International environmental agreements with the formation of multiple coalitions

ABSTRACT This paper examines the formation of multiple coalitions under the umbrella of an international environmental agreement. Using a repeated game, we demonstrate that in any coalition structure consisting of only two or three countries, cooperation can be sustained through punishments inflicted on the deviating country from agreement rules by fellow coalition members if the discount factor is greater than . Furthermore, our rules of agreement can be effective even in the case of heterogeneous countries if coalitions are formed among the same type of countries. With this method, a coalition consisting of two or three countries is sustained whenever the discount factor is greater than , as is the case with symmetric countries.


I. Introduction
The United States (U.S.) announced its withdrawal from the Paris Agreement in 2015 and its re-entry in 2021. The U.S. also withdrew from the Kyoto Protocol in 2001. However, the past withdrawals of the U.S. indicate that the design of international environmental agreements (IEAs), which is restricted to a single coalition, does not function well. Barrett (1994) and Carraro and Siniscalco (1993), as well as other recent studies of IEA with one-shot game models, show that an IEA cannot have many participants without imposing considerable restrictions on countries' behaviours and payoff structures (e.g. Al Khourdajie and Finus 2020;Barrett 2006;Karp and Simon 2013;Weikard 2009).
The existing IEA literature presents agreement rules called strategies to sustain a coalition through the use of repeated games. In a repeated game model, a game is infinitely repeated, and the signatories to an agreement are forced to cooperate during subsequent stages based on a strategy. The strategy specifies the involved countries' behaviours, including punishments inflicted in response to deviations from the agreement. This equilibrium concept is called weakly renegotiation-proof (WRP) equilibrium (in the sense of Farrell and Maskin 1989, pp. 330-331).
Much of the literature on IEAs with repeated games examine the formation of a single coalition. In Barrett (2002), based on a strategy called consensus treaty, a single grand coalition can be sustained with linear benefit and cost functions. However, each signatory must choose an abatement level that is less than the efficient level. Additionally, Froyn and Hovi (2008) and Asheim and Holtsmark (2009) examine the formation of the grand coalition and present the penance-m strategy, which limits the number of countries that are permitted to punish a deviator. In the setting of linear benefit and quadratic cost functions, Asheim and Holtsmark (2009) show that the abatement levels of signatories and the number of punishing countries, which are required to sustain a grand coalition, depend on the level of discount factor. By using penance strategy, Asheim et al. (2006) present a pioneering work on the formation of two coalitions with linear benefit and cost functions. Using this functional form, they show that the number of signatories to IEA can be doubled if two coalitions are sustained and that the size of a stable coalition is related to the benefit-cost ratio from the abatement. Finus (2008) indicates that multiple coalitions may be useful to lead to a comprehensive agreement, as with international trade regimes in which regional trade agreements are often sustained under the umbrella of the World Trade Organization. In recent years, in fact, many countries have actively established bilateral and regional trade agreements. 1 Nevertheless, in the field of IEAs, the existing investigations and discussions of the formation of multiple coalitions are utterly inadequate.
Furthermore, the multiple-coalition structure can be the key for the success of IEA consisting of heterogeneous countries. The development levels of countries can be categorized into more than two types. For example, the Kyoto Protocol categorizes countries as developed countries, economies in transition, and developing countries. The World Bank divides countries into four types based on income: low, lower-middle, upper-middle, and high. 2 In general, there is a difficulty in sustaining a grand coalition in which several types of countries participate and abate greenhouse gas emissions because the levels of countries' payoffs consisting of abatement benefits and costs can differ depending on the types of countries. Therefore, the Paris Agreement, as well as several theoretical studies of IEAs based on a grand coalition, adopts the assistance to adjust the payoffs among countries (e.g. Barrett 2001;Biancardi and Villani 2010;Weikard 2009). However, the formation of a grand coalition consisting of heterogeneous countries is fragile, in that its sustainability relies on the assistance mechanisms. Zhang et al. (2017) mention that the cut off or decreased financial support from the U.S. to international cooperation on climate change will delay the progress in meeting the goal of the Paris Agreement. Unfortunately, little consideration has been given to the design of IEA consisting of heterogeneous countries, which is sustained without relying on the assistance mechanisms.
This article presents a new benchmark model for IEA consisting of heterogeneous countries without any assistance mechanisms based on the multiple-coalition formation. If coalitions are formed among the same type of countries, the signatories can behave myopically in the coalition to which they belong under the umbrella of an IEA. This paper proposes a new strategy to sustain an IEA with multiple coalitions that are formed with linear benefit and quadratic cost functions based on the penance presented by Asheim et al. (2006). The penance is a novel and fundamental strategy in that it sets a simple punishment rule: a deviator is punished by the other signatories in its coalition. Based on the new strategy, this article aims to clarify the effectiveness of the multiple-coalition formation in sustaining the IEA in the cases of symmetric and heterogeneous countries.

II. The model
Let N ¼ 1; . . . ; n f g denote a set of symmetric countries. The abatement benefit is assumed to be a linear function, namely, of the quantity of individual abatement q i . We assume thatb; c > 0. The net benefit of country i is In every period of the game, each country can choose whether to cooperate on abatement. Nonsignatories always choose the abatement level that maximizes their own payoff. Each signatory to the coalition must choose to cooperate (i.e. choose the abatement level that maximizes the aggregate payoff of the countries in the coalition to which it belongs) or to defect exhibiting the same behaviour as nonsignatories.

Noncooperation
This paper considers the formation of multiple coalitions. We define S as a partition of the set of players N such that S 1 ; . . . ; S m f g includes the singletons. Therefore, for g ¼ 1; . . . ; m, the number of be the abatement level of a country in coalition S g . Hereafter, we regard any S g that satisfies S g � � � � � 2 as a coalition. We first consider the abatement level in cases of noncooperation. Each nonsignatory chooses the q i that maximizes π i . We assume an interior solution to solve the country's optimization problem. Differentiating π i with respect to q i and based on the assumption of an interior solution, we determine that the first-order condition for maximizing an individual country's payoff is Based on (2), the optimal abatement level q 1 of a nonsignatory is The value of q 1 corresponds to the unique Nash equilibrium strategy adopted by each country during the stage game. This value is the same as the abatement level of a country that deviates from the coalition because the deviator also chooses the abatement level to maximize its payoff.

Cooperation
Next, we consider the abatement level of a cooperating country. Rearranging (1), we find that the net benefit of country i in S g is which is shown in the first brackets, is the aggregate abatement level of all the countries. This paper assumes that signatories choose the abatement level that maximizes the aggregate payoff of the countries in the coalition to which they belong rather than the one that maximizes the aggregate payoff of all the countries in all coalitions (i.e. IEAs). Thus, our model allows signatories to behave myopically.
We assume an interior solution to solve the country-level optimization problem. Differentiating P i2S g π i with respect to q i , we find that the firstorder condition for maximizing the total payoff is In (5), the optimal abatement level q S g j j when all the countries in coalition S g choose to cooperate is meaning that the optimal abatement level of a signatory when all signatories choose to cooperate differs based on the size of the coalition to which it belongs. From (3), (4), and (6), we have From (7), it can be observed that each signatory in coalition S g S g � � � � � 2 À � has an incentive to deviate in the stage game. Moreover, (4) shows that the payoff of signatory i to coalition S g when all the signatories in the coalition choose to cooperate is Using (3) and (6), Equation (8) can be rearranged as

III. Analysis
This section examines the conditions under which signatories to coalitions choose to cooperate through the use of a repeated game.

Weakly renegotiation-proof (WRP) equilibrium
The equilibrium concept of a repeated game model is referred to as a WRP equilibrium. To qualify as a WRP equilibrium, a strategy must satisfy two requirements: (1) The strategy profile must be subgame perfect (i.e. a player cannot increase its payoff by changing its behaviours). In any repeated game with discounting, no player can gain from a one-period deviation after any history. 3 (2) The strategy profile must be renegotiation-proof. This requirement is fulfilled if not all the players of a given coalition can strictly gain by collectively restarting cooperation instead of carrying out a punishment when a signatory has unilaterally deviated during the previous period because the punishment for the deviation lasts only one period.
Based on item 1, this study assumes that each country discounts its future payoffs using a common discount factor, δ (0 < δ < 1). Based on item 2, a punishment implies that all the punishing countries, but not the deviator, must choose to defect in the punishment phase.

IEA rules (strategy)
This paper assumes that all signatories commit to IEA rules called strategy in preplay communication (i.e. period 0), as is the case with other studies on IEAs using repeated game frameworks. We specify the following strategy: (i) Any participating country chooses to cooperate except when it has been the sole deviator from the coalition to which it belongs during the previous period. (ii) If a deviation occurs, the deviator will be punished by the other countries in its coalition during the next period, but not by countries in other coalitions. Punishing countries choose the abatement level that maximizes their individual payoffs as punishment.

(iii) If (ii) is satisfied and a deviator chooses to
cooperate after deviating (i.e. the punishment phase), then the cooperative relationship will be re-established in the next period.
The members of each coalition behave in accordance with the 'common' IEA rules above. Therefore, our model considers an IEA under which coalitions are formed. Nonsignatories always choose the abatement level (q 1 ) that maximizes their own payoff after any historical action (see Section II) without committing to behave in accordance with the strategy. Our strategy is based on the fundamental penance strategy presented by Asheim et al. (2006), in which a deviator is punished by the other signatories in its coalition. Our strategy assumes single-period punishments following penance.

Equilibrium outcome
This section examines the conditions under which the signatories to a coalition choose to cooperate in accordance with our strategy. A strategy must satisfy two requirements to enable coalitions to sustain WRP equilibrium: they must be subgame perfect and renegotiation-proof. We make the following proposition regarding WRP equilibrium.

Proposition I
In any coalition structure consisting of coalitions of size two or three countries only, cooperation can be sustained as WRP equilibrium through punishments inflicted by fellow coalition members whenever δ � 1 2 . (Proof) See Appendix A in the Online appendices.▀ From Proposition I, we know that the discount factor condition for a coalition to be sustained at WRP equilibrium is δ � 1 2 . A coalition must consist of two or three countries, in keeping with the results of seminal works on IEAs based on a grand coalition formation (e.g. Barrett 1994Barrett , 2002Barrett , 2006Karp and Simon 2013). 4 However, a small coalition consisting of two or three countries can lead to broad cooperation if multiple coalitions are formed. Furthermore, our result based on multiple-coalition model is partially consistent with the grand coalition model of Asheim and Holtsmark (2009). Assuming that signatories can choose the abatement level that is less than that maximizes the aggregate payoff of all members in the coalition, they show that the abatement level in cases of cooperation, the number of punishing countries, and the punishment level, which are needed to sustain a grand coalition, are decided depending on the discount factor level. 5 We illustrate a numerical example of our main result while comparing the results of Asheim and Holtsmark (2009) and Barrett (2002) in Appendix B of the Online appendices.
However, compared with the penance-m, our strategy based on the penance provides the clear way to select the punishing countries, in that it prescribes that fellow coalition members punish a deviator. Each signatory is likely to have an incentive to be selected as the punishing country because it can enhance its payoff during the punishment phase. Therefore, the clear rule for the selection of punishing countries is one of important elements of IEA rules.

Multiple coalitions and punishment
Here, we discuss the effects of punishment on IEAs. Punishments in repeated games entail decreased future abatement on the part of punishing countries, which damages the environment. Our multiple-coalition model involves punishments inflicted by only one or two coalition members, irrespective of the number of coalitions concerned. If a deviation occurs in coalition S g , each country except for the punishing countries decreases its payoff during the punishment phase compared to that corresponding to the equilibrium path by That is, the reduction shrinks as the number of punishing countries (i.e. S g � � � � À 1) decreases. In general, punishments are not implemented on the equilibrium path because it functions as a credible threat. However, such a punishment could be a significant factor for an IEA. For example, punishments with little environmental damage could be effective in cases where accidental deviations could occur irrespective of a country's intentions, as noted by Takashima (2017). Next, we discuss the motivation underlying the equilibrium concept, WRP. As with other studies (e.g. Asheim et al. 2006;Asheim and Holtsmark 2009;Barrett 2002), this paper adopts the concept of WRP equilibrium, assuming that countries that are renegotiating only consider returning to the original state of cooperation. That is, a coalition structure cannot be revised through renegotiation. Without the concept of WRP, a coalition structure could be revised to the benefit of all the countries involved to avoid punishment. 6

Heterogeneous countries
We discuss cases involving heterogeneous countries based on the formation of multiple coalitions. If the parameters of abatement benefits and costs are heterogeneous, there can be differences in the payoffs of each signatory (in (4)). The abatement levels of a signatory that chooses to defect (in (3)) or cooperate (in (6)) also differ depending on those parameters. Furthermore, from (7), we know that the incentive for each country to deviate can differ depending on those parameters. Thus, the subgame perfection and renegotiation-proof requirements cannot be the same for all signatories. Using a one-shot game, several studies on IEAs consisting of heterogeneous countries adopt assistance mechanisms to adjust countries' abatement benefits and costs (e.g. Barrett 2001;Biancardi and Villani 2010;Weikard 2009).
This article presents a new benchmark model of IEAs consisting of heterogeneous countries without any assistance mechanisms based on a repeated game model. Based on the analysis in Section III, we consider the formation of multiple coalitions in heterogeneous countries. Suppose that there are t 2 N types of countries. The parameter of the abatement benefit of country type t is b t and that of its abatement cost is c t . To generalize the heterogeneous benchmark model of IEAs with multiple coalitions, this article considers a simple coalition formation involving IEAs with individual coalitions consisting of the same type of country. That is, in our model, asymmetric groups of symmetric countries can be formed internationally. If there are n types of countries, each country remains a singleton. According to (8), the payoff of signatory i of type t to coalition S g when all the signatories in the coalition choose to cooperate is According to (3), the optimal abatement level of a deviator and type t nonsignatories is Moreover, from (6), we have From (10), (11), and (12), we have According to (13), each signatory in coalition S g S g � � � � � 2 À � has an incentive to deviate in the stage game. Based on (12), Equation (10) can be rearranged as Based on equations (10)- (14), by analysing in a manner similar to the analysis shown in Appendix A in the Online appendices, we make Proposition II.

Proposition II
A coalition consisting of the same type of countries can be sustained as WRP equilibrium through punishments inflicted by fellow coalition members whenever δ � 1 2 if it comprises two or three countries.
Proposition II shows that if a coalition consists of the same type of countries, its size is equal to 2 or 3 whenever δ � 1 2 even in the case of heterogeneous countries. An intuitive explanation can be given as follows. Forming a coalition involving one country type makes participants behave myopically and partially equalizes countries' payoffs. Under this method, the subgame perfect and renegotiationproof requirements are the same for all the signatories in a coalition.
We clarify the contributions of our multiplecoalition model. Our analysis justifies that grouping countries with common abatement benefits and costs is effective in terms of sustaining the multiple coalitions in the world with heterogeneous countries. Our model provides a way to establish a realistic IEA consisting of heterogeneous countries.

V. Conclusions
This paper provides a new benchmark model of IEA under which multiple coalitions are formed using a repeated game. Our model shows that if the size of a coalition is equal to 2 or 3, each signatory cooperates to abatement in accordance with agreement rules if the discount factor is greater than or equal to 1 2 . Furthermore, the formation of multiple coalitions can also be effective in cases involving heterogeneous countries. If a coalition is formed by a group consisting of one country type, it can be sustained if it consists of two or three countries and the discount factor is greater than or equal to 1 2 , even in the case of heterogeneous countries. In conclusion, our model constructs a theoretical foundation for IEAs consisting of multiple coalitions and suggests a possible and effective approach to the sustainability of a realistic IEA in which heterogeneous countries participate.