1 Introduction

Climate change is the leading global concern of the 21st century, making global actions more compelling than ever. The actions required to protect the environment involve several actors, at several levels, cooperating and competing on several purposes. Given that context, game theory is a useful tool to analyze these interactions and to provide useful insights. A rich literature has emerged on many aspects of the problem, like common pool resources [see Biancardi and Maddalena (2018) for groundwater management; Radi et al. (2021) for the harvesting problem] or the literature on shallow lakes [see e.g. Mäler et al. (2003)].

Another important issue is that the global actions needed to face climate change have to be coordinated in some way. And, historically, that occurs through International Environmental Agreements (IEAs). The United Nations were very active in the last three decades, creating the Intergovernmental Panel on Climate Change (IPCC), with the aim to provide regular reports on science, technology, and the socio-economic implications of climate change, also providing options to reduce future risks. These reports are often used as a basis for discussion in the Conference of Parties (COP), the annual meetings organized by the United Nations Framework Convention on Climate Change (UNFCCC). COPs started in the early 90 s and represented the main means by which countries have signed IEAs. The most famous, and in some way successful, examples of these agreements are COP-3 (Kyoto, 1997) and COP-21 (Paris, 2015).

The model proposed in this paper belongs to the non-cooperative games literature, but IEAs were extensively examined in the context of cooperative games. In this literature, it is assumed that a grand coalition is constituted. Two-step models are then considered: (i) in the first step, the collective optimal emissions are computed, (ii) in the second step, the cooperative solution concepts known in literature are used to share the emissions reduction responsibility among countries [see, e.g. Petrosjan and Zaccour (2003) for the Shapley value; Germain et al. (2003) for the core].

Instead, the non-cooperative literature started from the assumption that participation in an IEA must be voluntary, beacuse of the absence of a supranational authority that can force countries to reduce their greenhouse gas emissions. That requires that the agreements have to be self-enforcing in order to be stable. The concepts usually adopted to deal with the enforcement are the stability conditions proposed by d’Aspremont et al. (1983). Then an agreement is stable when none of the members has an incentive to withdraw and all non-members have no incentive to join. This literature is mainly based on two stage games: (i) in the first stage (the membership game) countries autonomously choose whether to join the coalition or not; (ii) in the second stage (the emissions game) countries set their emissions level.

The two main variables, then, are the gain achieved from cooperation in terms of emissions and the size of a stable coalition. The first articles in the literature [see e.g., Carraro and Siniscalco (1993), Barrett (1994)] clearly highlight that self-enforcing agreements can be possible in two cases: (i) with a small number of countries involved, or (ii) with a small gain in emissions abatement with respect to non-cooperation. In the last three decades a large number of scholars have worked on the small coalition puzzle, analyzing the problem through several approaches and proposing different solutions. One possible solution is the idea that the signatory countries have the power to punish the defectors (i.e., by imposing a carbon tax). This approach has been followed, among others, by Hoel and Schneider (1997) and Breton et al. (2010). The reverse approach tries to increase participation in an IEA by considering a transfer scheme, through which signatories transfer money to non-signatories to compensate for a lower emissions level [see e.g., Fuentes-Albero and Rubio (2010), Pavlova and de Zeeuw (2013)]. The issue linkage literature proposes the idea to link the environmental agreement with another agreement (i.e., trading or R &D), to induce reluctant countries to join the coalition (Botteon and Carraro 1998; Hübler and Finus 2013). Other approaches consider the introduction of a minimum participation clause or a ratification threshold (Rubio and Casino 2005; Carraro et al. 2009), or the benefits deriving from the creation of a network through the environmental agreement (Cabon-Dhersin and Ramani 2006; Sacco and Zaccour 2019).

The mechanisms described above work by punishing defectors or rewarding signatories, but the goal is the same: to compensate the free-ride incentive. Free riding is a well-known problem in economics; that is, each country takes advantage from an emissions reduction by other countries and that generates the incentive to let others take care of the environment, enjoying the benefits and avoiding the costs. The psychological motivation behind free riding lies in the assumption that agents (in this case countries) are selfish.

This selfishness is nested in the decision rules normally used in the two-stage game. In the membership game players choose to be in coalition if their reward is higher than if they defect. The Nash equilibrium concept used in the emissions game assumes that players (or coalitions of them) maximize their own payoff. Then a strategy profile is a Nash equilibrium if all the players have no incentive to change their own strategy. Nevertheless, many scientists (especially biologists and sociologists) have argued that often humankind has an attitude to act in an altruistic way.

Altruism can be classified in different ways. An initial division investigates the motivation of altruism, which is between pure and impure altruism. Pure altruism regards agents that act for the pure wellbeing of others, while in the case of impure altruism the behavior of the agents is altruistic for the possible rewards [Andreoni (1990) studied how it can affect public goods]. In this class lies reciprocity: cooperation is rewarded and non-cooperation is punished. In other words, people accept to help someone, even if it would cost them, because they think that they would be helped if needed [see Trivers (1971) about reciprocal altruism]. Another kind of impure altruism is reputational altruism, in which agents choose altruistic behavior to improve their public reputation. Milinski et al. (2002) studied how this form of altruism can help to solve the tragedy of the commons. Pure and impure altruism can also be divided into impartial altruism (each agent has the concern of every other agent) and community altruism (agents only care about other agents in their group).

The Berge–Zhukovskii equilibrium (BZ-equilibrium) is a first attempt to introduce into game theory an equilibrium concept that allows altruistic behavior. The idea, introduced in Berge (1957) and formalized in Zhukovskii (1985), is that a strategy profile is a BZ-equilibrium if the utility gained by each player is maximized only if other players do not have an incentive to deviate. That means, in the simplest case of two players, that the BZ-equilibrium is achieved when both players choose their own strategies, not looking at their own utility function, but looking at that of the other. Even if BZ-equilibrium is a suitable concept to describe games with altruistic behavior, it is not really useful in a global emissions game: the payoff structure of this kind of game is such that the BZ-equilibrium is unique and is given by all emissions being set to zero.

Nevertheless, some work has been done in the last few years to introduce altruism in environmental economics. Mazalov et al. (2021) define an altruistic equilibrium as a midway point between the Nash and Berge equilibria, in a dynamic model of renewable common pool resources. The assumption of their model is that players want to maximize global benefits, minimizing their own costs. In the paper by Van der Pol et al. (2012), altruism is introduced in a non-cooperative (static) game of IEA, to study how it may overcome the free-ride incentive. The authors study both impartial altruism and community altruism. In their framework altruism only concerns the membership game, but it does not affect the abatement game.

Following this stream of literature, the model presented in the paper assumes that players are to some degree altruistic. The aim is to provide an answer to two main questions:

  1. 1.

    Can altruism be a sufficient motivation to enable agreements to be signed by many countries?

  2. 2.

    Can altruistic behavior be a key factor in the effectiveness of the agreement?

To do that, the definition of altruism is split into the two stages of the game to study the effects on both membership and emissions reduction. The idea is that players need to take two different decisions, one for each stage of the game. Then they need two different decision rules to confront the problem.

More specifically, in the membership game the conditions of stability are modified, to describe agents that do not look to their own reward, but to the payoff of the entire coalition. These results are contrasted with the results obtained using the usual cartel stability conditions. The results presented in the paper show that this approach can guarantee significantly larger coalitions’ respect for the non-altruistic case, even if the grand coalition is not stable.

In the emissions game the paradigm shift introduced with the Paris Agreement in 2015 is followed: from a top-down to a bottom-up approach [see Winkler et al. (2018)]. The signatory countries do not jointly decide the optimal emissions level, but each country chooses the actions needed to achieve a general goal on global warming. In that spirit, and in contrast with the larger part of the literature, in the model the coalition does not act as a single player: each country maximizes its own payoff [for a similar approach see Colombo et al. (2022)]. The idea in this case is to study if reciprocity between players can be a factor in overcoming the free-ride incentive. The focus, then, is on community altruism, i.e. any player who chooses to join the coalition sets the emissions, taking care of the environmental cost of the other signatories. Numerical simulations show that this kind of altruism alone is not sufficient to enlarge the coalition, but that another kind of altruistic thinking is needed in the membership game. In this sense, assuming impartial altruism would lower the coalition’s emissions even further, with the effect of increasing the free-ride incentive. The closest model to the one presented in the paper is Van der Pol et al. (2012), although there are significant differences.

The first difference is that the altruistic behavior directly affects the emissions game [these effects are also studied in Kemfert and Tol (2002)]. Another relevant difference is that here the model is a differential game, and that allows us to also take into account the dynamic of pollution. Moreover, while they assume that all players are somehow altruistic, in our model only those in coalition are altruistic in both stages of the game.

As Finus and McGinty (2019) point out, asymmetry may be beneficial for the stability of an IEA. Besides, historically environmental agreements have recognized responsibilities. One of the main political points of the Kyoto Protocol was to divide the signatories into two sets: Annex I contained the developed countries, while Annex II contained the developing countries. The rationale behind that was to address developed countries with the responsibility of the current pollution level, and to make them pay the cost for the reduction. The model follows the same approach: the players of the game are then divided into two homogeneous groups, representing developed and developing countries. The paper is divided as follows: Sect. 2 introduces the model. In Sect. 3 the optimal emissions of members and non-members are computed, while Sect. 4 presents stability results and sensitivity analysis. Section 5 concludes.

2 The model

The model presented in this section is a global emissions game defined by a triple \(G=\{I, A, \pi _{i} \}\). The set \(I=\{1,2, \dots , n\}\) is the set of the n players, each of them representing a country. This set I is split into two subsets, denoted by \(I_{1}\) and \(I_{2}\), that contain the \(n_{1}>0\) developed countries and the \(n_{2}>0\) developing countries, respectively. Even if more asymmetry would be more realistic, the division into two homogeneous groups is suitable to take differences into account and is largely used in the literature [see e.g., Pavlova and de Zeeuw (2013)]. An environmental coalition is then a subset \(C=(C_{1}\cup C_{2}) \subseteq I\), where \(C_{1} \subseteq I_{1}\) is the set of developed countries in coalition and \(C_{2} \subseteq I_{2}\) is the set of developing countries in coalition. The second element of the triple G is the set of strategies A. Also this set can be written as the union of two disjoint sets \(A=A_{1}\cup A_{2}\), where \(A_{1}\) and \(A_{2}\) contain the strategies of developed and developing countries, respectively. The strategies contained in each set \(A_{i}\) are providen by the emissions’s function of each player i, that are the functions of time \(e_{i}(t)\) such that \(e_{i}(t)\ge 0 \,\,\forall t \in [0,+\infty )\). The third element of the triple G is the payoff (or welfare) function \(\pi _{i}\), \(i=\{1,2\}\), that is a map that, for every possible strategy profile, determines each player’s gain.

The production of goods and services generates benefits to the citizens of a country and, as a by-product, emissions of pollution as well. Calling by \(y_{i}(t)\) the total production of goods and services for country i at time t, it is possible to write the emissions of the country i as a function of its own production: \(e_{i}(t)=h(y_{i}(t))\), where h is an increasing function that satisfies \(h(0)=0\). If the function h is also smooth, then it is possible to express the relation between production and benefit in term of emissions directly. A very well-known form in the literature [see e.g., de Frutos et al. (2018)] for the benefit function of the player i, expressed by \(B_{i}(e_{i}(t))\), is the quadratic and concave function

$$\begin{aligned} B_{i}(e_{i}(t))=\alpha _{i}e_{i}(t)-\frac{1}{2}e_{i}^{2}(t), \end{aligned}$$
(1)

where \(\alpha _{i}\) is a strictly positive parameter. The assumption of two homogeneous kinds of players means that there are only two different values for the parameter \(\alpha \); \(\alpha _{1}\) for each \(i \in I_{1}\) and \(\alpha _{2}\) for each \(i \in I_{2}\). Moreover, the usual presumption on these parameters is that \(\alpha _{1}>\alpha _{2}\). The simple idea is that developed countries are able to produce more goods and services for the unity of pollution with respect to developing countries.

Polluting emissions accumulate in the atmosphere over time. The stock of pollutants is defined by the variable S(t), that is the solution of the dynamical system

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{S}}(t)=\mu E(t)-\delta S(t)\\ S(0)=S_{0}, \end{array}\right. } \end{aligned}$$
(2)

where \(E(t)=\sum _{i=1}^{n}e_{i}(t)\) is the total amount of emissions at time t, \(\mu \) is a positive parameter representing the absorption rate of the current emissions and \(\delta \) is a positive parameter representing the decay rate of the stock of past pollution. The accumulation of polluting substances generates costs (i.e., costs for public health or the destruction of common pool resources). Then in the welfare function, each country has to enter a damage-cost function, denoted, for each player i, by \(D_{i}(S(t))\). This function is chosen linearly with respect to the state variable and it is given by

$$\begin{aligned} D_{i}(S(t))= \beta _{i}S(t), \end{aligned}$$
(3)

where \(\beta _{i}>0\), \(\beta _{i}=\beta _{1}\,\, \forall i \in I_{1}\) and \(\beta _{i}=\beta _{2}\,\, \forall i \in I_{2}\). Here, the assumption is that \(\beta _{1}<\beta _{2}\). Even if developed countries produce more pollution, they are also financially and technologically more resilient to the damage generated by this pollution. As an example, one can consider extreme weather events; for a developing country it is harder to rebuild (more costs) than a developed country after it is hit by an extreme weather event. The choice of a linear-state structure is made for reasons of tractability. A linear-quadratic function would be more suitable to describe this kind of costs, assuring a Nash equilibrium that depends on the state variable. The problem of the linear-quadratic assumption is that, jointly with asymmetric players and partial cooperation, it does not allow us to analytically find the Nash equilibrium of the game. Even if the linear-state structure will lead to a feedback Nash equilibrium that is constant with respect to time and does not depend on the state variable, the effects of the dynamics are still relevant in two aspects: (i) the optimal emissions will depend on the inter-temporal discount factor and on the absorption rate and on the decay rate of the stock of pollution; (ii) the decision rules in the membership game will depend on the stock of pollution, that is the solution of the differential Eq. (2). Moreover, the linear-state structure is also a very common choice in literature [see Benchekroun and Taherkhani (2014), Breton et al. (2010), Masoudi and Zaccour (2013)]. As Labriet and Loulou (2003) pointed out, the difference with a quadratic form is more quantitative than qualitative.

Hereafter, the assumption of an infinite horizon is made. Then the discounted payoff function of each player i is provided by

$$\begin{aligned} \pi _{i}=\int _{0}^{\infty }e^{-\rho t}\biggl [B_{i}(e_{i}(t))-D_{i}(S(t))\biggr ]dt, \end{aligned}$$
(4)

\(\forall i \in I\), where \(\rho >0\) is the discount factor. The assumption of a symmetric discount factor is common in the literature and guarantees the time consistency of the aggregate payoff [see Marín-Solano (2015) for the time inconsistency problem]. In a full non-cooperative environment, each player \(i \in I\) maximizes the integral in (4), subject to the constraint provided by the dynamics in (2).

Instead, partial cooperation models assume that a subset of k players chooses to sign an environmental agreement. In this case, the k players in a coalition maximize joint welfare, acting as if they were a single player. In this sense, the game is like a non-cooperative game played by \((n-k+1)\) players: the \((n-k)\) non-signatories, that solve the same optimization problem as above; and the player “coalition” that sets the emissions levels, optimizing the sum of the welfare of all the signatories. As usual, the game is solved backwards: first, it is assumed that a coalition of k players is formed and optimal emissions are computed; second, given the optimal emissions, the size of a stable coalition is measured.

3 Optimal emissions

This model lies in the partial cooperation literature to the extent that it is assumed that a number of countries \(k\le n\) join an environmental coalition. In particular, \(k=c_{1}+c_{2}\), where \(c_{1}\) is the number of developed countries and \(c_{2}\) is the number of developing countries. But, following the bottom-up approach of the Paris Agreement, it is not assumed that the coalition acts as a single player. Instead, the emissions game is fully non-cooperative and each country chooses its own emissions level [see also Colombo et al. (2022)]. The difference with respect to the non-cooperative solutions is that here countries in coalition are altruistic in some way.

3.1 Non-members optimal emissions

The case of non-members is the classic non-cooperative optimization problem, in which every player maximizes its own payoff. Then, each of the \((n-k)\) countries outside the coalition has to solve the problem

$$\begin{aligned}{} & {} \max _{e_{i}(t)\in A_{i}}\int _{0}^{\infty }e^{-\rho t}\biggl [\alpha _{i}e_{i}(t)-\frac{1}{2}e_{i}^{2}(t)-\beta _{i}S(t)\biggr ]dt,\nonumber \\{} & {} \text{ subject } \text{ to } \nonumber \\{} & {} {\left\{ \begin{array}{ll} {\dot{S}}(t)=\mu E(t)-\delta S(t)\\ S(0)=S_{0}, \end{array}\right. } \end{aligned}$$
(5)

where \(i=\{1,2\}\). We look for the feedback Nash equilibrium, that is unique and is characterized in the following proposition.Footnote 1

Proposition 1

The non-members optimal emissions, obtained as a solution of the problem (5) assuming a linear value function, are given by

$$\begin{aligned}{} & {} e_{i}^{nm}= \alpha _{1}-\beta _{1}\frac{\mu }{\rho +\delta }, \quad \forall i \in I_{1}\backslash C_{1},\nonumber \\{} & {} e_{i}^{nm}= \alpha _{2}-\beta _{2}\frac{\mu }{\rho +\delta }, \quad \forall i \in I_{2}\backslash C_{2}. \end{aligned}$$
(6)

Proof

See Appendix A. \(\square \)

3.2 Members optimal emissions

The signatories do not maximize joint welfare, but they act non-cooperatively in the emissions game. The difference between members and non-members is that the former perform some kind of altruism when they have to decide their own emissions level. This altruism is expressed in the objective function by increasing the damage cost function. That is, each of the k players who signs the agreement has to solve

$$\begin{aligned}{} & {} \max _{e_{i}(t)\in A_{i}}\int _{0}^{\infty }e^{-\rho t}\biggl [\alpha _{i}e_{i}(t)-\frac{1}{2}e_{i}^{2}(t)-\beta _{i}\biggl (1+\sum _{h\in C\backslash \{i\}}\theta _{i,h}\biggr )S(t)\biggr ]dt,\nonumber \\{} & {} \text{ subject } \text{ to } \nonumber \\{} & {} {\left\{ \begin{array}{ll} {\dot{S}}(t)=\mu E(t)-\delta S(t)\\ S(0)=S_{0}, \end{array}\right. } \end{aligned}$$
(7)

where \(i=\{1,2\}\). Some considerations can be made with respect to the optimization problem in (7).

First, it is assumed that each country recognizes that its emissions have a negative impact on the cost structure of other countries. This awareness is achieved through the parameters \(\theta _{i,h}\) in the welfare function. These parameters, assumed to be greater than zero, represent the weights of the environmental cost of the others that country i internalizes in its welfare function. In the benchmark model it is assumed that \(\theta _{1,h}>\theta _{2,h}, \, \forall h\in C\backslash \{i\}\). The idea is that developed countries take responsibility for reducing emissions more than developing countries.

Second, it is assumed that member countries agree to practice community altruism. Signatories do not take into account the cost of all the other countries, but just the one amongst the others who signed the agreement. That means that, for each \(j \in (I \backslash C)\), \(\theta _{i,j}=0\) must hold. The rationale is that being reciprocal between members can be an incentive to join the coalition.

Third, the altruism can be asymmetric. In general, it is assumed that a country \(i \in C\) may assign different weights \(\theta _{i,h}\), for each \(h \in (C\backslash \{i\})\). The only assumption here is that homogeneous weights are chosen in each group of cooperators \(C_{1}\) and \(C_{2}\).

Finus and Maus (2008) proposed a game in which a modesty parameter is included by cooperators to weight the damage cost function. Even if the functional form is close to the one presented above, there are some significant differences. First, the model presented in Finus and Maus (2008) is a static game with symmetric countries. Second, they assume that the coalition acts as a single player, maximizing the joint payoff. In this context, the modesty parameter is not meant to take into account the damage caused to others, but to allow countries to disregard part of their damage costs. Third, while we assume that parameters \(\theta _{i,h}\) are exogenous, they study endogenously the properties of the modesty parameter that guarantee increasing payoffs and decreasing emissions with respect to the number of cooperators.

The members’ optimal emissions are characterized in the following proposition.

Proposition 2

The members’ optimal emissions, obtained as a solution of the problem (7) assuming a linear value function, are given by

$$\begin{aligned}{} & {} e_{i}^{m}= \alpha _{1}-\beta _{1}\biggl (1+\sum _{h\in C\backslash \{i\}}\theta _{i,h}\biggr )\frac{\mu }{\rho +\delta }, \quad \forall i \in C_{1},\nonumber \\{} & {} e_{i}^{m}= \alpha _{2}-\beta _{2}\biggl (1+\sum _{h\in C\backslash \{i\}}\theta _{i,h}\biggr )\frac{\mu }{\rho +\delta }, \quad \forall i \in C_{2}. \end{aligned}$$
(8)

Proof

See Appendix B. \(\square \)

Remark 1

The classic partial cooperative approach, in which it is assumed that the coalition acts as a single player maximizing joint welfare, brings to the cooperative solutions given by

$$\begin{aligned}{} & {} e_{i}^{m}= \alpha _{1}-\frac{\mu }{\rho +\delta }\sum _{h\in C}\beta _{h}, \quad \forall i \in C_{1},\nonumber \\{} & {} e_{i}^{m}= \alpha _{2}-\frac{\mu }{\rho +\delta }\sum _{h\in C}\beta _{h}, \quad \forall i \in C_{2}. \end{aligned}$$
(9)

Looking at the solutions in (8), if each player i weights others’ cost as \(\theta _{i,h}= \beta _{h}/\beta _{1}, \,\, \forall h \in C_{1}\) and \(\theta _{i,h}=\beta _{h}/ \beta _{2}, \,\, \forall h \in C_{2}\), then the altruistic emissions are equal to the classic partial cooperative solutions. Nevertheless, the altruistic approach highlights a higher level of “flexibility” regarding the level of optimal emissions. The classic solutions, reported in Eq. (9), depend entirely on the parameters \(\alpha _{i}\) and \(\beta _{i}\), which are determined by the economical and technical status of each country. On the other side, the altruistic solutions in (8) set an emissions level that depends on parameters \(\theta _{i,h}\), which are a measure of the commitment of each countries and that can be influenced more easily.

As expected, the members’ optimal emissions are decreasing functions of altruistic behavior: \(\partial e_{i}^{m}/ \partial \theta _{i,h}<0, \forall i \in C\) and \(\forall h \in C \backslash \{i\}\). Moreover, since every parameter \(\theta _{i,h}>0\), the members’ optimal emissions are decreasing with respect to the number of countries that join the coalition.

Assuming a coalition formed by \(c_{1}+c_{2}\) countries, then the total emissions, E, are given by the sum of emissions set by members and non-members:

$$\begin{aligned} E= & {} n_{1}\biggl (\alpha _{1}-\frac{\mu }{\rho +\delta }\beta _{1}\biggr )+n_{2}\biggl (\alpha _{2}-\frac{\mu }{\rho +\delta }\beta _{2}\biggr )+\nonumber \\{} & {} -c_{1}\frac{\mu }{\rho +\delta }\beta _{1}\sum _{h\in C\backslash \{i\}}\theta _{1,h}-c_{2}\frac{\mu }{\rho +\delta }\beta _{2}\sum _{h\in C\backslash \{i\}}\theta _{2,h}, \end{aligned}$$
(10)

in which \(n_{1}\) is the total number of developed countries and \(n_{2}\) is the total number of developing countries. The first thing to note is that the total emissions E is a constant function of time. That follows from the assumptions of linear value functions and an infinite horizon, which lead to constant optimal emissions, both for signatories and non-signatories. Moreover, Eq. (10) explains very clearly how the altruism works. The total emissions are determined by two elements, namely \(E=E^{nc}-E^{c}\):

  1. 1.

    The sum of the first two terms on the right side of the Eq. (10) represents total emissions in a world without cooperation:

    $$\begin{aligned} E^{nc}=n_{1}\biggl (\alpha _{1}-\frac{\mu }{\rho +\delta }\beta _{1}\biggr )+n_{2}\biggl (\alpha _{2}-\frac{\mu }{\rho +\delta }\beta _{2}\biggr ); \end{aligned}$$
  2. 2.

    The sum of the other two terms of the Eq. (10) represents how the fully non-cooperative emissions are reduced by partial cooperation

    $$\begin{aligned} E^{c}=c_{1}\frac{\mu }{\rho +\delta }\beta _{1}\sum _{h\in C\backslash \{i\}}\theta _{1,h}+c_{2}\frac{\mu }{\rho +\delta }\beta _{2}\sum _{h\in C\backslash \{i\}}\theta _{2,h}. \end{aligned}$$

    The emissions’ reduction effect is increasing with respect to altruism: the more the parameters \(\theta _{i,h}\) increase the more the function \(E^{c}\) increases and the total emissions decrease as a consequence. Of course, if signatories are non-altruistic, that is \(\theta _{i,h}=0\) for all \(h \in C\backslash \{i\}\), then the model become a full non-cooperative model since \(E^{c}=0\).

Considering a coalition of \(c_{1}+c_{2}\) players, the solution of the differential equation in (2) is provided by

$$\begin{aligned} S(t)=\biggl (S_{0}-\frac{\mu }{\delta }E\biggr )e^{-\delta t}+ \frac{\mu }{\delta }E. \end{aligned}$$
(11)

Given that the total emissions E is constant in time, the steady state of the stock of pollutant is given by

$$\begin{aligned} \lim _{t \rightarrow +\infty }S(t)= \frac{\mu }{\delta }E. \end{aligned}$$

4 Stability

Going backwards, after computation of the optimal emissions, the next step is to derive which coalitions of size \(k=c_{1}+c_{2}\) can be formed. The usual approach to the membership game is to consider it a meta-game: there is no coalition formation process, but for each possible coalition of size k it must be checked whether that coalition is stable or not.

To do that, the IEA literature borrowed the stability conditions introduced by d’Aspremont et al. (1983) for a cartel’s formation. These stability conditions are given by two sets of inequalities: (i) internal stability conditions, that guarantee that each member has no incentive to withdraw; (ii) external stability conditions, that guarantee that each non-member has no incentive to join the coalition. Denoting with \(\pi _{i}^{m}(k)\) and \(\pi _{i}^{nm}(k)\) the discounted payoff of a member and a non-member when the coalition is formed by k players, then that coalition is stable if k satisfies the following sets of inequalities

  • for each player \(i\in C\) must hold \(\pi _{i}^{m}(k)\ge \pi _{i}^{nm}(k-i)\);

  • for each player \(i\in I\backslash C\) must hold \(\pi _{i}^{nm}(k)\ge \pi _{i}^{m}(k+i)\).

The first set of inequalities is called the internal stability conditions, while the second set represents the external stability conditions.

When some degree of symmetry is assumed, i.e., dividing players into two sets of homogeneous agents, it is not necessary to check the condition of every player, but it is possible to consider only two internal and two external stability conditions [see also Pavlova and de Zeeuw (2013), Mallozzi et al. (2015), Sacco and Zaccour (2019)]. Then, when a coalition of \(k=c_{1}+c_{2}\) players is considered, the stability conditions are given by the following system of inequalities

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\pi _{1}^{m}(c_{1}, c_{2})\ge \pi _{1}^{nm}(c_{1}-1,c_{2}) \\ &{}\pi _{2}^{m}(c_{1}, c_{2})\ge \pi _{1}^{nm}(c_{1},c_{2}-1)\quad \text{(internal } \text{ stability), } \\ &{}\pi _{1}^{nm}(c_{1}, c_{2})\ge \pi _{i}^{m}(c_{1}+1, c_{2}) \\ &{}\pi _{2}^{nm}(c_{1}, c_{2})\ge \pi _{i}^{m}(c_{1}, c_{2}+1)\quad \text{(external } \text{ stability) }. \end{array}\right. } \end{aligned}$$

In the model presented, the discounted payoffs are given by

$$\begin{aligned} \pi _{i}^{m}(c_{1}, c_{2})= & {} \int _{0}^{+\infty }e^{-\rho t}\biggl (\alpha _{i}e_{i}^{m}-\frac{1}{2}e_{i}^{m}-\beta _{i}S(t)\biggr )dt\nonumber \\ \pi _{i}^{nm}(c_{1}, c_{2})= & {} \int _{0}^{+\infty }e^{-\rho t}\biggl (\alpha _{i}e_{i}^{nm}-\frac{1}{2}e_{i}^{nm}-\beta _{i}S(t)\biggr )dt, \end{aligned}$$
(12)

where \(i=\{1,2\}\) and S(t) is the solution of the system (2), computed with the optimal emissions. The dependence of these two functions with respect to the number of signatories, \(c_1\) and \(c_2\), is embedded in the optimal emissions \(e_{i}^{m}\) and, then, in the solution of the stock of pollutant S(t).

To evaluate the stability conditions, the members’ discounted payoff function, \(\pi _{i}^{m}\), in (12) is assumed to be different with respect to the one used in the emissions game. Again, each country is called upon to make two different and separate decisions in the two stages of the game and therefore, to do so, they can use two different welfare functions. There are two levels of reasoning then. In the emissions game, players that are in coalition are altruistic between themselves and choose their optimal emissions incorporating at least part of the cost that pollution generates for other countries.

In the membership game players compute the actual cost they pay and decide whether for them it is convenient to join the coalition or not. At this stage they do not consider the damage that pollution procures for others for the simple reason that they will not actually pay for that damage. This extra-cost is embedded in the optimal emissions \(e_{i}^{m}\) and countries can take it into account in the membership stage because they are assumed to be rational.

The discounted payoff of the members also depends on the cardinality of the coalition, since it affects the sum of the parameters \(\theta \) in the optimal emissions \(e_{i}^{m}\). For the sake of tractability, parameters \(\theta _{i,h}\) are assumed to be unique within each group: countries in set \(I_{i}\), \(i=\{1,2\}\), apply the same \(\theta _{i}\) to all other countries. Simulations that consider more asymmetry, namely considering \(\theta _{i,1}\) and \(\theta _{i,2}\) for each \(i=\{1,2\}\), have been made but the qualitative results are very close to the ones obtained in the simulations presented in this section. Then the optimal emissions of members countries considered in this section are provided by

$$\begin{aligned} e_{i}^{m}=\alpha _{i}-\frac{\beta _{i}[1+(c_{1}+c_{2}-1)\theta _{i}]}{\rho +\delta }\mu , \quad i=\{1,2\}. \end{aligned}$$

Internal and external stability is hard to check analytically, since it depends on parameter value (see Appendix C). Then a useful approach is to run simulations to deduce some conclusions. We used the World Bank data set on CO2 emissions,Footnote 2 expressed in kg per purchasing power parity (PPP) of Gross Domestic Product (GDP), to calibrate the parameters \(\alpha _{i}\) and \(\beta _{i}\), with \(i=\{1,2\}\). The parameters of players in group \(I_{1}\) are calibrated using the aggregate data for developed countries. For developing countries, the players in group \(I_{2}\), the set chosen is the aggregate data set for upper-middle-income countries. This group is formed by all countries classified as developing, but with a significant industrial structure and, hence, with a significant emissions level. Classic examples of these countries are the ones in the BRICS group. Moreover, the starting point S(0) is assumed to be equal to zero.Footnote 3 The calibration of parameters \(\mu \) and \(\delta \) in the state dynamics are based on Nordhaus (1993). The parameters \(\theta _{i}\), \(i=\{1,2\}\) are chosen to guarantee non negative emissions. Table 1 gives the parameter values.

Table 1 Benchmark parameters for developed (Type 1) and developing (Type 2) countries
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Fig. 1
figure 1

Internal stable coalitions with \(\theta _{1}=0.5\) and \(\theta _{2}=0.3\)

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To test the stability conditions, it is assumed that the world is made up of \(n_{1}=10\) developed countries and \(n_{2}=10\) developing countries. Hereafter only the internal stability conditions are discussed, since the interest is in the largest stable coalitions possible, but the external stability conditions are also checked. As shown in Fig. 1, developed countries have a great incentive to freeride. The colored areas in Fig. 1a and in Fig. 1b represent all the couples \((c_{1}, c_{2})\) such that the internal stability conditions hold. Then with the parameter values chosen, to be stable a coalition must be no larger than four players, otherwise developed countries have an incentive to withdraw.

Since the optimal emissions are invariant with respect to the sum \(c_{1}+c_{2}\), then every coalition of four players leads to non-cooperative emissions provided by \(e_{1}^{nm}=3.271\) and \(e_{2}^{nm}=2.152\), while the signatories’ emissions are given by \(e_{1}^{m}=3.109\) and \(e_{2}^{m}=2.001\). The total emissions for each instant of time t (that is the total emissions E), slightly changes considering the coalitions of four players. The best result achievable in terms of environmental protection is provided by the coalition that is made up of four developed countries, which produces total emissions E of 53.582. Compared with the case of non-cooperation (that gives an E of 54.23), this coalition guarantees a reduction of about 1.1% of pollution.

Some considerations need to be made, regarding the use of these stability conditions in the context of a model in which some degree of altruism is assumed. The fact is that these conditions are inherently selfish. Each player looks at its own payoff and decides whether it is more convenient for it to join the coalition or not. If this criterion is picked out as the stability mechanism in the membership game, then the free-ride incentive is present, even if players are altruistic in the emissions game. Changing perspective, it sounds quite natural that if players are altruistic in the emissions stage, they have to be altruistic in the membership stage too. Then stability conditions are modified to take into account both altruism and self-interest.

The most immediate way to do that, is to consider a modified version of internal stability. Instead of measuring its own payoff, inside and outside the coalition, and evaluating what is better for itself, each player i accepts to stay in the coalition if the joint payoff of the players in the coalition is greater or equal to the joint payoff of defectors when the \(i-th\) player withdraws from the agreement. The basic idea is that each player evaluates if its contribution is pivotal in the confrontation between the coalition of signatories and the “non-coalition” of defectors. Consider a coalition formed by \(k=c_{1}+c_{2}\) players. Then that coalition is said to be stable if the sum of the discounted payoff of signatories is greater than or equal to the sum of the discounted payoff of non-signatories when the coalition is formed by \(k-1\) players:

$$\begin{aligned} {\left\{ \begin{array}{ll} c_{1}\pi _{1}^{m} + c_{2}\pi _{2}^{m}\ge (n_{1}-c_{1}+1) \pi _{1}^{nm} + (n_{2}-c_{2})\pi _{2}^{nm} \\ c_{1}\pi _{1}^{m} + c_{2}\pi _{2}^{m}\ge (n_{1}-c_{1}) \pi _{1}^{nm} + (n_{2}-c_{2}+1)\pi _{2}^{nm}, \end{array}\right. } \end{aligned}$$
(13)

where

$$\begin{aligned} \begin{aligned} \pi _{i}^{m}(c_{1}, c_{2})&=\int _{0}^{+\infty }e^{-\rho t}\biggl (\alpha _{i}e_{i}^{m}-\frac{1}{2}e_{i}^{m}-\beta _{i}S(t)\biggr )dt\\ \pi _{i}^{nm}(c_{1}, c_{2})&=\int _{0}^{+\infty }e^{-\rho t}\biggl (\alpha _{i}e_{i}^{nm}-\frac{1}{2}e_{i}^{nm}-\beta _{i}S(t)\biggr )dt, \end{aligned} \end{aligned}$$

with \(i=\{1,2\}\) and S(t) is the solution of the system (2), computed with the optimal emissions. Again, the dependence with respect to \(c_{1}\) and \(c_{2}\) is embedded in the optimal emissions \(e_{i}^{m}\) and then in the stock of pollution S(t). To distinguish the condition in (13) with the stability conditions in d’Aspremont et al. (1983), we call them altruistic stability conditions. With regard to external stability, the conditions chosen are the classic ones. Players outside the coalition do not adopt altruistic decision making in either step of the game, therefore it can be assumed rather naturally that in the membership game they have no incentive to join the coalition if they can not increase their own welfare.Footnote 4

Given the same parameters value as above, now stable coalitions include all developed countries. The colored areas in Fig. 2a and in Fig. 2b represent all the couples \((c_{1}, c_{2})\) such that the altruistic stability conditions hold. The largest coalition achievable with this set of parameters is the one that includes all the developed countries and five developing countries. That is a bigger coalition with respect to the one achievable with the usual stability conditions.

In this case, the emissions of signatories are provided by \(e_{1}^{m}=2.513\) and \(e_{2}^{m}=1.447\). Looking at the total emissions for each instant of time t, that means a reduction of 20.5% with respect to the full non-cooperative case. That result is much better than the result obtained with the classic stability conditions. E in this case is equal to 43.1.

Fig. 2
figure 2

Altruistic stability conditions with \(\theta _{1}=0.5\) and \(\theta _{2}=0.3\)

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4.1 Sensitivity analysis

In this subsection a sensitivity analysis is carried out with two main goals: (i) to evaluate if it is possible to increase participation; (ii) to evaluate how different coalitions impact on total emissions.

The sensitivity analysis is formulated on the two main sets of parameters: (i) \(\theta _{i}\), that governs the altruism in the emissions game and then the total emissions of the coalition and (ii) \(\beta _{i}\), since the marginal cost has a significant impact on countries’ decisions in the membership game. The analysis regards both the classic stability conditions and the altruistic stability conditions.

The first thing to notice is that a variation of parameters \(\theta _{i}\), \(i=\{1,2\}\), has a different impact on the two types of stability conditions.Footnote 5

Table 2 Sensitivity analysis for parameters \(\theta _{1}\) and \(\theta _{2}\) under classic stability conditions
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4.1.1 Classic stability conditions

This subsection computes how changes in parameters \(\theta \) affect the stability of coalitions under classic stability conditions.

The results of these simulations are summarized in Table 2. In the first two lines parameter \(\theta _{1}\) is at the same level as in the benchmark case, while parameter \(\theta _{2}\) is decreased to 0.15 and then increased to 0.5. In both cases, the largest stable coalition is made up of four players. More specifically, in the first case every coalition formed by four countries is stable, except for the coalition (2, 2). Then the coalition (4, 0) is the best for environmental protection in this framework, while in the second case the coalition (2, 2) is the only coalition of four players to be stable.

In the following two lines \(\theta _{2}\) is as in the benchmark case, while parameter \(\theta _{1}\) is decreased to 0.25 and then increased to 0.8. In the first case the largest stable coalition is slightly increased, since it is given by the couple (3, 2). Interestingly, the total level of emissions E is very slightly affected by the enlargement of the coalition. Even worse is the second case, in which the largest stable coalition is made up of three developed countries and the total emissions are equal to 53.711. To increase participation that both parameters \(\theta _{1}\) and \(\theta _{2}\) need to decrease. In fact, considering the couple \(\theta _{1}=0.25\) and \(\theta _{2}=0.15\), then all the coalitions made up of eight members are internally stable, with optimal emissions provided by \(e_{1}^{m}=3.081\) and \(e_{2}^{m}= 1.976\). Again, the best option for environmental protection is the coalition which consists only of developed countries, that guarantees an E of 52.71.

Two considerations can be made looking at these results. The first one is that there is a clear trade-off between altruism in the emissions game and the stability of a coalition. Optimal emissions are decreasing, both with respect to the parameters \(\theta _{1}\) and \(\theta _{2}\) and with respect to the number of players in coalition \(c_{1}\) and \(c_{2}\). Then an increase of \(\theta _{1}\) and \(\theta _{2}\) on the one hand lowers members’ optimal emissions, but on the other it also reduces the size of a stable coalition, having a negative impact on the emissions level. Proof of that is the fact that the lower level of total emissions is obtained by reducing both \(\theta _{1}\) and \(\theta _{2}\), increasing participation.

The second consideration is that if altruism is only present in the emissions game, then it is not sufficient to avoid the free-riding incentive. The results show that it is not possible to have a large coalition, consisting of both kind of players, and to preserve the environment. This phenomenon derives directly from the assumption that players are selfish in the membership game.

4.1.2 Altruistic stability conditions

The altruistic stability conditions are robust with respect to variations of the parameters \(\theta _{i}\), \(i=\{1,2\}\). Letting \(\theta _{1}\) vary in the interval [0, 1] and \(\theta _{2}\) in the interval [0, 0.6], that guarantees that every coalition sets non-negative emissions, the result is always that the largest stable coalition is \((c_{1}, c_{2})=(10,5)\).

This highlights how, in this context, two kinds of altruism are needed for the agreement to work. First, countries have to think globally to decide whether to join the agreement or not. Second, they can choose their level of emissions, internalizing different levels of concern for others. As has been said, with the parameters chosen, a coalition consisting of all developed countries and five developing countries is stable. Even if it is not the grand coalition, it involves 75% of countries considered. Then environmental protection depends on how countries are sensitive to others. If parameters \(\theta _{1}\) and \(\theta _{2}\) converge to zero, then members’ optimal emissions converge to non-cooperative emissions and the agreement is not effective. On the other side, the more the values of \(\theta _{1}\) and \(\theta _{2}\) increase, the greater the result achieved. If, for example, they reach the levels \(\theta _{1}=1\) and \(\theta _{2}=0.6\), then we have \(e_{1}^{m}=1.754\) and \(e_{2}^{m}= 0.742\), with an E of 32.01, that is 41% less than the full non-cooperative case. This result seems to be a good explanation for what happens in the real world: agreements signed by many countries are often realized; whether they are effective or not depends on how much signatories are willing to commit.

Sensitivity analysis on parameters \(\beta \) The altruistic stability conditions are more sensitive to changes in parameters \(\beta _{i}\). Table 3 are summarize the results obtained in terms of the largest stable coalitions, the emissions of each country, and total emissions for different combinations of parameters \(\beta _{1}\) and \(\beta _{2}\).

The first three lines consider the cases in which the marginal cost increases for at least one kind of country. More specifically, in the first line it is \(\beta _{1}\), to increase to 0.0062. The effect is that the largest stable coalition is smaller than the benchmark case and total emissions increase slightly.

The second case is when it is \(\beta _{2}\), to increase to 0.0096. The largest stable coalition is still (10, 5) and the total emissions E are lower than in the benchmark case, due to the increase in marginal costs given, the same coalition.

In the third line, it is assumed that both parameters \(\beta _{1}\) and \(\beta _{2}\) increase. In this case, the largest stable coalition is smaller than the benchmark case; it is (10, 3) here, but the optimal emissions are so much lower that their reduction overcompensates for the shrinking of the coalition. In fact, this is the case with the lower level of total emissions E obtained in the simulations.

Table 3 Sensitivity analysis for parameters \(\beta _{1}\) and \(\beta _{2}\) under altruistic stability conditions
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On the other side, a decrease of parameters \(\beta \) leads to an increase in cooperation, and these scenarios are considered in the last three lines of Table 3.

Line four is the case in which \(\beta _{1}\) decreases. In this case the largest stable coalition involves seven developing countries beside all the developed countries, but the increase in emissions overcomes the positive effect of the enlargement of the coalition and total emissions are higher than the benchmark case.

A better result is obtained if it is \(\beta _{2}\), to decrease. In this case the grand coalition is stable and the total emissions are lower than the benchmark.

Finally, when both \(\beta _{1}\) and \(\beta _{2}\) decrease, the effect is that the grand coalition is still stable, but the total emissions are much higher than the benchmark.

Some considerations can be drawn from these simulations:

  • The model captures some important reality factors. For example, to achieve a grand coalition is a difficult task and to have this you may accept a downward agreement in terms of environmental protection. Nevertheless, it is possible to have stable and large coalitions with almost all countries and still have better results in terms of emissions reduction.

  • In every case, the largest stable coalition involves all the developed countries. That is confirmation of the importance for environmental agreements to be led by developed countries.

  • Even if there is no room for discussion about the dynamics of the parameters, that are assumed to be constant, if the objective is to achieve the grand coalition, then the better way to do it is to decrease the value of \(\beta _{2}\). Working on the capacity of developing countries to react to environmental damage, makes it possible to have the grand coalition and to significantly reduce emissions. The other way to have a stable grand coalition is to decrease both \(\beta _{1}\) and \(\beta _{2}\), but in this case the increasing effect on emissions due to lower costs is greater than the decreasing effect, due to the enlargement of the coalition. In terms of pollution this case gives the worst result in our simulations.

  • Achieving grand coalition may not be the best option for the protection of the environment. In fact, we experienced the best results in terms of total emissions per time in two cases: i) when countries have higher environmental costs or ii) when cooperators have a high level of altruism. The common result is that we have smaller stable coalitions with respect to the grand coalition, and they guarantee a high level of pollution abatement. In all the simulations made, the lowest level of emissions is achieved when \(\theta _{1}\) and \(\theta _{2}\) are set to their maximum level. In the case of high values of \(\beta _{i}\), the results differ between developed and developing countries. Two major facts can be highlighted. The first one is crystal clear: when both betas are doubled the environmental concern effect significantly overcomes the negative effect provided by smaller stable coalitions. The second one is perhaps more interesting: even if it is not historically fair, shifting the burden of environmental protection onto developing countries also works. In fact, when the marginal cost of developing countries is much higher, namely \(\beta _{2}\) is doubled, the third best result in term of pollution’s level is obtained in simulations.

  • In terms of feasibility, the best way to protect the environment would of course be to work on countries’ altruism. Parameters \(\theta _{1}\) and \(\theta _{2}\) depend on the awareness that each country has of the environmental damage it produces and on the sense of urgency it feels to remedy this. On the other side, \(\beta _{1}\) and \(\beta _{2}\) depend more on the economic and technical capacities of each country. Then, making again the abuse of thinking about parameter dynamics, to consider an increase in these parameters means to consider that their status would get worse, while to increase \(\theta _{1}\) and \(\theta _{2}\) would be easier, since they depend on the attitude of each country.

5 Conclusions

The aim of this paper was to study whether altruism is a possible factor to achieve a successful environmental agreement. Two different types of altruistic behavior were then considered. During the emissions game, it was assumed that countries in coalition choose their own optimal emissions, incorporating in the cost function the damage made by emissions to other countries. In this stage, community altruism was assumed. The optimal emissions obtained in the model were decreasing with respect to altruism and coincide with the classic partial cooperative solutions for a particular choice of the vector of parameters \(\theta \).

In the membership game, simulations showed that, with classic stability conditions, the free-ride incentive is not avoided and the agreement is not effective. Instead, with altruistic stability conditions, it is possible to achieve large coalitions. An interesting result highlighted by the sensitivity analysis was that the grand coalition is not necessarily the best possible outcome. We found that it is stable only when costs are low and this implies that cooperative emissions increase. Then simulations have shown that the best results in terms of pollution abatement are possible with coalitions that involve most of the countries, even if not all of them.

This model belongs to a stream of literature that proposes a different point of view on the behavior of economic agents, including altruism in their decision making [see Van der Pol et al. (2012), Mazalov et al. (2021)). Further steps need to be taken in that direction. What was done in the emissions game was to include altruism in the objective function of the players, incorporating it into a selfish decision rule. The idea behind the model is that when agents have to make a choice, strategic or not, for them it is perfectly rational to implement a decision making process characterized by self-interest and altruism. Then it would be logical to think of a possible next step that involves the definition of a new equilibrium concept that recognizes that agents are, in general, neither selfish nor fully altruistic, but they are both.