is a decision of one player. More precisely, we assume that initially players (countries) agree to manage cooperatively the common stock of pollution. As a shortcut to the constitutional aspects of the coalition, we assume that each country enters the coalition with a given fixed share of the (intertemporal) payoff of the coalition. We do not include splitting costs (formalizing possible penalties paid by the splitting country) for simplicity. As it will be clear in the main text and in the Appendix, the algebraic developments needed without this additional ingredient are already huge. Note that a splitting fixed sunk cost can hardly change the qualitative results in terms of the sustainability of time-invariant constitutional rules. If a country splits at time T , a non-cooperative game sets in between the country and the group of countries remaining in the coalition. Within a fully linear-quadratic model, we characterize the optimal affine Markovian subgame perfect strategies for a given split time T . We later solve for the whole sequence starting with the initial cooperative game phase where all coalition members play as if the coalition will last for ever. As argued above, if this assumption fails, then a different solution setting via backward induction should be applied reflecting the forward-looking nature of splitting in such a case. We ultimately uncover the conditions under which splitting occurs at finite time. We also study the determinants of the coalition duration with particular attention to the role of technological vs constitutional heterogeneity across players. It's worth pointing out that the choice of strategies can be extended to non-Markovian ones. For example, open-loop Nash equilibria or heterogeneous Nash equilibria à la Zou (2016) may match better some situations. For example, after the U.S. withdrawal from the Paris agreement, the remaining coalition stayed committed to the initial decarbonization objectives. Nevertheless, the techniques developed in this paper are general enough to study different kinds of choices of strategic spaces. 4Technically, our analytical approach combines multistage optimal control tools with the typical techniques used to solve differential games. There exist an increasing number of papers using multistage optimal control to characterize optimal/ equilibrium regime transitions and the inherent optimal regime shift timings [START_REF] Boucekkine | Optimal regime switching and threshold effects[END_REF][START_REF] Moser | History-dependence in production-pollution-trade-off models: a multi-stage approach[END_REF][START_REF] Saglam | Optimal pattern of technology adoptions under embodiment: A multi-stage optimal control approach[END_REF]Zampolli et al. 2016; etc.). 5 In contrast, much fewer papers merging multi-stage optimal control and dynamic games have come out. 6 We shall show how the latter avenue can also be taken safely in our paper.

Three key aspects drive the paper's results: the technological gap as an indicator of heterogeneity across players, the Constitution of the coalition (captured by a single parameter, the payoff share accruing to countries under coalitions) and the pollution damage. Thanks to these parsimonious specifications, we are able to provide with a full analytical solution to the two-stage differential game under scrutiny. We do cover all the set of parameterizations taken by the three indicators listed above, which results in a highly nontrivial mathematical analysis (despite parsimony). In particular, we characterize the intermediate parametric cases leading to optimal finite time splitting. We specially highlight the requirement that the payoff share accruing to the splitting country should be large enough in the latter case. Consistently, we prove that constraining the payoff share to be low enough by Constitution may lead to optimal everlasting coalitions only provided initial pollution is high enough, which may cover the emergency cases we are witnessing nowadays.

The paper is organized as follows. Section 2 presents the general specification of our gametheoretical setting. Section 3 analyzes a specialized linear-quadratic version of the game, providing in particular the optimal players' strategies for given splitting times. Section 4 characterizes the existence of an optimal splitting time, discusses its drivers and delivers some policy insights. Section 5 addresses the case of an uncommited coalition where the players anticipate splitting and behave in a forward-looking way in this respect. Finally, Section 6 concludes.

The model

Suppose that at time 0 there is a given coalition of players, say a pro-environmental coalition managing a common stock of pollution, denoted y. Suppose that one of the coalition members, named player i, can potentially quit it at some future date T , where 0 ≤ T ≤ ∞. The rest of the coalition is assumed to be roughly homogenous to the point that we can label it as a single player J. Both players i and J differ in a number of characteristics, technological and constitutional as mentioned in the Introduction. Our analysis will identify a posteriori the specific characteristics that lead player i to split. Within the coalition, players i and J choose jointly the level of variables x i , x J ∈ [0, X] ⊂ [0, +∞), which provide them with a joint utility or payoff. The players' actions increase the level of the public bad, y, resulting in a drop in welfare, which corresponds to the pollution externality in environmental economics. In our model, we assume that at time 0, players play cooperatively until time T , when player i decides eventually to quit the coalition. Note that at time T , player J may also switch her strategy in response. We shall concentrate on the Markovian perfect equilibria of the game after T . Finally, we introduce a further simplification by assuming that actions x i , x J ∈ [0, X] ⊂ [0, +∞), while determining the level of players' utilities does also increase the level of CO 2 emissions by exactly x i + x J . The model can be then straightforwardly interpreted in a one-good economy: the good x is consumed and produced with a linear technology, and there is a one-to-one relationship between input and output, and between output and pollution emissions. In the end, player j can obtain utility directly from x j , but she also suffers from pollution, since y brings a (partially external) pollution disutility.

Initially, the objective of the players in the coalition is therefore to maximize joint overall welfare or payoff, which is defined for everlasting coalitions as max

x i ,x J W (∞) = +∞ 0 e -rt [u i (x i ) + u J (x J ) -c i (y) -c J (y)] dt, (1)
where r is the time discount rate, u i (.) and u J (.) are utility functions of players i and J, respectively, which are strictly increasing and concave; and c i (y), c J (y) are their respective individual disutility due to pollution, which are strictly increasing and convex functions. Note that the objective function is simply the aggregate payoff of the two players. We shall discuss how the optimal payoff is shared across players when we come to the constitutional bases of the initial coalition.

Finally, decisions are subject to the dynamic constraint:

ẏ(t) = x i (t) + x J (t) -δy(t), (2)
and δ ∈ [0, 1] is the depreciation rate. In our example, y stands for the stock of CO 2 , so that δ would stand for the natural reabsorption rate of CO 2 in the atmosphere. The initial condition y(0) = y 0 ≥ 0 is given.

We get now into the constitutional aspects of the coalition. Essential aspects are sharing rules (of the aggregate payoff) and penalties in case of splitting. As argued in the Introduction, we shall focus on the first aspect. Concretely, we suppose that player i's share in the total payoff is α ∈ (0, 1) and the remaining share, 1 -α, belongs to the rest of the coalition, i.e., welfare of players i and J are W i = αW (∞) and

W J = (1 -α)W (∞).
Two points are worth doing at this stage. First of all, while α is defined as a fraction of the intertemporal and discounted payoff, it indeed applies at any period of time since it's constant.

We can therefore interpret it also as an instantaneous share. This said, the payoff to be shared in our game-theoretic framework is the intertemporal one: when splitting is an option, player i will consider the share of the intertemporal payoff from the start of the coalition to its (potential) end at date T , that is αW (T ) where:

W (T ) = T 0 e -rt [u i (x i ) + u J (x J ) -c i (y) -c J (y)] dt.
Second, beside this technical point, one would inquire about the particular meaning of such a sharing rule in a pollution problem like this one. One would be naturally tempted to bring this aspect closer to the standard literature of environmental agreements where the enforcement of a Pareto efficiency criterion would require transfers from certain countries to others (see the early contribution of [START_REF] Tahvonen | Carbon dioxide abatement as a differential game[END_REF]. However, this is precisely what we don't do in this model, the weight α is by no way a Pareto weight nor the Shapley value: it is stricto sensu a constitutional parameter, it's fixed initially with the birth of the coalition according to the initial political, demographic or economic relative powers of the members.

Of course, α would be generally dependent on the characteristics of each country member of the coalition, that is, on the shapes of the national preferences and technologies. But in reality each country's weight also depends on more complex characteristics like global and regional history, geography and the resulting regional and global geopolitics, which can hardly be recovered unequivocally from technological differentials or cultural differences. In this paper, we define the constitutional rule as being independent from the latter to clearly discriminate between the constitutional aspects of the coalition and the more purely technological diversity. We consider this case as the natural benchmark to explore. Moreover, we assume that renegotiation (of α) is impossible or too costly, which is far from unrealistic if one recalls the political and constitutional foundations of international coalitions like those mentioned in the Introduction: often splitting may prove less costly than renegotiation. 7We now move to some preliminary technical considerations. If player i quits the coalition at time T , then she obtains a share α of overall welfare until time T . Accordingly from time T onwards, player i's objective becomes

W i,II = max x i +∞ T e -rt [u i (x i ) -c i (y)] dt, (3)
and player J faces

W J,II = max x J +∞ T e -rt [u J (x J ) -c J (y)] dt, (4)
subject to the same state equation:

ẏ(t) = x i (t) + x J (t) -δy(t), t ≥ T, (5)
where the initial condition y(T ) is determined (by continuity) from the outcomes of the first (coalition) period.

The optimal switching time for player i is defined as

max T αW (T ) + +∞ T e -rt [u i (x * i ) -c i (y * )] dt = max T (αW (T ) + W i,II (T )) . (6)
Intuitively, a coalition between i and J established at time 0 can last over the period of time [0, T ] if the first term in ( 6) is non-decreasing in T . That is, the longer player i stays in, the higher is joint social welfare. Otherwise, if the first term in ( 6) is decreasing in T , then player i would exit immediately the coalition with J and T will be 0. Similarly, player i may consider to quit the coalition with J if the second term in ( 6) is non-increasing with T . Otherwise, if the second term was also increasing in T , then it would always be optimal to set T = +∞ and somehow very surprisingly, this a priori non-optimal (almost ad-hoc) coalition would be stable and last forever.

Obviously, the precise optimal choice of T relies on the game that is played after the splitting, or more precisely, on the strategy space after the splitting. As one can deduce from all the above, the optimal choice of T can be 0, ∞ or take any other finite value between 0 and ∞, depending on the parameter set. If it exists, the interior optimal switching time T is obtained by taking the first order condition of ( 6), that is, T is the solution to

α dW (T ) dT + dW i,II dT = 0, (7)
provided the second order optimality condition

α d 2 W (T ) dT 2 + d 2 W i,II dT 2 < 0 holds.
In the next section using a linear-quadratic model, we first study the situation in which the coalition lasts forever, that is, when T = +∞. Then, we will analyze the conditions ensuring the existence of a unique interior solution for T , 0 < T < +∞. If splitting occurs in finite time, then applying the implicit function theorem to [START_REF] Boucekkine | Optimal coalition splitting with heterogenous strategies[END_REF] shows that

∂T ∂α = - dW (T ) dT α d 2 W (T ) dT 2 + d 2 W i,II dT 2 > 0.
That is, the larger the payoff share player i gets, the later she will quit the coalition. Similarly, the smaller the stake of player i in the coalition, the sooner she quits the coalition to potentially gain more freedom of choice.

It is worth mentioning that differently from most of the optimal switching literature, the players before and after time T are different in our setting. Indeed, before time T , there is a single player: the coalition. After T , there are two competing players. Thus special care should be taken when employing the usual necessary optimal switching conditions at T . These difficulties may come mainly from the choice of different strategic spaces after the coalition splits. We shall be more explicit in this respect below.

The linear-quadratic differential game with initially committed coalition members

We start with the case where coalition members act as if the coalition lasts forever. As it's traditional in differential games, we resort to linear-quadratic functional forms for analytical tractability (see [START_REF] Dockner | International pollution control: cooperative versus noncooperative strategies[END_REF][START_REF] Dockner | Differential Games in Economics and Management[END_REF][START_REF] Bertinelli | Carbon capture and storage and transboundary pollution: A differential game approach[END_REF]etc). In this section, we focus on the strategies at each stage of the game for a given splitting time, Section 4 will address the optimal splitting time issues. Section 5 considers the case of un-commited coalition members as mentioned in the Introduction.

In our linear-quadratic setting, the utility functions are given by

u i (x i ) = a i x i - x 2 i 2 , u J (x J ) = a J x J - x 2 J 2 .
If x j is the pollution emission that player j employs to produce the final consumption good, then a j is here the efficiency parameter which converts pollution into the consumption good. Note that a higher a j indicates a more advanced economy, meaning that it can generate more of the consumption good from the same unit of pollution. 8The pollution damage functions are

c j (y) = by 2 2 , j = i, J.
The pollution damage is the same for both players independent of any individual characteristic and in particular, independent of the agent's development level. In the following, we assume that coefficients a j , j = i, J, are sufficiently large, such that the utility functions are always positive and increasing in x j and that the long-run steady state of pollution is positive.

The cooperative stage

Let us start by solving the problem of the ever lasting coalition. As motivated in the Introduction, each coalition member acts as if the coalition lasts forever, and will do so till splitting (if any) occurs. The joint payoff function is max

x i ,x J W (∞) = +∞ 0 e -rt a i x i + a J x J - x 2 i + x 2 J 2 -by 2 dt, (8)
subject to the following state equation ( 2).

We can readily summarize the main results of the optimization problem faced by any coalition member acting as if the coalition is everlasting, in the following proposition.

Proposition 1 For any positive constants b, r, δ, then for any state trajectory y(t), the choices for player i and J are x * j (y) = a j + B + Cy, j = i, J, where

C = r + 2δ -(r + 2δ) 2 + 16b 4 (< 0), B = (a i + a J )C r + δ -2C (< 0).
The trajectory of state is: ∀t ≥ 0,

y(t) = (y 0 -y * )e (2C-δ)t + y *
where y * is the asymptotically stable long-run steady state given by

y * = a i + a J + 2B δ -2C (> 0).
The long-run steady state y * depends on all the parameters, especially the sum of the technology are required to be increasing in the control domains. This amounts to having the controls x k in the intervals [0 a k ], k ∈ {i, J}. These conditions are checked for the optimal and equilibrium solutions computed hereafter.

levels, a i and a J . A higher technology level, which translates into higher consumption, leads to a higher level of long-run pollution. Consistently with the standard linear-quadratic model considered, the convergence speed, (2C -δ), is independent of the technology levels, it rather depends on time preference r, the unit damage of pollution, b, and Nature's regeneration rate δ.

We notice that the two players' aggregate consumption at the long-run steady state is obviously a function of y * and is always positive:

x * i (y * ) + x * J (y * ) = a i + a J + 2B + 2Cy * = δ δ -2C (a i + a J + 2B) > 0,
where the last inequality comes from the fact that a i + a J + 2B > 0. Furthermore, aggregate consumption is always positive along the optimal trajectory path, that is,

x * i (y) + x * J (y) = a i + a J + 2B + 2Cy = a i + a J + 2B + 2C[(y 0 -y * )e (2C-δ)t + y * ] = x * i (y * ) + x * J (y * ) + 2C(y 0 -y * )e (2C-δ)t > 0
∀y and ∀r, δ, a i , a J , b > 0, provided y 0 < y * , which is a natural assumption. If there exists a unique finite solution to [START_REF] Boucekkine | Optimal coalition splitting with heterogenous strategies[END_REF], then using the Proposition above, the pollution stock would reach the value y(T ) given by

y(T ) = (y 0 -y * )e (2C-δ)T + y * . (9)
Accordingly, the total payoff of player i just before the splitting is thus

αW (T ) = α (a 2 i + a 1 J -2B 2 )(1 -e -rT ) 2r -2BC T 0 e -rt y(t)dt -(C 2 + b) T 0 e -rt y 2 (t)dt . ( 10
)
It is straightforward that

dW (T ) dT = e -rT a 2 i + a 2 J -2B 2 2 -2BCy(T ) -(C 2 + b)y 2 (T ) > 0 (11)
if and only if

y(T ) = (y 0 -y * )e (2C-δ)T + y * ∈ (0, y)
where

y = -2BC + 4B 2 C 2 + 2(C 2 + b)(a 2 i + a 2 J -2B 2 ) 2(C 2 + b) (> 0). ( 12
)
Incidentally, the analysis above provides an upper-bound condition for remaining in the coalition in terms of pollution:

Corollary 1 Under the assumptions of Proposition 1, and provided that α > 0:

• if the initial condition checks y 0 > y, then T = 0;

• if the coalition potential long-run steady state checks (y 0 <)y * < y, then T = +∞.

The above corollary can be written in a more compact manner:

T      = 0, if y < y 0 , ∈ (0, +∞), if y 0 < y < y * , = +∞, if y > y * .
From [START_REF] Dockner | International pollution control: cooperative versus noncooperative strategies[END_REF], it is easy to see that lim a i ,a J →0 y = 0, for all b > 0. By continuity when a i , a J are sufficiently small, and for any y 0 > 0, it follows that y < y 0 , thus according to Corollary 1 it must necessarily be that T = 0. In other words, when both players in the initial coalition have a low enough development level, the coalition hardly exists. It is intuitive to see why. Both players here are heavy polluters, in the sense that they do not extract much consumption from the pollutant. None of them can make a remarkable effort to reduce pollution and alleviate the damage from the common bad. Since player i does not perceive the gains of staying in the coalition, she will exit immediately. Notice that here the motive for quitting the coalition is not free-riding in a strict sense since player i also takes into account J's welfare when they remain in the coalition. Here the coalition does not last because player i does no perceive any advantage for neither of them.

Additionally, it can be shown that for any a i , a J not both zero, lim b→0 y ≥ +∞ > lim b→0 y * = a i + a J δ . 9 When pollution damage is low and a i and a J are not both close to zero, the coalition will remain together forever. Intuitively, when pollution damage is negligible, then one would have expected T to be 0 since there is no incentive to stay in the coalition. But, what we prove here is just the opposite. The reason for this seemingly contradicting result comes from the fact that when b is close to zero, both cooperation in the coalition and competition bring nearly identical welfare to player i. Indeed, note that the limit case shows lim b→0

x * j = a j = lim b→0

x m j for both j = i, J. Therefore, when damage from pollution is low and if the coalition is already established, then the coalition will last forever. However, if there was no coalition at time 0, then there is no incentive to form one either.

Let us add some further comments on our results. A given level of pollution stock provides on 9 It is easy to check that

y = 2(C 2 + b)(ai + aJ ) -4bB 2 -2BC 2(C 2 + b) ≥ C 2 + b -(4bC 2 /(r + δ -2C) 2 ) -2C 2 2(C 2 + b) (ai + aJ ) ≡ S(b)(ai + aJ ) and y * = r + δ δ(r + δ) + 4b (ai + aJ ).
Thus, one sufficient condition for y > y * is S(b) > y * , for any ai and aJ not both zero. By l'Hopital's rule, lim b→0 S(b) = +∞.

the one hand with utility via consumption, but on the other hand, it also generates disutility.

The positive effect on welfare is linear on emissions while the disutility generated by the stock of pollution is quadratic. Hence, participating in a coalition makes sense especially when b is small enough relative to the technological parameters a i and a J . As a result, if there is no technological progress (at least for one player), then coalitions make no sense if b > 0 whatever α ̸ = 0. Conversely, if b tends to zero, and technological progress is nonzero, then coalitions would last forever whatever α ̸ = 0. These mechanisms will play an important role in all the cases we study in the following sections.

To study the interior situation where splitting happens in finite time, we impose the following conditions on the parameters:

Assumption 1
The model parameters ensure that the following inequalities hold:

y 0 < y < y * .
Unfortunately, it's not possible to explore analytically how this condition relates to the deep parameters of the model given the expressions of y, y * , B and C. We shall fortunately obtain interpretable expressions for the interior splitting conditions (and optimality) in the next section under the above assumption.

Remark 1 Though it is not easy to clearly see if the parameter domains in which Assumption 1 holds are non-empty, it is not difficult to check it through numerical exercises. For example, let r = 0.015 and δ = 0.0005. Considering that the damage parameter b is comparable with δ, we take b ∈ [0.0001, 0.0002]. Furthermore, given that a i and a J appear in Assumption 1 in the form of a i + a J , we set a i + a J ∈ [0.3, 0.4]. It can be shown that with the above parameters values, y * ∈ [19.7, 32] and y < y * . Obviously, y 0 can be chosen such that Assumption 1 holds. Nevertheless, as mentioned in the last footnote, for any a i + a J > 0, lim b→0 y = +∞ > y * = a i + a J δ . Thus, with a sufficiently small damage parameter b, Assumption 1 fails to hold. It is also not so difficult to see that any combination of the parameters such that the numerator in y is negative leads to y < 0 < y 0 , and thus violates Assumption 1.

Optimal strategies in the non-cooperative stage (after T )

Suppose player i quits the coalition at time T , and that after that both players play Markovian. Consider Markovian subgame perfect strategies: the strategies are such that the choice variables x j for player j = i, J, depend upon time and the current state: x i (t) = x i (t, y(t)), for all y. Since the game is autonomous, we can directly study the stationary Markovian perfect equilibrium (MPE) via the stationary HJB equations. If we denote the value functions of player j = i, J as U j (y), they must check check the following HJB equations for t ≥ T rU j (y) = max

x j a j x j - x 2 j 2 - b y 2 2 + U ′ j (y) (x i + x J -δy) , j = i, J.
From these HJB equations, Appendix A.1 demonstrates the following existence results of the MPE.

Proposition 2 Suppose that player i quits the coalition at a finite time T , and that both players i and J adopt Markovian strategies after the split. Then there exists a stable affine Markovian subgame perfect Nash equilibrium

(x m i , x m J ) = (a i + B m + C m y, a J + B m + C m y) , ∀y,
with coefficients

C m = (r + 2δ) -(r + 2δ) 2 + 12b 6 (< 0), B m = (a i + a J )C m r + δ -3C m (< 0).
For a given initial condition at T , the corresponding optimal state trajectory is

y m (t) = (y(T ) -y m )e (2C m -δ)(t-T ) + y m , ∀t ≥ T,
where y m = a i + a J + 2B m δ -2C m (> 0) is the asymptotically stable long-run steady state.

The following corollary can be then obtained.

Corollary 2 Under the assumptions of Propositions 1 and 2, it follows

y * < y m , ∀r, δ, b > 0.
The proof is detailed at Step 3 of Appendix A.1. Even if the coalition breaks down at T because the option of staying in the coalition does not provide player i with higher welfare, the coalition does better in terms of pollution. This a standard outcome in the environmental agreements literature. Obviously, the decision to split can hardly be in general determined by the steady state pollution criterion, especially if players go Markovian after the split like in our case. This will be crystal clear in the next section devoted to the determination of the optimal splitting time.

Optimal splitting time and its drivers with initially committed coalition members

Under the above Markovian perfect Nash equilibrium, it is easy to obtain player i's welfare in the second period

W m i,II = +∞ T e -rt a i x i - x 2 i 2 - by 2 2 dt = a 2 i -(B m ) 2 2 +∞ T e -rt dt -B m C m +∞ T e -rt y m (t)dt - ((C m ) 2 + b) 2 +∞ T e -rt (y m ) 2 dt, (13)
where y m (t) also depends on the splitting time T . In order to assess how the splitting time affects player i's welfare, we compute dW m i,II (y(T )) dT using (13): 10

dW m i,II (y(T )) dT = {[-rA m i + B m (a i + a J + 2B)] + [B m (2C -δ -r) + (a i + a J + 2B)C m ]y(T ) + [(2C -δ) -r/2]C m y 2 (T ) e -rT = [ a + by(T ) + cy 2 (T )]e -rT    < 0 for 0 ≤ y(T ) < y, > 0 for y(T ) > y, (14)
with y the positive root of the second degree polynomial

dW m i,II (y(T )) dT = 0 11 and A m i = a 2 i 2r + (a i + a J )B m r + 3(B m ) 2 2r .
If y(T ) > ȳ, then

dW m i,II (Y (T )) dT > 0.
Hence the later the splitting happens, if there is splitting at all, the higher player i's welfare in the second period. If this is the case, then player i would postpone the splitting as much as possible. In other words, splitting never happens when the stock of pollution is high enough, namely if y(t) > ȳ.

Recall that Assumption 1 explicitly states the precise condition under which the stock of pollution is increasing over time. Thus if we assume that players i and J are initially in a coalition and that y 0 > y, then y(t) > y for all t ≥ 0, and there will be no splitting. We conclude in the following corollary 10 See Appendix A.2 for the details. 11 The positive root is given by

y = -b -b 2 -4 a c 2 a where a = [-rA m i + B m (ai + aJ + 2B)] < 0, c = [(2C -δ) -r/2]C m > 0 and b = B m (2C -δ -r) + (ai + aJ + 2B)C m .
Corollary 3 Suppose Assumption 1 holds and that y 0 > y, then splitting will never happen, that is, T = +∞.

Optimal finite splitting time

In order to focus on the situation where splitting can happen in finite time, we must complete Assumption 1 with the following (given the properties established just above):

Assumption 2 Suppose the initial condition and the parameter set check y 0 < y.

Obviously, if y * < y, then splitting may happen in finite time. If instead y 0 < y < y * , then splitting can only take place before the pollution stock reaches the upper limit y. Otherwise, player i cannot afford the damage cost from the accumulated pollution and would rather stay with player J.

Substituting the first and second periods' welfare derivatives with respect to T , i.e. ( 11) and ( 14), into the first order condition α dW (T ) dT + dW m i,II dT = 0, it follows that the first-order optimality condition is equivalent to:

Λ (y m T ) 2 + Σ y m T + Γ = 0, (15)
where y m T = y m (T ) stands for the stock of pollution at the switching time and the coefficients in [START_REF] Gancia | A theory of economic unions[END_REF] are

     Λ = -α C 2 + b + C m 2C -δ -r 2 , Σ = -2αBC + B m (2C -δ -r) + C m (a i + a J + 2B) , Γ = 1 2 α a 2 i + a 2 J -2B 2 -rA m i + B m (a i + a J + 2B) . (16)
The roots of (15), if they exist, are given by

y m T = -Σ ± √ Σ 2 -4ΛΓ 2Λ . (17)
Obviously, the existence of real roots is granted if and only if Σ 2 -4ΛΓ ≥ 0. The last inequality condition is ensured by Λ ≤ 0 and Γ ≥ 0, which are equivalent to

α ≥ C m (2C -δ -r/2) C 2 + b ≡ G(b) (18)
and

α ≥ 1 + 3(C m ) 2 (r+δ-3C m ) 2 - 4CC m (r+δ-3C m )(r+δ-2C) a J a i + 1 2 a J a i 2 + 1 - 2C 2 (r+δ-2C) 2 a J a i + 1 2 ≡ F a J a i , b . (19)
In other words, there exists a finite splitting time T m ∈ (0, ∞) if the payoff share is large enough, for given b. It can be readily shown that G(b) is increasing in b and that G(0) = 1 2 (see the Appendix for all related computations). Consequently, the larger b, the larger the share α needed to make finite time splitting possible. Moreover, as function G(.) is increasing, α should be always bigger than one-half whatever b. As explained below Corollary 1, the larger b, the more reluctant player i is to stay in the coalition. Only when the payoff share α is large enough, would player i remain in the coalition (even if she will eventually leave it at some later time).

Again, as for condition [START_REF] Mayer | The cost of non-Europe, revisited[END_REF], finite time splitting is granted if the payoff share is large enough, although the involved lower bound in this case is different: in contrast to condition [START_REF] Mayer | The cost of non-Europe, revisited[END_REF], the lower bound also depends on the technological gap, a J a i . We shall examine the implications below. It should be noted that the above conditions result from the first-order condition, and we now move to the analysis of the second-order condition. Given that parameters Λ, Σ and Γ are independent of the switching time T , the second-order sufficient condition, α d 2 W (T )

dT 2 + d 2 W m i,II dT 2 < 0, holds if and only if [2Λy m (T ) + Σ] ∂y m (T ) ∂T < 0.
Since the pollution stock is increasing over time, we also have that ∂y m (T ) ∂T > 0 for any T . Then the second-order sufficient condition holds if and only if

2Λy m T + Σ < 0. ( 20
)
The above second-order condition is equivalent to

± Σ 2 -4ΛΓ < 0.
If Λ < 0, then the unique optimizer of αW (T ) + W m i,II (T ) is at

y m T = -Σ - √ Σ 2 -4ΛΓ 2Λ .
So the optimal finite splitting time, T , is unique. At the minute notice that combining Conditions ( 18) and ( 19) ensures the existence of a unique optimal finite splitting time T . We summarize this important result in the following proposition, and its detailed proof is reported in Appendix A.3.

Proposition 3 Let Assumptions 1 and 2 hold. Suppose player i quits the coalition at time T , and after that players i and J adopt MPE given by Proposition 2. Suppose the sharing parameter α

checks max F a J a i , b , G(b) < α < 1 ( 21
)
where functions G(b) and F a J a i , b are defined in [START_REF] Mayer | The cost of non-Europe, revisited[END_REF] and [START_REF] Moser | History-dependence in production-pollution-trade-off models: a multi-stage approach[END_REF]. Furthermore, suppose that the pollution quantity

y m T = -Σ - √ Σ 2 -4ΛΓ 2Λ (22) satisfies y 0 < y m T < y * ,
where Λ, Σ, Γ are given by [START_REF] Hoel | Intertemporal properties of an international carbon tax[END_REF]. Then player i optimally quits the coalition at a finite time T:

T = 1 2C -δ ln y m T -y * y 0 -y * . ( 23
)
If Condition (21) fails to hold, then it can happen that either T = 0 and splitting is immediate, or T = +∞ and there is no splitting at all. We shall pay more attention to the "corner" solution, T = +∞ in the policy implications part of this section.

Note that Proposition 3 delivers an explicit solution for the optimal splitting time as none of the terms involved in ( 23) depends on T . It should also be noted that both the splitting time and the stock of pollution depend on the sharing parameter. Indeed, the optimal splitting time, given by ( 23), depends on α through y m T ; and so do the level of the pollution stock at T and the conditions that ensure its existence (in short, the second-order optimality conditions). Interestingly enough, Condition [START_REF] Petrosjan | Stable Solutions of Differential Games with Many Participants[END_REF] shows that the three fundamental ingredients of our model do matter in the duration of coalitions: the sharing parameter, the pollution damage parameter, b, and the technological gap, a J a i . The economic interpretation of Condition [START_REF] Petrosjan | Stable Solutions of Differential Games with Many Participants[END_REF] involves the technological gap: for given technological gap and pollution damage parameter, the payoff share under coalition is required to be large enough for player i to engage in a coalition and to stay in for a finite time. Again, the constitutional parameter α is key in the optimal institutional dynamics: it's key for the existence of an optimal finite time splitting, and it's also key for the duration of the coalition (through the level of pollution at the splitting time, y m T as explained above). We shall devote the next subsection to the latter point. Meanwhile, we shall clarify the implications of Proposition 3 by specializing in two cases depending on the ratio a J a i :12 first, the case of a technologically lagged country i, a J a i > 1, and then a case where this country is more advanced.

Corollary 4 Under the assumptions of Proposition 3, and provided a J a i > 1 and α > 1 2 , player i optimally quits the coalition at time T if and only if α > G(b).

A technologically lagged country may remain in the coalition for any value of b provided the reward, as captured by α, is large enough, and in any case, larger than 1 2 . Notice that this is true whatever the value of the technological gap, provided it's bigger than one. Suppose now that country i is more advanced than J, what would the outcome be? We provide below a simple illustrative case.

Corollary 5 Let the assumptions of Proposition 3 be satisfied and let α > 1 2 . Assume that

a J a i < √ 3 -2 + 12 -2 √ 3 2 + √ 3 ≈ 0.7, ( 24
) then F a J a i , b ≥ G (b)
holds for all b ≥ 0. Therefore, ( 21) holds if and only if

F a J a i , b < α.
Condition ( 21) is the single most important requirement in Proposition 3 since it provides a range for α for the player to remain in the coalition, even temporarily. The condition states that player i needs to retain a sufficiently large share of the total payoff to stay. In the case of a technologically advanced country i, the technological gap shows up in the existence and optimality Condition [START_REF] Petrosjan | Stable Solutions of Differential Games with Many Participants[END_REF] contrary to the case of the lagged player i studied in Corollary 4. This is hardly surprising: a lagged country will always benefit from a coalition if the pollution damage is small enough, at least for a while, if her payoff share under coalition is good enough. However, as Corollary 5 shows, the tradeoffs are more involved if the country is more advanced than the other coalition members (on average). Suppose that the pollution damage parameter, b, is small, then the benefits for player i to remain in the coalition are rather thin. In this case one may expect that the more advanced player i's technology, that's the smaller the technological gap a J a i , the larger the payoff share requested by player i to remain in the coalition. The contrary also holds true, if the pollution damage is large enough: in such a case, the more advanced the country, the lower the payoff share requested to remain in the coalition. 13 Proposition 3 and its corollaries show that our model is indeed able to generate finely all the possible institutional configurations depending on three key parameters, (α, b, a J a i ).

Payoff sharing and the duration of coalitions

This section clarifies how the sharing parameter α affects the duration of a coalition. We already know that the duration of the coalition is increasing in y m T , the pollution stock at the splitting time, according to equation [START_REF] Sampson | Brexit: The Economics of International Disintegration[END_REF]. A critical point is that the pollution stock at the splitting time, y m T , is increasing in α if y m T ≤ y and decreasing in α if y m T > y, where y is defined in [START_REF] Dockner | International pollution control: cooperative versus noncooperative strategies[END_REF] and y m T in [START_REF] Saglam | Optimal pattern of technology adoptions under embodiment: A multi-stage optimal control approach[END_REF]. We can go a step further, Appendix A.5 shows the following results.

Theorem 2 Suppose there exists α 0 ∈ [0, 1], such that, y m T (α 0 ) satisfies 0 ≤ y m T (α 0 ) < y, then y m T (α) is increasing in α in a neighborhood of α 0 in [0, 1]. Similarly, if there exists a value α 0 such that y m T (α 0 ) > y, then y m T (α) is decreasing in α in a neighborhood of α 0 in [0, 1].

To better grasp the importance of Theorem 2, let us link Assumption 1 and the last statement in Proposition 3. Assumption 1 delivers a condition under which a coalition may split in finite time.

The first assumption of Theorem 1 states that if there exists a sharing parameter, α 0 , such that the coalition can split in finite time, then increasing player i's payoff share raises the stock of pollution upon splitting y m T , leading in turn to a more durable coalition (by Proposition 3). Therefore, when the stock of pollution generated by the coalition is relatively small, one obtains that the larger the payoff share of player i, the later splitting (and the larger the subsequent pollution stock at the splitting time). Recall that postponing T increases the joint payoff in the first stage of the game, thus lengthening the coalition duration benefits both players i and J.

The opposite also holds true. If the payoff share of player i is such that the stock of pollution y m T is relatively high, then the coalition is actually no longer beneficial (for player i) 14 and an increase in α fastens the splitting process.

The above results are based on the existence of one particular sharing strategy α 0 , which may be difficult to find. The following corollary extends the conditions in Theorem 2 from one particular point into an interval, which is easier to find and apply. A detailed proof is given in Appendix A.6.

Corollary 6 Suppose y m T (α) is real and nonnegative in a subinterval (α 1 , α 2 ) ⊂ [0, 1]. Then, either y m T (α) ≤ y in the entire subinterval (α 1 , α 2 ), or y m T (α) ≥ y in the entire subinterval (α 1 , α 2 ). In particular, if

max C m (2C -δ -r/2) C 2 + b , 2 [rA m i -B m (a i + a J + 2B)] a 2 i + a 2 J -2B 2 < 1,
then either y m T (α) ≤ y or y m T (α) ≥ y for all α that satisfies (21).

Policy insights

We shall now point at some potentially interesting policy insights one can extract from our theoretical analysis. Let's first mention that while our model may seem too stylized to tackle "real" splitting problems, it's not more stylized than the typical two-stage or repeated games devoted to this topic, surveyed in the Introduction: while it has some ingredients of differential games, it's essentially a two-stage game, the first stage being the coalition stage. There are two main differences with respect to the vast majority of dynamic or two-stage games in this topic. First, the initial coalition is defined by a profit sharing scheme that is not necessarily an equilibrium of any sort, so that the coalition is not necessarily stable. And second, the duration of the first stage is itself the direct result of the individual decision of a particular player. It's very hard to argue that this latter trait is unrealistic, this is exactly how the environmental agreements/protocols and other political coalitions have been unravelled.

Our analysis brings several interesting results along Sections 3, 4.1 and 4.2. Let's stress one of them: as clearly stated in Corollaries 4 and 5, only "big" enough countries may under certain conditions quit the coalition. More precisely, optimal finite time splitting requires α > 1 2 . Recall that the payoff share is determined by the Constitution of the coalition, reflecting in particular the relative historical, geographic, demographic and economic weight of the countries. If the Constitution is also meant to guarantee no-splitting, two avenues can be taken within our framework. One is to counterbalance the impact of too large payoff share (in the sense of Corollaries 4 and 5) by adding penalties to the constitution, making sure that penalties are increasing enough in the payoff share to discourage splitting. The second (non-exclusive) solution is to limit by constitution the payoff share of all individual players, which guarantees that everlasting coalitions are the unique optimal institutional arrangement. In our theory, such an arrangement can be possible under the following conditions summarized in the following proposition.

Proposition 4 Suppose α is chosen according to

α < min G (b) , F a J a i , b ,
thus violating Condition [START_REF] Petrosjan | Stable Solutions of Differential Games with Many Participants[END_REF]. Then, the coalition optimally lasts forever provided the initial pollution, y 0 is large enough.

Technically speaking, Proposition 4 delivers conditions under which the "corner" solution, T = ∞, is optimal. With respect to the optimal finite time splitting case, not only Condition ( 21) is violated: the stability of the everlasting coalition also requires the initial pollution to be large enough, which indeed also violates the second condition stated in Proposition 3 (that is, y 0 < y m T < y * ). 15 Therefore, a Constitution which limits the payoff share to the upper bound identified in Proposition Finally, one can show that there is an interval of α on which the coalition lasts forever, without assuming any lower bound on initial pollution. Instead, we require a lower bound for the share α.

Let α = C m (2C -δ -r/2) C 2 + b
and let α satisfies the linear equation

Λ (α) (y * ) 2 + Σ (α) y * + Γ (α) = 0
where y * is defined by [START_REF] Saglam | Optimal pattern of technology adoptions under embodiment: A multi-stage optimal control approach[END_REF]. Furthermore, let α be the first root greater than α such that

(Σ (α)) 2 -4Λ (α) Γ ( α) = 0.
If there is no such α, we let α = +∞. Then, it can be shown that 0 < α < min {1, α, α} and

Proposition 5 Suppose y 0 < y * . Then T = ∞ if α satisfies α < α ≤ min {1, α, α} . ( 25
)
The proof is given in Appendix A.8.

The linear-quadratic game with an initially uncommitted coalition member

In this section we consider that player i does not behave as if the coalition would last forever. That is, the player is forward-looking in what concerns splitting and inherent timing. To unburden the presentation and get the comparison with the "committed" case at glance, we shall focus on the identification of the coalition break-up point denoted in this section y * T , which is indeed the counterpart of y m T given by ( 22) in Subsection 4.1. Similarity between the two expressions will readily show up and confirm our claim in the Introduction.

Without the commitment assumption, the optimal controls (x * i (t) , x * J (t)) at the cooperative stage are not given by Proposition 1 because the joint value function at this stage is not a quadratic function of y. As a consequence the coalition breaks up at a different point. The next Proposition gives expression of y * T .

Proposition 6 Let λ = -α 4 (C m ) 2 + b + C m 4C m -δ -r 2 σ = -8αB m C m + B m (4C m -δ -r) + C m (a i + a J + 4B m ) γ = α 2 a 2 i + a 2 J -8 (B m ) 2 -rA m i + B m (a i + a J + 4B m ) ( 26
)
where B m , C m are given in Proposition 2, and A m i is given in [START_REF] Zampolli | Optimal monetary policy in a regime switching economy: the response to abrupt shifts in exchange rate dynamics[END_REF] with j = i. Suppose that all players jointly optimize the utility

T 0 e -rt a i x i + a J x J - x 2 i + x 2 J 2 -by 2 dt
in the coalition stage until the splitting time, T , and that the joint value function W (y) for the coalition is differentiable. Then, the breaks up point y * T = y (T ) satisfies the equation

y * T = -σ -σ 2 -4λγ 2λ ( 27
)
provided the right-hand side of ( 27) falls on the interval (0, ∞).

The detailed proof is given in Appendix C. Comparing ( 26) and ( 27) with ( 16) and ( 22), we see that y m T becomes y * T if B and C in ( 16) are changed to 2B m and 2C m , respectively. As a consequence, a similar sequence of results and implications can be derived with the same constructive approach as in the "committed" case, starting with a counterpart to the main Proposition 3. By construction and given the similarity of the algebra, the "uncommitted" and "commited" cases will deliver similar qualitative results. In particular, for coalitions to break down, the share α has to be large enough. Of course, the threshold values for α are no longer the same but the policy implications remain the same. 16

Conclusion

In this paper, we have presented an alternative view of the coalition breakdown problem. Motivated by the most recent related events (in particular, unilateral withdrawal of several countries from institutional or environmental agreements), we have built up an alternative framework in which the sharing rules under the initial coalitions may not be necessarily optimal, and in which, because the coalitions are initially viewed as essential or existential, members act as if they will stay for ever. We have formulated and solved the corresponding, endogenous splitting problem assuming the renegotiation of the initial coalition consitutional rules are impossible or too costly.

We have solved the specific two-stage differential game induced by our alternative theory, and derived several highly nontrivial results on the technological/constitutional characteristics of the splitting coalitions and on the way the sharing rules could be set in order to prevent coalition breakdown. We have also solved the counterpart dynamic game where the players do anticipate splitting from t = 0 and act accordingly in a forward-looking manner. Despite the induced solution scheme is opposite to our alternative theory, the ultimate optimal splitting problems are shown to be analogous as well as the main policy (qualitative) implications. That's, while alternative, our framework is far from generating crazy outcomes. Of course, our analysis is preliminary in that it is for example restricted to time-invariant coalition payoff sharing rules. While our methodological approach can be still applied, analytical tractability is likely to be less conclusive than in our current benchmark.

A Appendix

A.1 Proof of Proposition 2 and Corollary 2

The proof is completed in three steps: step 1 demonstrates the existence of affine-linear Marvkovian Nash equilibrium; step 2 shows the stability and step 3 provides steady states comparison.

Step 1. Existence of Markovian Nash equilibrium Define the Bellman Value function of player j = i, J as U j (y), which must check the following HJB equation: for t ≥ T , rU j (y) = max

x j a j x j - x 2 j 2 - b y 2 2 + U ′ j (y) (x i + x J -δy) , j = i, J.
Then the first order condition yields

x m j (t) = a j + U ′ j (y(t)). ( 28
) Guess U j (y) = A j + B j y + C j 2 y 2 , and j = i, J, then U ′ j (y) = B j + C j y. Substituting x i = a i + B i + C i y and x m J (t) = a J + B J + C J y(t)
into the HJB equations, comparing coefficients on both hand sides, it yields

           rA i = (a i +B i ) 2 2 + (a J + B J )B i , (r + δ -C i -C J )B i = C i (a i + a J + B J ), (r + 2δ)C i = C 2 i + 2C i C J -b, (29) and 
          rA J = (a J +B J ) 2 2 + (a i + B i )B J , (r + δ -C i -C J )B J = C J (a i + a J + B i ), (r + 2δ)C J = C 2 J + 2C i C J -b, (30)
Remark. More generally, if b i ̸ = b J , then the b in the last two equations should be b i and b J respectively.

Solving the above two group equations system simultaneously, it follows that the only coefficients which yields valid Bellman value functions are

C i = C J = (r + 2δ) -(r + 2δ) 2 + 12b 6 ≡ C m , and B i = B J = (a i + a J )C m r + δ -3C m ≡ B m , and
A m j = a 2 j 2r + (a i + a J )B m r + 3(B m ) 2 2r , j = i, J. (31)
Thus the Markovian Nash equilibrium is given by

(x m i , x m J ) = a i + U ′ i (y), a J U ′ J (y) = (a i + B m + C m y, a J + B m + C m y) , ∀y.
Step

Stability

The stability is straightforward by substituting the above Markovian Nash equilibrium into the state equation, it yields

ẏ = (a i + a J + 2B m ) + (2C m -δ)y ∀t ≥ T
with y(T ) coming from the first period cooperation and T unknown. The explicit solution is thus straightforward as given in the Proposition. Furthermore, it is easy to obtain that long-run steady state y m (t) = (y(T ) -y m )e (2C m -δ)t + y m (> 0).

Given 2C m -δ < 0, for any y(T ), the trajectory asymptotically converges to this steady state.

Step 3. Proof of Corollary 2

We can easily rewrite two steady states of pollution as

y * = a i + a J + 2B δ -2C = (a i + a J ) r + δ (δ -2C)(r + δ -2C) and y m = a i + a J + 2B m δ -2C m = (a i + a J ) r + δ -C m (δ -2C m )(r + δ -3C m )
.

Therefore, in order to compare which steady state yields higher pollution, we only need to compare

h 1 ≡ r + δ (δ -2C)(r + δ -2C) = r + δ δ(r + δ) -2(r + 2δ)C + 4C 2 and h 2 ≡ r + δ -C m (δ -2C m )(r + δ -3C m ) = r + δ -C m δ(r + δ) -2(r + 2δ)C m + 6(C m ) 2 -δC m . It is easy to see that C < C m < 0, ∀r, δ, b > 0. Thus, -C > -C m > 0, C 2 > (C m ) 2 > 0,
and

-2C(r + 2δ) + 4C 2 > -2C m (r + 2δ) + 4(C m ) 2 .
So it is straightforward that

h 1 = r + δ δ(r + δ) -2C(r + 2δ) + 4C 2 < r + δ δ(r + δ) -2C m (r + 2δ) + 4(C m ) 2 .
Furthermore, simple algebra yields that

r + δ δ(r + δ) -2C m (r + 2δ) + 4(C m ) 2 < r + δ -C m δ(r + δ) -2(r + 2δ)C m + 6(C m ) 2 -δC m = h 2 .
Hence, h 1 < h 2 and y * < y m , ∀r, δ, b > 0.

That completes the proof.

A.2 The first order condition

In this section, we obtained the derivative of second period's welfare with respect to time, [START_REF] Dutta | A strategic analysis of global warming: Theory and some numbers[END_REF], and the first order condition at the same time.

Suppose rA m i > B m (a i + a J + 2B) and Assumption 1 holds. The switching time T must be given by the FOC

α dW (T ) dT + dW m i,II dT = 0,
provided the second order sufficient condition holds:

α d 2 W (T ) dT 2 + d 2 W m i,II (T ) dT 2 < 0.
By definition,

W m i,II = +∞ T e -rt a i x i - x 2 i - by 2 2 dt = +∞ T e -rt 2 a 2 i -(B m ) 2 -2B m C m y (t) -(C m ) 2 + b y (t) 2 dt
where y(t) is a function of y m (t), which depends on T . Direct calculation yields:

dW m i,II dt = e -rT 2 -a 2 i + (B m ) 2 + 2B m C m y (T ) -(C m ) 2 + b y (T ) 2 + +∞ T e -rt 2 -2B m C m ∂y m ∂T -2 (C m ) 2 + b y m ∂y m ∂T dt.
In order to obtain explicit result, we try to get rid of the term ∂y m ∂T in the above first order derivative. Let V σ j (y) be the value function of Player j in Mode σ when the value of the state variable is y, for j = i, J and σ = 1, 2, where σ = 1 represents the mode before splitting, and σ = 2 after splitting. In Mode 1 the players have the unchangeable Markovian strategies

x j = a j + B + Cy for j = i, J.
In addition, Player i has the impulse control in Mode 1 to exit the coalition. In Mode 2, the players have the value functions V 2 j (y) = U j (y) for j = i, J. The optimal strategy of Player i's impulse control results in maximization of the value V 1 i (y) for y < y m , where y m is the value of y when Player i breaks up with the coalition. Hence, we have

V 1 i (y m ) = V 2 i (y m ) ≡ U i (y m ) (32)
and ∂ y m V 1 i (y) = 0 for any y < y m . Since in Mode 1, Player i has the instantaneous utility

α a i x i + a J x J - x 2 i + x 2 J 2 -by 2 = α a 2 i + a 2 J -B 2 -2BCy -C 2 + b y 2 ,
and the equation of dynamics is ẏ = x i + x J -δy ≡ a i + a J + 2B + (2C -δ) y.

The HJB equation for V 1 i is

rV 1 i = α a 2 i + a 2 J -B 2 -2BCy -C 2 + b y 2 + dV 1 i dy [a i + a J + 2B + (2C -δ) y]
for y < y m subject to the terminal condition [START_REF] Zhang | US withdrawal from the Paris Agreement: Reasons, impacts, and China's response[END_REF]. The solution can be written in the integral form

V 1 i (y) = U i (y m ) + y y m dV 1 i (z) dy dz = U i (y m ) + y y m rV 1 i (z) -α a 2 i + a 2 J -B 2 -2BCz -C 2 + b z 2 a i + a J + 2B + (2C -δ) z dz
for y < y m . Differentiating both sides with respect to y m and using the condition

∂ y m V 1 i (y) = 0 we find 0 = U ′ i (y m ) - rV 1 i (y m ) -α a 2 i + a 2 J -B 2 -2BCy m -C 2 + b (y m ) 2 a i + a J + 2B + (2C -δ) y m . Since V 1 i (y m ) = U i (y m ) = A m i + B m y m + C m 2 (y m ) 2 and
U ′ i (y m ) = B m + C m y m ,
we obtain that the first order condition is equivalent to

(B m + C m y m ) [a i + a J + 2B + (2C -δ) y m ] = r A m i + B m y m + C m 2 (y m ) 2 -α a 2 i + a 2 J -B 2 -2BCy m -C 2 + b (y m ) 2 ,
and

dW m i,II dT = -r A m i + B m y (T ) + C m 2 y (T ) 2 + (B m + C m y (T )) (a i + a J + 2B + (2C -δ) y (T ))} e -rT .
Combining with [START_REF] Colombo | A dynamic analysis of international environmental agreements under partial cooperation[END_REF], we obtain

α dW (T ) dT + dW m i,II dT = e -rT Λy m (T ) 2 + Σy m (T ) + Γ
where the coefficients are given in [START_REF] Hoel | Intertemporal properties of an international carbon tax[END_REF].

That completes the proof.

A.3 Proof of Proposition 3

In the last subsection, the above first order condition can be rewritten as the following second degree polynomial in term of y(T ):

Λ y m (T ) 2 + Σ y m (T ) + Γ = 0.
The roots, if they exist, are given by

y m (T ) = -Σ ± √ Σ 2 -4ΛΓ 2Λ . (33)
Given that the parameters Λ, Σ and Γ are independent of switching time T , the second order sufficient condition holds if and only if (2Λy(T ) + Σ) y ′ (T ) < 0.

Given the assumption that pollution accumulation is increasing over time, that is, y ′ (T ) > 0 is always true, then the second order sufficient condition holds if and only if

2Λy(T ) + Σ < 0. ( 34
)
Combining the second order condition (34) and the explicit solution [START_REF] Latorre | Trade and foreign direct investment-related impacts of Brexit[END_REF], it follows that 2Λy(T ) + Σ = ± Σ 2 -4ΛΓ < 0 if and only if the negative sign is taken in the explicit solution [START_REF] Latorre | Trade and foreign direct investment-related impacts of Brexit[END_REF]. Taking into account that only positive pollution level is possible, then

y m (T ) = - Σ + √ Σ 2 -4ΛΓ 2Λ > 0,
which is true if Λ < 0, Γ > 0 (these two inequality implicitly guarantee the existence of a real positive solution from FOC) and regardless the sign of Σ. Condition Λ < 0 is equivalent to

α > C m (2C -δ -r/2) C 2 + b ≡ G(b)
and Γ > 0 if and only if

α > 2 [rA m i -B m (a i + a J + 2B)] a 2 i + a 2 J -2B 2 ≡ F (a J , a i , b).
To finish the proof, from the explicit solution,

y(T ) = (y 0 -y * )e (2C-δ)T + y * = y m .
rearranging terms, it yields that

T = 1 2C -δ ln y m -y * y 0 -y * .
Recall Assumption 1 guarantees that y 0 < y(T ) < y * , thus, 0 < y m -y * y 0 -y * < 1 and T ∈ (0, +∞).

That completes the proof.

A.4 Profs of Corollaries 4 and 5.

We first prove

Lemma 1 Let the assumptions of Proposition 3 be satisfied. Then the following properties hold.

1. For a J and a i that satisfies [START_REF] Shiryaev | Optimal Stopping Rules[END_REF], relation

F a J a i , b ≥ G (b)
holds for all b ≥ 0.

2. For a J and a i that satisfies

√ 3 -2 + 12 -2 √ 3 2 + √ 3 < a J a i < 1, (35)
there is b * (a J /a i ) such that

F a J a i , b > G (b) if and only if b < b * a J a i .
Hence, [START_REF] Petrosjan | Stable Solutions of Differential Games with Many Participants[END_REF] holds if

F a J a i , b < α for b < b * a J a i and
G (b) < α for b > b * a J a i .
3. For a J and a i that satisfies

a J a i ≥ 1 (36) relation F a J a i , b ≤ G (b)
holds for all b > 0.

Proof. Part 1. We rewrite F (a J , a i , b) as

F a J a i , b = 1 + 3(C m ) 2 (r+δ-3C m ) 2 - 4CC m (r+δ-3C m )(r+δ-2C) a J a i + 1 2 a J a 1 2 + 1 - 2C 2 (r+δ-2C) 2 a J a i + 1 2 . For shorter notation, denote x = a J a i , H = 3(C m ) 2 (r+δ-3C m ) 2 , K = 4CC m (r+δ-3C m )(r+δ-2C) and L = 2C 2 (r+δ-2C) 2 , then F (x, b) = F a J a i , b = 1 + (1 + x) 2 (H -K) x 2 + 1 -(1 + x) 2 L , (37)
with H -K < 0 and L < 1/2. Thus, the condition on α can be shortened as:

max{G(b), F (x, b)} < α < 1.
Straightforward algebra yields that ∀r, δ > 0,

lim b→0 G(b) = 1 2 , lim b→+∞ G(b) = 1 √ 3 , and
dG(b) db > 0, lim b→0 dG(b) db = 0, thus G(b) ∈ 1 2 , 1 √ 3 , ∀b > 0.
Again straightforward, though cumbersome, algebra yields that

lim b→0 F (x, b) = 1 x 2 + 1 , lim b→+∞ F (x, b) = 1 -(1 + x) 2 /3 x 2 + 1 -(x + 1) 2 /2 . (38) Furthermore, ∂F (x, b) ∂b > 0 if x < √ 2 2 , ∂F (x, b) ∂b < 0 if x > 3 5
for all b > 0 and there is b > 0 such that

∂F (x, b) ∂b =    > 0, if 0 < b < b, < 0, if b > b, if √ 2 2 ≤ x ≤ 3 5 . ( 39
)
Note that 1 -

(1 + x) 2 /3 x 2 + 1 -(x + 1) 2 /2 > 1 √ 3
if and only if [START_REF] Shiryaev | Optimal Stopping Rules[END_REF] holds with x = a J /a i . Therefore

F (x, b) > 1 √ 3 ≥ G (b)
for any b > 0 if [START_REF] Shiryaev | Optimal Stopping Rules[END_REF] holds. This proves Part 1.

Part 2. Since x = a J /a i satisfies the reversed inequality in [START_REF] Shiryaev | Optimal Stopping Rules[END_REF],

lim b→∞ F (x, b) = 1 -(1 + x) 2 /3 x 2 + 1 -(x + 1) 2 /2 < 1 √ 3 .

In addition lim

b→0 F (x, b) = 1 x 2 + 1 > 1 2 = lim b→0 G (b) ,
by the intermediate value theorem, there is a b * (x) such that

G (b) = F (x, b * (x)) .
We show that b * (x) is the only solution to the above equation and

G(b) < F (x, b), if 0 < b < b * (x) , G(b) > F (x, b), if b > b * (x) . ( 40
) Since 1 √ 2 < √ 3 -2 + 12 -2 √ 3 2 + √ 3 < 3 5 , F (x, b
) is either decreasing in b for all b or there is a b > 0 such that (39) holds if

x > √ 3 -2 + 12 -2 √ 3 2 + √ 3 .
In the former case, since G(b) is increasing, it is obvious that b * (x) is the only solution and (40) holds. In the latter case, F (x, b) is bell-shaped. Note that

lim b→0 F (x, b) = 1 x 2 + 1 > 5 8 > 1 √ 3 = lim b→∞ G (b) if x < 3 5 , it follows that F (x, b) ≥ lim b→0 F (x, b) > lim b→∞ G (b) if b < b.

A.5 Proof of Theorem 2

We first show that y is a positive real number. Since C < 0, it follows that

|B| = (a i + a J ) |C| r + δ + 2 |C| < a i + a J 2 .
Hence,

a 2 i + a 2 J -2B 2 > a 2 i + a 2 J - (a i + a J ) 2 2 = 1 2 (a i -a J ) 2 ≥ 0.
Therefore the numerator in [START_REF] Dockner | International pollution control: cooperative versus noncooperative strategies[END_REF] is positive. This proves the assertion.

Note that Λ, Σ, Γ and y m all depend on α. We use Λ (α), Σ (α), Γ (α), and y m (α) to indicate the dependence.

By Proposition 3, y m (α) is a solution to the quadratic equation

Λ (α) (y m ) 2 + Σ (α) y m + Γ (α) = 0. Let F (y, α) = Λ (α) y 2 + Σ (α) y + Γ (α) .
Then, F (y m (α) , α) = 0 and

dy m (α) dα = - F α (y m (α) , α) F y (y m (α) , α) . (41)
By differentiation,

F α (y m (α) , α) = Λ α (α) (y m (α)) 2 + Σ α (α) y m (α) + Γ α (α) = -C 2 + b y m (α) 2 -2BCy m (α) + 1 2 a 2 i + a 2 J -2B 2 , F y (y m (α) , α) = 2Λ (α) y m (α) + Σ (α) .
It is shown in the Proof of Proposition 3 that 2Λ (α) y m (α) + Σ (α) < 0 (see [START_REF] Nong | Effects on the U.S. economy of its proposed withdrawal from the Paris Agreement: A quantitative assessment[END_REF]). Note that y is the only positive solution to the quadratic equation

-C 2 + b y 2 -2BCy + 1 2 a 2 i + a 2 J -2B 2 = 0.
Hence, since y m (α 0 ) < y,

F α (y m (α 0 ) , α 0 ) = -C 2 + b (y m (α 0 )) 2 -2BCy m (α 0 ) + 1 2 a 2 i + a 2 J -2B 2 > 0.
It follows that dy m /dα > 0 at a 0 . Hence y m (α) is nondecreasing in α in a neighborhood of α 0 .

This proof for the other case is similar. This completes the proof.

A.6 Proof of Corollary 6

Suppose there are points α, α ∈ (α 1 , α 2 ) such that y m (α) < y < y m (α). Then y m (α) is increasing in α at α and it is decreasing in α at α. It is not possible that α < α because otherwise there would exist an ᾱ, α < ᾱ < α such that y m (ᾱ) is the minimum of y m (α) between α and α. Thus dy m (ᾱ) /dα = 0.

However, by (41), F α (y m (ᾱ) , ᾱ) = 0, which implies that y m (ᾱ) = y. Therefore, y m (ᾱ) = y > y m (â), which is a contradiction. So it is necessary that α < α. In this case there is ᾱ such that α < ᾱ < α and y m (ᾱ) is the maximum of y m (α) between α and α. Hence, again dy m ( ᾱ) /dα = 0 and we have y m (ᾱ) = y < y m (α). This is again a contradiction. Therefore no such points α and α exist.

This completes the proof.

A.7 Optimal splitting time: interior vs corner solutions

The optimal splitting time, T , satisfies the equation

α dW (T ) dT + dW m i,II (T ) dT = 0.
By ( 11) and ( 14), the left-hand side can be written as e -rT Λy m (T ) 2 + Σy m (T ) + Γ where Λ, Σ, and Γ are given by ( 16). We define η (y) = Λy 2 + Σy + Γ.

In the case where Λ ̸ = 0, η (y) has two roots

y m 1 = -Σ - √ Σ 2 -4ΛΓ 2Λ , y m 2 = -Σ + √ Σ 2 -4ΛΓ 2Λ which are real if Σ 2 ≥ 4ΛΓ,
and in the case where Λ = 0, η (y) has one root

y m 0 = -Γ/Σ provided that Σ ̸ = 0.
Using G (b) and F (a J /a i , b) defined in ( 18) and [START_REF] Moser | History-dependence in production-pollution-trade-off models: a multi-stage approach[END_REF],

Λ < 0 ⇐⇒ α > G (b) , Γ > 0 ⇐⇒ α > F a J a i , b .
There are four possible cases.

Case 1: α > max {G (b) , F (a J /a i , b)} .
In this case, Λ < 0 and Γ > 0. So η has one positive root, y m 1 . Furthermore, η (y) > 0 if y < y m 1 and η (y) < 0 if y > y m 1 . So if y (0) < y m 1 , coalition can be formed and lasts until y (T ) = y m 1 , and if y (0) ≥ y m 1 , coalition cannot be formed.

A.8 Proof of Proposition 5

By computation,

α = r + 2δ + (r + 2δ) 2 + 16b 2 r + 2δ + (r + 2δ) 2 + 12b . ( 42
)
It can be shown that the right-hand side is increasing in b. Hence

1 2 < α < 1 √ 3 < 1.
We show that Σ (α) > 0. This is equivalent to showing that

B m (2C -δ -r) + C m (a i + a J + 2B) > 2αBC. ( 43
)
By computation

B m (2C -δ -r) = 2b (a i + a J ) r + (r + 2δ) 2 + 16b r + (r + 2δ) 2 + 12b r + 2δ + (r + 2δ) 2 + 12b , C m (a i + a J + 2B) = -4b (a i + a J ) (r + δ) r + (r + 2δ) 2 + 16b r + 2δ + (r + 2δ) 2 + 12b , BC = 32b 2 (a i + a J ) r + (r + 2δ) 2 + 16b r + 2δ + (r + 2δ) 2 + 16b 2 .
Substituting the above and (42) into (43) and cancel common factors, the inequality is equivalent to

r + (r + 2δ) 2 + 16b 2 r + (r + 2δ) 2 + 12b - 16b r + 2δ + (r + 2δ) 2 + 16b > 2 (r + δ) . (44)
Note that -16b

r + 2δ + (r + 2δ) 2 + 16b = r + 2δ -(r + 2δ) 2 + 16b.
The left-hand side of (44) can be written as

r + (r + 2δ) 2 + 16b 2 + r + 2δ -(r + 2δ) 2 + 16b r + (r + 2δ) 2 + 12b r + (r + 2δ) 2 + 12b . ( 45
) Since r + 2δ -(r + 2δ) 2 + 16b < 0, (r + 2δ) 2 + 12b < (r + 2δ) 2 + 16b, it follows that r + 2δ -(r + 2δ) 2 + 16b r + (r + 2δ) 2 + 12b > r + 2δ -(r + 2δ) 2 + 16b r + (r + 2δ) 2 + 16b = r 2 -(r + 2δ) 2 -16b + 2δ r + (r + 2δ) 2 + 16b .
Thus, the numerator of the quotient in (45) is greater than

r 2 + (r + 2δ) 2 + 16b + 2r (r + 2δ) 2 + 16b + r 2 -(r + 2δ) 2 -16b + 2δ (r + 2δ) 2 + 16b = 2r 2 + 2 (r + δ) (r + 2δ) 2 + 16b + 2δr = 2 (r + δ) r + (r + 2δ) 2 + 16b > 2 (r + δ) r + (r + 2δ) 2 + 12b .
As a result, the quotient in (45) is greater than 2 (r + δ). This proves (44), which is equivalent to Σ (α) > 0.

Since Σ (α) > 0, Λ (α) = 0 and Λ (α) < 0 for α > α, it follows from ( 22) that y m T (α) exists for α > α and is near α. In addition, lim

α→α + y m T (α) = ∞.
Hence, for α > α and is near α, y m T (α) > y. By Theorem 2 and Corollary 6, y m T (α) is decreasing in α for all α > α at which y m T (α) exists. As α increases, y m T (α) continues to exists until either α reaches α if α is finite, or α reaches α, or α reaches 1, whichever arrives earlier. If α ≤ min {α, 1}, then for any α ∈ (α, α)

y m T (α) > y * . ( 46
)
By Proposition 3, T = ∞. If α ≤ min {α, 1}, then (46) holds for α ∈ (α, α). Hence, T = ∞. Finally, if 1 ≤ min {α, α}, (46) holds for α ∈ (α, 1). So, T = ∞.

This completes the proof.

B Heterogeneous strategies

In this section, we illustrate one "out of equilibrium" outcome in which one player re-optimizes its behaviour after the fall of a coalition but the remaining members don't. The motivation draws from the facts that Canada withdrew from Kyoto Protocol on December 13, 2011, the U.S. ceased its participation in the 2015 Paris Agreement on climate change mitigation in 2017 and the United Kingdom withdrew from the European Union on January 31, 2020. In all three examples, the remaining coalition did not change its strategy: neither did so the CO 2 emission targets in the Kyoto or Paris agreements, nor the trading rules within the EU.

More precisely, suppose player J stays with her original commitment to the coalition after T i , x * J (t). Thus, the differential game is reduced after T i to a standard optimal control problem for player i:

max x i W i II ≡ +∞ T i e -rt a i x i - x 2 i 2 - b y 2 2 dt,
subject to ẏ = x i + x * J -δy, ∀t ≥ T i , with y(T i ) = (y 0 -y * )e (2C-δ)T i +y * and x * J (y) = a J +B +Cy given. The system is still autonomous and it is defined over an infinite time horizon. The same calculation as above yields the following: Proposition 7 Suppose that player i quits the coalition at time T i , and that player J keeps her initial commitment. Then for any t ≥ T i , the optimal Markovian strategy of player i is

x i i (y) = a i + B i + C i y, ∀y. (47)
Furthermore, given the initial condition y(T i ), the corresponding state variable y i (t) is given by y i (t) = y(T ) -y i e (C i +C-δ)(t-T i ) + y i ∀t ≥ T i , where y i is the asymptotically stable long-run steady state and it is given by

y i = a i + a J + B + B i δ -C -C i ,
and parameters

C i = -2(C -δ) -4(C -δ) 2 + 4b(1 -r) 2(1 -r) (< 0), B i = (a i + a J + B)C i r + δ -C -C i (< 0).
Obviously, since player J does not update her choice after the collapse of the coalition, the pair (x * J , x i i ) may not be a Nash equilibrium given that x * J may not be the optimal response from player J's point of view, after the collapse of the coalition. Nonetheless, as discussed in Section 1, this is one possible choice among others.

Straightforwardly, the separation time T i can be obtained via the same technique as the one of Proposition 3 with similar, though different, parameter conditions. The detailed calculations of the separation time T i , as well as the comparison between T i and the Markovian splitting time, T , together with their respective impacts on welfare can be found in [START_REF] Boucekkine | Optimal coalition splitting with heterogenous strategies[END_REF].

C Proof of Proposition 6

To derive ( 26) and ( 27), we use dynamic programming to obtain the equation rW (y) = max

x i ,x J a i x i + a J x J - x 2 i + x 2 J 2 -by 2 + W ′ (y) [x i + x J -δy] . (48)
It is straighforward that the maximizers x * i and x * J of the right-hand side take the form

x * j (y) = a j + W ′ (y) for j = i, J.

If the breaking up point y * T ∈ (0, ∞) exists, W also satisfies the transition condition W (y * T ) = U i (y * T ) + U J (y * T ) .

Since W , U i and U J are all differentiable, we also have

W ′ (y * T ) = U ′ i (y * T ) + U ′ j (y * T ) . (50)
To find the optimal breaking up point for Player i, we let W (y; y * ) to denote the solution of (48)-(49) for y ∈ (0, y * ) and the boundary condition

W (y * ; y * ) = U i (y * ) + U J (y * ) . (51)
Consider a time t before breaking up when the stock of pollution is y. Let t * be the time when the breaking up occurs. Then t * > t. By dynamic programming, W (y; y * ) = t * t e -r(τ -t) g (y (τ ) , x * i (y (τ )) , x * J (y (τ ))) dτ + e -r(t * -t) [U i (y * ) + U J (y * )] (52) where g (y, x i , x J ) = a i x i + a J x J -

x 2 i + x 2 J 2 -by 2 (53)
and x * i (y) and x * J (y) are given by (49). On the other hand, the total discounted benefit that Player i receives for the time interval (t, ∞) is α t * t e -r(τ -t) a i x * i (τ ) + a J x * J (τ ) -

x * i (τ ) 2 + x * J (τ ) 2 2
-by (τ ) 2 + e -r(t * -t) U i (y * ) .

We let W i (y; y * ) to denote this quantity. Using (52) we find W i (y; y * ) = αW (y; y * ) + e -r(t

Also, by differentiating both sides of (52) with respect to t * , we find where g (y * ) denotes g (y * , x * i (y * ) , x * J (y * )). Note that U i and U J are quadratic functions

U j (y) = A m j + B m y + C m 2 y 2 , (60)
By ( 49) and (50),

x * j (y * T ) = a j + W ′ (y * T ) = a j + U ′ i (y * T ) + U ′ J (y * T ) for j = i, J

are linear functions of y * T . Hence, f (y * T ) is a linear function of y * T and g (y * T ) is a quadratic function of y * T . This implies that equation ( 59) is quadratic in y * T . Thus we can write the equation in the form λ (y * T ) 2 + σy * T + γ = 0, and find the coefficients λ, σ, and γ using (60) and (61). The result is [START_REF] Tomiyama | Two-stage optimal control problems and optimality conditions[END_REF]. This leads to

y * T = -σ ± σ 2 -4λγ 2λ . ( 62
)
It remains to show that the sign in front of the square root is negative. For this purpose we use the second order condition in (55). Using (58) we write (56) and (57) in the form

∂W i ∂y * f = α ∂W ∂y * f -re -r(t * -t) [(1 -α) U i -αU J ] + e -r(t * -t) (1 -α) U ′ i -αU ′ J f. (63)
and ∂W ∂y * f = e -r(t * -t) {g -r [U i + U J ]} + e -r(t * -t)

U ′ i + U ′ J f, (64)
respectively. Differentiating the two sides of ( 63) and (64) with respect to t * , and using (55), we obtain

0 ≥ α ∂ 2 W ∂y * 2 f 2 + ∂W ∂y * f ′ f + r 2 e -r(t * -t) [(1 -α) U i -αU J ] -2re -r(t * -t) f (1 -α) U ′ i -αU ′ J +e -r(t * -t) 1 -αU ′′ i -αU ′′ J f 2 + (1 -α) U ′ i -αU ′ J f ′ f and ∂ 2 W ∂y * 2 f 2 +
∂W ∂y * f ′ f = -re -r(t * -t) g + e -r(t * -t) g ′ f + r 2 e -r(t * -t) [U i + U J ] -2re -r(t * -t) U ′ i + U ′ J f + e -r(t * -t) U ′′ i + U ′′ J f 2 + U ′ i + U ′ J f ′ f at y * = y * T , where f ′ = f ′ (y * ), g ′ = g ′ (y * ). These two relations lead to

α -rg + g ′ f + r 2 U i + U ′ i f -2r + f ′ + U ′′ i f 2 ≤

F

  Therefore, b * (x) > b. Since F (x, b) is decreasing in b for b > b, we again find b * (x) is the only solution and (40) holds. Part 3. Since x ≥ 1, it follows that lim b→0

Furthermore, since x > 3 / 5 ,F

 35 F (x, b) is decreasing in b. Hence, since G (b) is increasing in b, we find F (x, b) ≤ lim b→0 (x, b) ≤ lim b→0 G (b) ≤ G (b) . This completes the proof. Now, Corollary 4 follows directly from Part 3 of Lemma 1, and Corollary 5 follows directly from Part 1 of Lemma 1.

  * = e -r(t * -t) {g (y * , x * i (y* ) , x * J (y * )) -r [U i (y * ) + U J (y * )]} +e -r(t * -t) [U ′ i (y * ) + U ′ J (y * )] dy * dt *(57)where dy * /dt * follows from the system of dynamic equationsdy * dt * = x * i (y * ) + x * J (y * ) -δy * . (58)We denote the right-hand side as f (y * ). At y * = y * T , the first order condition in (55) and the above two equations lead to αg (y * T ) -rU i (y * T ) + U ′ i (y * T ) f (y * T ) = 0 (59)

  -t) [(1 -α) U i (y * ) -αU J (y * )] . (54)Clearly, if y * T is the optimal breaking up point for Player i, and if 0 < y * T < ∞, then the first and second order conditions ∂W i ∂y * (y; y * T ) = 0,∂ 2 W i ∂y * 2 (y; y * T ) ≤ 0 (55) both hold, since W i is smooth.We find ∂W i /∂y * as follows. Differentiating both sides of (54) with respect to t * , we obtain

	∂W i ∂y

* * dy * dt * = α ∂W ∂y * dy * dt * -re -r(t * -t) [(1 -α) U i (y * ) -αU J (y * )] +e -r(t * -t) [(1 -α) U ′ i (y * ) -αU ′ J (y * )] dy * dt * .

  0 at y * = y * T . We can write the left-hand side as-r αg -rU i + U ′ i f + f αg -rU i + U ′ i f

	′
	Using (59) and assuming
	f (y * T ) > 0,
	it follows that
	αg (y

* ) -rU i (y * ) + U ′ i (y * ) f (y * ) ′ ≤ 0

(65)

In Appendix B, we illustrate this point by showing how our setting can be readily adapted to heterogeneous strategies after the splitting.

Commonly, all these studies rely onTomiyama (1984). It is worth mentioning here that our stopping time problem differs from the one explored in the literature under stochastic setting (see[START_REF] Shiryaev | Optimal Stopping Rules[END_REF][START_REF] Albrecht | Search by Committee[END_REF]. The main difference derives from the fact that our stopping time is a contingent event, not following any random observations.

An exception is for example[START_REF] Boucekkine | Environmental quality versus economic performance: a dynamic game approach[END_REF].

One way to avoid that player i quits the coalition is to redefine α as the Shapley value, allow renegotiation and letting α be a function of the common resource, i.e. α = α(y).

Consistently with the general conditions on the preferences given in Section 2, the LQ utility functions posited

A more general result is stated and proved in Lemma 1 in the Appendix.

We refer the reader to Appendix A.4 for a complete description and more general results.

See the discussion after[START_REF] Colombo | A dynamic analysis of international environmental agreements under partial cooperation[END_REF].

depending on the pollution damage and the relative technological position of each country would, in our model, allow for coalition splitting if the level of pollution is threatening enough. Of course, if the pollution level is small, then coalitions are much less attractive, and they would not even exist. That is, the corner solution T = 0 would arise in such a case.[START_REF] Gancia | A theory of economic unions[END_REF] In the Appendix, we show that yo large enough corresponds indeed to y0 > y m T .

Comparison of the coalition durations in the two cases is not possible analytically. Our numerical exercises show however that the coalition survival is higher in the "committed" case, which is far from surprising.

In memory of Ngo Van Long. Financial support from the French National Agency (grant ANR-17-EURE-0001) is gratefully acknowledged by Carmen Camacho.

Case 2:

In this case, Γ ≥ 0 and Λ ≥ 0. There are four subcases, (1) Λ = 0 and Σ < 0, (2) Λ = 0 and Σ ≥ 0, (3) Λ > 0 and Σ 2 ≤ 4ΛΓ, and (4) Λ > 0 and Σ 2 > 4ΛΓ.

In subcase [START_REF] Albrecht | Search by Committee[END_REF], η (y) is linear and has one positive root, y m 0 . Also, η (y) > 0 if y < y m 0 and η (y) < 0 if y > y m 0 . So if y (0) < y m 0 , the coalition lasts until y (T ) = y m 0 , and if y (0) ≥ y m 0 , a coalition cannot be formed.

In subcases ( 2) and (3), η (y) ≥ 0 for all y > 0. Therefore, coalition lasts forever.

In subcase (4), η (y) has two positive roots y m 1 and y m 2 if Σ < 0. It is clear that y m 1 < y m 2 , and η (y) > 0 for y < y m 1 or y > y m 2 , and η (y

, a coalition cannot be formed, and if y (0) > y m 2 , the coalition lasts forever.

Case 3:

In this case Λ ≤ 0 and Γ ≤ 0. There are four subcases, (1) Λ = 0 and Σ ≤ 0, (2) Λ = 0 and Σ > 0, (3) Λ < 0 and Σ 2 ≤ 4ΛΓ, and (4) Λ < 0 and Σ 2 > 4ΛΓ.

In subcases (1) and (3), η (y) is nonpositive for all y > 0. So a coalition cannot be formed.

In subcase (2) η (y) < 0 if y < y m 0 and η (y) > 0 if y > y m 0 . So if y (0) < y m 0 , a coalition cannot be formed, and if y (0) > y m 0 , the coalition lasts forever.

In subcase (4), both y m 1 and y m 2 are nonnegative, and η (y) < 0 for y < y m 1 or y > y m 2 and η (y) > 0 for y m 1 < y < y m 2 . So if y (0) < y m 1 or y (0) > y m 2 , a coalition cannot be formed, and If y m 1 ≤ y (0) ≤ y m 2 , the coalition continues until y (T ) = y m 2 .

Case 4:

In this case Γ < 0 and Λ > 0. So η has one positive root, y m 2 . Also, η (y) changes from negative to positive as y passes through y m 2 . So if y (0) < y m 2 , a coalition cannot be formed, and if y (0) ≥ y m 2 , the coalition lasts forever.

Note that Proposition 4 follows from this conclusion. at y = y * T . In terms of coefficients λ, σ, and γ, the left hand side is the same as d dy * λ (y * ) 2 + σy * + γ .

Hence, 2λy * T + σ ≤ 0.

This proves that the sign of the square root in (62) is negative.

The proof is complete.

Declarations

Ethical Approval This declaration is not applicable.

Competing interests All the co-authors declare no conflit of interest.

Authors' contributions The co-authors have contributed equally to the development of the research project and to the drafting of the resulting paper.

Funding Financial support from the French National Agency (grant ANR-17-EURE-0001) is gratefully acknowledged by Carmen Camacho.

Availability of data and materials This declaration is not applicable.