Self-Enforcing International Environmental Agreements and Altruistic Preferences

Schopf, Mark Article — Published Version Self-Enforcing International Environmental Agreements and Altruistic Preferences Environmental and Resource Economics Provided in Cooperation with: Springer Nature Suggested Citation: Schopf, Mark (2024) : Self-Enforcing International Environmental Agreements and Altruistic Preferences, Environmental and Resource Economics, ISSN 1573-1502, Springer Netherlands, Dordrecht, Vol. 87, Iss. 9, pp. 2309-2359, https://doi.org/10.1007/s10640-024-00885-8 This Version is available at: https://hdl.handle.net/10419/315261 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Environmental and Resource Economics (2024) 87:2309–2359 https://doi.org/10.1007/s10640-024-00885-8 1 3 Self‑Enforcing International Environmental Agreements andAltruistic Preferences MarkSchopf1 Accepted: 10 May 2024 / Published online: 25 June 2024 © The Author(s) 2024 Abstract This paper analyses the effects of altruism on the formation of climate coalitions in the standard two-stage game of self-enforcing international environmental agreements with identical countries. Altruism implies that each country values, to some extent, every other country’s welfare when deciding on its coalition membership and emissions policy. In the Nash [Stackelberg] game, the fringe [coalition] countries exploit the altruism of the coalition [fringe] countries so that altruism decreases [increases] the coalition size. In any case, global emissions and global welfare are close to the non-cooperative values. However, altruism narrows the gap between the individually optimal emissions and the socially optimal emissions, so altruism increases global welfare. The effects of altruism on the formation of climate coalitions crucially depends on its modelling: If altruism affects the membership decision but not the policy decision, or if each coalition country is more altruistic toward other coalition countries than toward fringe countries, altruism can stabilise large coalitions up to the grand coalition. Finally, altruism can stabilise small coalitions but destabilises large coalitions with asymmetric countries. Keywords Climate coalition· Climate policy· Moral behaviour· Social norms JEL Classification C72· D64· Q54· Q58 1 Introduction The Paris Agreement, negotiated by 196 parties at the 2015 United Nations Climate Change Conference, aims to limit global warming to well below 2°C compared to preindustrial levels (UN 2015). Although there is thus broad consensus on the international goal of climate policy, a continuation of current policies would result in global warming of about 3°C above pre-industrial levels (UN 2023).1 Consequently, the Paris Agreement with its nationally determined contributions does not constitute an effective international * Mark Schopf [email protected] 1 Department ofEconomics, University ofHagen, Universitätsstr. 41, 58097Hagen, Germany 1 If current policies continue through 2030 and the implied carbon price in 2030 increases with the global growth rate through 2100, there is a 50% [4%] median chance of limiting global warming to 2.7°C 2310 M.Schopf 1 3 environmental agreement in terms of the international political narrative. On the other hand, some world regions have introduced rather high carbon prices despite facing negative social costs of carbon (see Table1). Although these carbon prices are still well below the global social cost of carbon (418/tCO2 from Ricke etal. 2018), this behaviour can hardly be explained with perfect selfishness.2 Instead, it may reflect the important effects of altruistic values on environmental behaviour found in the psychological literature (see, e.g., Dietz etal. 2005; Steg 2016; Lades etal. 2021).3 In particular, there is evidence that climate policy negotiators have social preferences regarding burden-sharing rules: If climate policy negotiators from rich countries were perfectly selfish, they would support the grandfathering rule, i.e. equal percentage reduction of emissions, and oppose the egalitarian rule, i.e. equal per capita emissions. However, gross domestic product per capita is not positively [negatively] correlated with support for the grandfathering [egalitarian] rule (Lange etal. 2007; Meulemann and Ziegler 2015). Furthermore, climate policy negotiators from industrialized countries state that the egalitarian rule should be the most important burden-sharing rule in international climate agreements (Kesternich etal. 2021). These results suggest that climate policy negotiators are not driven by perfect selfishness.4 This paper analyses the formation of climate coalitions with altruistic preferences. In particular, each country values, to some extent, every other country’s welfare when deciding on its coalition membership at the first stage of the game and emissions policy at the second stage of the game. In order to be able to compare our results with the standard literature (Carraro and Siniscalco 1993; Barrett 1994), we apply the canonical model of self-enforcing international environmental agreements with identical countries, concave utility from own emissions and convex costs from global emissions. Without altruistic preferences, the standard (Nash or Stackelberg) game with linear-quadratic emissions benefits and linear marginal emissions damages predicts either small or ineffective climate coalitions.5 Beyond the linear-quadratic case, Barrett (2013) and Nkuiya et al. (2015) show that climate thresholds can stabilise the grand coalition, Nkuiya (2020) finds that 2 With strategic climate policy, terms-of-trade effects could explain part of the gap between regional carbon prices and regional social costs of carbon (Markusen 1975; Hoel 1996). Furthermore, the presence of an international carbon market with endogenous permit choice (Helm 2003; Holtsmark and Weitzman 2020) incentivises regions with low abatement costs to mitigate emissions and sell permits, which can reduce global emissions and raise global welfare. This has been shown for an exogenous carbon market with an endogenous climate coalition (Altamirano-Cabrera and Finus 2006; Lessmann etal. 2014), an endogenous carbon market without climate coalition (Carbone etal. 2009; Holtsmark and Midttømme 2021) and an endogenous carbon market with an endogenous climate coalition (Yu and Wu 2022). 3 Further examples include Kotchen and Moore (2007) and Ziegler (2020) [Engler etal. (2022) and Andre etal. (2024)], who find that altruistic values are significantly positively correlated with participation in green-electricity programs [pro-climate donations]. 4 Hjerpe etal. (2011) argue that the ability-to-pay rule, in terms of gross domestic product per capita, “has the greatest potential to serve as a basis for agreement in negotiations on allocating mitigation commitments” because it is supported by many and opposed by few climate negotiators. Furthermore, it is not opposed by any climate negotiator from industrialized countries. 5 In the Nash game with linear-quadratic emissions benefits and quadratic emissions damages, Finus (2001,p. 232) finds that climate coalitions consist of no more than two countries. In the Stackelberg game with linear-quadratic emissions benefits and quadratic emissions damages, Finus (2001,p. 232) finds that Footnote 1 (continued) [2°C] above pre-industrial levels (UN 2023,Chapter4.5, Appendix C). However, the very high emissions scenario of the IPCC (2021), which best reflects the cumulative emissions from 2005 to 2020 (Schwalm etal. 2020), even predicts a temperature increase of 4.4 °C above pre-industrial levels (median; fifth to ninety-fifth percentile: 3.3–5.7°C) by the end of the century (IPCC 2021,Chapter4.3). 2311 Self‑Enforcing International Environmental Agreements and… 1 3 climate coalitions can be large and effective with isoelastic emissions benefits in the Stackelberg game, and Eckert and Nkuiya (2022) show that convex marginal emissions damages can stabilise large coalitions up to the grand coalition in the Nash game. With general functional forms, the second stage of the Nash game and the Stackelberg game have been analysed in detail by Bayramoglu etal. (2018) and Finus etal. (2021a, 2021b), respectively. Finally, Finus etal. (2023) show that the stable coalition is always weakly larger in the Stackelberg game than in the Nash game.6 We distinguish between the coalition countries taking the fringe countries’ emissions as given (Nash game) and taking the reaction of the fringe countries’ emissions into account (Stackelberg game) when choosing their own emissions. In both cases, altruism reduces each fringe country’s emissions and raises global material welfare, i.e. global welfare in the absence of altruistic preferences. Furthermore, we get the typical results that global emissions decrease and each fringe country’s emissions and material welfare increase with the coalition size in the Nash game and above a critical coalition size in the Stackelberg game. By contrast, the effect of altruism on the equilibrium coalition size depends crucially on the game structure. In the Nash game with linear-quadratic emissions benefits and quadratic emissions damages, altruism weakly reduces the coalition size, and climate coalitions consist of no more than two countries. The direct effect of altruism, namely smaller global emissions and larger global material welfare for a larger coalition size, makes it worthwhile for all other countries if some country joins the coalition. However, the indirect effect of altruism, namely smaller global emissions and larger global material welfare for a given coalition size, makes it less costly for all other countries if some country does not join the coalition. This indirect effect outweighs the direct effect for small coalition sizes and explains the small climate coalition in equilibrium. In the Stackelberg game with linear-quadratic emissions benefits and quadratic emissions damages, altruism weakly raises the coalition size, and climate coalitions can consist of up to six countries. In this case, the coalition countries take advantage of the fringe countries’ altruism by becoming less ambitious in the fight against climate change, expecting the fringe countries to react by reducing their emissions more than they would without altruism. However, the coalition countries are not much more ambitious in the Stackelberg equilibrium than in the business-as-usual scenario without coalition formation. These results suggest that altruism cannot stabilise large and effective climate coalitions. However, altruism narrows the gap between the individually optimal emissions and the socially optimal emissions, so altruism increases global welfare. Thus, altruism affecting the membership decision and the policy decision appears to be more of a substitute than a complement for large climate coalitions. The economic literature has developed and tested several theories for imperfect selfishness. In the case of altruistic preferences (Becker 1974), one can distinguish between pure altruism, i.e. utility from others’ utility values (Becker 1981), paternalistic altruism, i.e. utility from others’ consumption bundles (Pollak 1988), and impure altruism, i.e. utility or warm glow from giving others (Andreoni 1990). Alger and Weibull (2010) show that pure altruism used in this paper is evolutionary stable, and Andreoni etal. (2010) 6 With linear abatement benefits, Karp and Simon (2013) find that a coalition of two or less [three or more] countries is stable with strictly convex [concave] marginal abatement costs in the Nash game. climate coalitions are either small or ineffective, and Diamantoudi and Sartzetakis (2006,p. 254) find that climate coalitions consist of no more than four countries when constraining the parameter space to ensure non-negative emissions. Footnote 5 (continued) 2312 M.Schopf 1 3 summarize the significant evidence for altruism in economic experiments. Other theories comprise reciprocal fairness (Rabin 1993), inequality aversion (Fehr and Schmidt 1999; Bolton and Ockenfels 2000) and Kantian behaviour (Alger and Weibull 2013; Roemer 2015). These theories have also been applied in the literature on self-enforcing international environmental agreements. Buchholz etal. (2018) and Nyborg (2018) analyse the effects of reciprocal fairness when countries decide on their membership in the coalition and on their emissions. They find that reciprocal fairness can stabilise the grand coalition, but it can also stabilise an interior coalition that is either weakly larger (Nyborg 2018) or even weakly smaller (Buchholz etal. 2018) than the interior coalition without reciprocal fairness. Lange and Vogt (2003) incorporate inequality aversion à la Bolton and Ockenfels (2000) into the canonical model of self-enforcing international environmental agreements and find that sufficiently large inequality aversion can stabilise the grand coalition. By contrast, Vogt (2016) applies inequality aversion à la Fehr and Schmidt (1999) and finds no stable coalition without transfers in his numerical model with heterogeneous countries. Recently, Eichner and Pethig (2022) and Ulph and Ulph (2023) analysed the effects of Kantian or moral behaviour when countries decide on their membership in the coalition and on their emissions. They find that membership moralism expands the climate coalition, and emissions moralism can expand the climate coalition only in the presence of membership moralism. Closest to our paper is vander Pol etal. (2012), who analyse the effects of altruism affecting the membership decision but not the policy decision. They find that this kind of partial altruism expands the climate coalition. We extend their model into different directions. First, we consider altruism on both stages of the game. Second, we analyse not only the Nash game but also the Stackelberg game. Third, while they solve their model numerically with heterogeneous countries, we solve our model analytically with homogeneous countries. Forth, we replicate their results analytically to discuss the differences from our results.7 Finally, we perform two model extensions. First, we analyse the effects of community altruism, that is, we distinguish between in-group altruism and out-group altruism. In this case, altruism can stabilise large coalitions up to the grand coalition. Second, we analyse the effects of altruism with linear climate damages and asymmetric countries. In this case, altruism can stabilise small coalitions but destabilises large coalitions. Table 1 Largest carbon pricing schemes representing 22% of global CO2 emissions Price: The World Bank (2023), SCC (social cost of carbon): Ricke etal. (2018) $/tCO2EU GBR CAN USA KOR ZAF CHN ARG MEX JPN KAZ UKR Price 73 58 38 28 19 10 10 5 4 2 1 1 SCC −4 −4 −8 48 −1 3 24 3 12 6 −1 −1 7 Daube (2019) and Goussebaïle et al. (2023) analyse the effects of altruism on climate policy with multiple countries. Daube (2019) shows that altruistic preferences lead to a partial internalization of the climate externality in the non-cooperative solution, and to a full internalization of the climate externality in the cooperative solution if and only if the altruistic preferences for all countries coincide. Goussebaïle etal. (2023) analyse the effects of altruistic foreign aid on climate change mitigation and find that paying transfers before abating emissions incentivises developing countries to choose efficient climate change mitigation and leads to the social optimum if altruistic preferences are sufficiently large. However, both papers abstract from coalition formation. 2313 Self‑Enforcing International Environmental Agreements and… 1 3 The remainder of the paper is organized as follows: Sect.2 introduces the model, and characterises the social optimum and the business-as-usual scenario. Section3 analyses the effects of altruism on the Nash game of coalition formation. This section also includes a comparison with the model of vander Pol etal. (2012). Section4 analyses the effects of altruism on the Stackelberg game of coalition formation with the coalition countries as Stackelberg leaders and the fringe countries as Stackelberg followers. Section5 discusses which realms of decision making might be influenced by social preferences. Section6 performs our two model extensions. Section7 concludes. 2 Model Consider a model with n≥3 identical countries.8 Each county i∈N derives consumption benefits B(ei) from its emissions ei , where B(0) ≥ 0 , B′>0 and B′′ <0 , and faces climate damages D(e) from global emissions e ∶= ∑i∈N e i , where D(0)=0 , D′>0 and D′′ >0 . Then, each country’s material welfare function is Wi=B(ei)−D(e) . Furthermore, each country is altruistic such that it values its own material welfare by 1 and every other country’s material welfare by 𝛼∈[0, 1] .9 Thus, the altruism parameter 𝛼=0 implies perfectly selfish countries, while 𝛼=1 implies perfectly altruistic countries. Then, each country’s moral welfare is where W∶= ∑i∈NWi is global material welfare, and the global moral welfare is Consequently, the socially optimal emissions (SO) are independent of the altruism parameter 𝛼 , while the individually optimal emissions, i.e. the business-as-usual emissions (BAU), are not (Daube 2019, Results 4 and 5). In particular, the socially optimal values and the individually optimal values coincide for 𝛼=1 . In AppendixA.1, we prove that global emissions decrease and global material welfare increases with the altruism parameter in the individually optimal solution. Consequently, the relative global emissions eBAU ∕ e SO decrease and the relative global material and moral welfare WBAU ∕WSO =VBAU ∕VSO increase with the altruism parameter. In the further course of the paper we analyse the two-stage game of self-enforcing environmental agreements. At the first stage of the game, countries decide on their membership in the coalition. Thereby, internal [external] stability implies that no country will leave [join] the coalition if this reduces its moral welfare (D’Aspremont etal. 1983). At the second stage of the game, there is a coalition of m countries, and countries decide on their (1) V i=Wi+𝛼 ∑ j∈N�i Wj=(1−𝛼)Wi+𝛼W , (2) V ∶= ∑ i∈N Vi= ∑ i∈N [ Wi+𝛼 ∑ j ∈ N � i Wj ] =[1+𝛼(n−1)]W . 8 We assume identical countries for analytical tractability with convex climate damages. In reality, countries benefit differently from own emissions and suffer differently from global emissions. For an analysis with asymmetric countries and linear climate damages, see Sect.6.2. 9 Instead, if each country values its own material welfare by 1 and every other country’s moral welfare by 𝛾∈[0, 1∕n] , then each country’s moral welfare function is V i=Wi+𝛾 ∑j∈N�i Vj=  Wi+𝛼 ∑j∈N�i W j with  Wi=Wi∕(1+𝛾) and 𝛼 =𝛾∕[1−𝛾(n−1)] ∈ [0, 1] , and our results do not change. For an analysis with community altruism (greater degree of in-group altruism than out-group altruism), see Sect.6.1. 2314 M.Schopf 1 3 emissions. Thereby, each fringe country maximizes its moral welfare (1), and each coalition country i∈M maximizes the sum of the coalition countries’ moral welfare Comparing (1) and (3), each fringe country’s policy weights its own material welfare by 1−𝛼 and global material welfare by 𝛼 , while each coalition country’s policy weights the coalition’s material welfare by 1−𝛼 and global material welfare by 𝛼m . In the following we distinguish between two game concepts. In Sect.3, we analyse the Nash game, and in Sect.4, we analyse the Stackelberg game with the coalition countries as Stackelberg leaders and the fringe countries as Stackelberg followers. The respective game is then solved by backward induction. 3 Nash Game At the second stage of the Nash game, each fringe country i=f maximizes its moral welfare (1) over its emissions ef , taking the other countries’ emissions as given, which yields Each fringe country equates marginal emissions benefits to its own marginal emissions damages D�(e) , plus all other countries’ marginal emissions damages weighted by the altruism parameter 𝛼(n−1)D�(e) . Furthermore, each coalition country i=c maximizes the sum of the coalition countries’ moral welfare (3) over its emissions ec , taking the other countries’ emissions as given, which yields10 For 𝛼=0 , each coalition country equates marginal emissions benefits to the coalition countries’ marginal emissions damages mD�(e) . For 𝛼>0 , altruism implies that each coalition country accounts for all other countries’ marginal emissions damages via 1+𝛼(n−1) , but it also implies that all other coalition countries account for each coalition country’s marginal emissions benefits via 1+𝛼(m−1) . Note that B� (e f )=B � (e c )=nD � (e ) for 𝛼=1 , so the Nash equilibrium and the social optimum then coincide. In the following we focus on 𝛼∈[0, 1) . Differentiating (4) and (5) yields the slopes of the aggregate reaction functions (3) ∑ i ∈M Vi= ∑ i∈M [ Wi+𝛼 ∑ j∈N�i Wj ] =(1−𝛼) ∑ i∈M Wi+𝛼mW . (4) B� (e f )=[1+𝛼(n−1)]D � (e)≤nD � (e) . (5) B �(ec)= 1+𝛼(n−1) 1+𝛼(m−1) mD�(e)≤nD�(e) . (6) R � f∶= d(n−m)e f dme c =− (n−m)[1+𝛼(n−1)]D��(e) (n−m)[1+𝛼(n−1)]D��(e)−B��(e f )∈ (−1, 0) , (7) R � c∶= dme c d(n−m)e f =− m 2 [1+𝛼(n−1)]D �� (e) m2[1+𝛼(n−1)]D��(e)−[1+𝛼(m−1)]B��(e c )∈ (−1, 0) . 10 The second-order conditions are fulfilled. 2315 Self‑Enforcing International Environmental Agreements and… 1 3 Consequently, emissions are strategic substitutes, and the slopes of the aggregate reaction functions ceteris paribus increase in absolute terms with the altruism parameter. Intuitively, altruism implies that countries react more sensitive to other countries’ emissions changes. Furthermore, we infer Consequently, each fringe country’s emissions are greater than each coalition country’s emissions. In AppendixA.2.1, we prove11 Proposition 1 (Comparison of Nash equilibrium and BAU) • ec <e BAU i <e f and e<eBAU , • Vf>Vc , • Wf >W c ,W BAU i . (8) implies that the coalition countries are ceteris paribus more ambitious in the fight against climate change than at BAU. This results in smaller coalition country’s emissions and global emissions, which raises the free-rider incentives and leads to greater fringe country’s emissions. Each fringe country’s emissions being greater than each coalition country’s emissions implies Vf>Vc and Wf>Wc . Finally, global emissions being smaller and each fringe country’s emissions being greater than at BAU implies Wf >W BAU i and, thus, Vf >V BAU i if Wc ≥W BAU i or if 𝛼 is sufficiently small. To prepare the analysis of the first stage of the Nash game, we prove in AppendixA.2.212 Lemma 1 (Effects of coalition size and altruism on emissions and welfare) • de f dm > 0 , de dm < 0 and dW f dm > 0 , • de f d𝛼 < 0 , de d𝛼 < 0 and dW d𝛼 > 0 . From the first bullet of the lemma, we get the typical results that each fringe country’s emissions increase but global emissions decrease with the coalition size, so free-rider incentives tend to increase as the coalition gets larger. The resulting higher consumption benefits and lower climate damages imply that each fringe country’s material welfare increases with the coalition size and, thus, that Vf increases with the coalition size if Wc increases with the coalition size or if 𝛼 is sufficiently small. The second bullet of the lemma reveals that each fringe country’s emissions and global emissions decrease with the altruism parameter and that global material welfare increases with the altruism parameter. Consequently, the relative global emissions e∕eSO decrease and the relative global material and moral welfare W∕WSO =V∕VSO increase with the altruism parameter. Finally, we prove in AppendixA.2.2 that the slope of the fringe countries’ aggregate reaction function increases in absolute terms with the altruism parameter for D′′′ ≤0 and B��� =0 . This slope corresponds to the leakage rate to the fringe countries (8) B� (ec) B �(e f )=m 1+𝛼(m−1)∈(1, m] . 11 For 𝛼=0 , the results of Proposition1 have been proven by Bayramoglu etal. (2018,p. 110) for the abatement game. 12 For 𝛼=0 , the results of Lemma 1 have been proven by Bayramoglu et al. (2018, p. 110) for the abatement game. 2316 M.Schopf 1 3 | R ′ f| , and the higher this leakage rate, the greater ceteris paribus the free-rider incentives in the Nash game (Carraro and Siniscalco 1993). Now we turn to the first stage of the Nash game. First note that Vf(m)>Vc(m) from Proposition1 implies that if a coalition is externally unstable, i.e. Vc(m+1) ≥ Vf(m) , then the corresponding expansion of the coalition is accompanied by a Pareto improvement, i.e. Vf(m+1)>Vc(m+1) ≥ Vf(m)>Vc(m) . For the detailed stability analysis, we use the following specification with linear-quadratic emissions benefits and quadratic emissions damages We constrain the parameter space to ensure non-negative emissions for all m∈[2, n] , which gives an upper bound for d. In particular, we formulate the following Assumption 1 d ≤  d∶= 4b (n−1)2 . This assumption is necessary and sufficient for non-negative emissions for all m∈[2, n] with 𝛼=0 .13 In AppendixA.2.3, we then prove Proposition 2 (Stability of coalitions with policy altruism) Consider the specification (9) and suppose altruism affects the membership decision and the policy decision. • Either the coalition m=2 is stable or no coalition is stable. • The coalition m=2 is [not] stable for 𝛼=0 , n≥12 and d≤  d [𝛼=0 , n<12 and d=  d] . • The coalition size weakly decreases with 𝛼 . From the first bullet of the proposition, we get the pessimistic result that at best a coalition of two countries is stable. The result from the second bullet of the proposition that m=2 is stable for 𝛼=0 when there is a sufficiently large number of countries may seem counterintuitive. In fact, there are two countervailing effects of n on the internal stability condition. On the one hand, the condition for m=2 to be stable becomes stricter as n increases for given parameter values a,b,d and 𝛼 , since the number of fringe countries increases with n, which raises marginal climate damages, reduces each coalition country’s emissions and, thus, the benefits of remaining in the coalition. On the other hand, the condition for m=2 to be stable also becomes stricter as d increases, since each coalition [fringe] country’s emissions decrease [increase] with the damage parameter, which reflects Barrett’s (1994) paradox of cooperation. These considerations imply that the upper bound for d to ensure non-negative emissions decreases with n, i.e. 𝜕 d∕𝜕n<0 , which relaxes the internal stability condition. This indirect effect of n on the internal stability condition outweighs the direct effect for m=2 and 𝛼=0 , which explains why the coalition m=2 (9) B(ei)=aei− b 2 e2 i,D(e)= d 2 e2, with a,b,d> 0. 13 Assumption1 is also sufficient for non-negative emissions for all m∈[2, n] with 𝛼≥0.75 . Further more, d ≤ b𝛼 2 ∕{(1−𝛼)[1+𝛼(n−1)−√1+𝛼(n−1)] 2 } is necessary and sufficient for non-negative emissions for all m∈[2, n] with 𝛼∈(0, 0.75) . 2323 Self‑Enforcing International Environmental Agreements and… 1 3 turn tends to increase with the altruism parameter from Proposition5. Via this mechanism, altruism could stabilise larger coalitions. For the detailed stability analysis, we use specification (9). We constrain the parameter space to ensure non-negative emissions for all m∈[2, n] , which gives an upper bound for d similar to the Nash game. In particular, we formulate the following Assumption 2 d ≤   d∶= 4b n(n−4) . This assumption is necessary and sufficient for non-negative emissions for all m∈[2, n] with 𝛼=0 .26 In AppendixA.3.5, we then prove27 Proposition 6 (Stability of coalitions) Consider specification (9) with n≥7 and d≤   d . • Some unique coalition m∈(m,m+2) is stable. • Some unique coalition m∈{2, 3} is stable for 𝛼=0 . • The coalition size weakly increases with 𝛼 . • Some unique coalition m∈{2, 3, 4, 5, 6} is stable for 𝛼>0 . Contrary to the Nash game, Proposition6 reveals that altruism stabilises larger coalitions. However, the coalition never comprises more than six countries. More importantly, the coalition is always smaller than m=m+2 . Since the Stackelberg equilibrium and BAU coincide for m=m , the emissions-reducing and welfare-enhancing effects of the coalition size from Lemma2 are negligible. In fact, the small coalitions stem from constraining the parameter space to ensure non-negative emissions for m∈[2, n] , which gives an upper bound for d/b and, thus, for m . From Proposition5, this upper bound increases with the altruism parameter, which is the driving force for larger coalitions with than without altruism. We use a numerical example to demonstrate there are economies in which the coalition is larger with than without altruism. Figure3 depicts each coalition country’s minimal emissions28 (left-hand side figure) and the internal stability condition (right-hand side figure) for m=3 (solid curve) and for m=4 (dashed curve) dependent on 𝛼 . In the numerical example, each coalition country’s emissions are positive for all m∈[2, n] . Furthermore, m=3 becomes stable for 𝛼≥0.223 , and m=4 becomes stable for 𝛼≥0.839 . Thus, there are economies in which the coalition is larger with than without altruism. Finally, Fig.4 shows that global emissions decrease and global material welfare increases with the altruism parameter in the numerical example. Furthermore, as the coalition gets larger at 𝛼=0.223 and at 𝛼=0.839 , global emissions jump downwards and global material welfare jumps upwards, but these jumps are (almost) not visible. 26 Assumption2 is also sufficient for non-negative emissions for all m∈[2, n] with 𝛼≥0.5 . 27 For 𝛼=0 , the results of Proposition6 have been proven by Diamantoudi and Sartzetakis (2006,p. 261). For n∈{5, 6} , they show that some unique coalition m∈(m,m+3) with m∈{2, 3, 4} [ m∈{2, 3} ] is stable for n=5 [ n=6 ]. 28 Using ec(m(𝛼),𝛼) with m( 𝛼 )=arg min ec(m, 𝛼 ) . 2324 M.Schopf 1 3 5 Discussion The previous sections have shown that altruism affecting the membership decision and the policy decision leads to small (Proposition2) or ineffective (Proposition6) climate coalitions. By contrast, altruism affecting the membership decision only (vander Pol etal. 2012) can stabilise the grand coalition (Proposition3).29 Given that the effects of altruism Fig. 3 Each coalition country’s minimal emissions (left-hand side figure) and the internal stability condition (right-hand side figure) for m=3 (solid curve) and for m=4 (dashed curve) dependent on 𝛼 with n=100, a=100, b=1 and d= 1 ∕4500 Fig. 4 Global emissions (left-hand side figure) and global material welfare (right-hand side figure) dependent on 𝛼 with n=100, a=100, b=1 and d=1∕4500 29 This was shown for the Nash game, but since the stable coalitions are always weakly larger in the Stackelberg game than in the Nash game (Finus etal. 2023), this also applies to the Stackelberg game. 2325 Self‑Enforcing International Environmental Agreements and… 1 3 on the formation of climate coalitions crucially depends on its modelling, this section discusses which realms of decision making might be influenced by social preferences in general.30 We start by looking at what assumptions the literature on self-enforcing international environmental agreements with social preferences makes and how it justifies them. First, the literature incorporating inequality aversion (Lange and Vogt 2003; Lange 2006; Vogt 2016; Rogna and Vogt 2022) considers social preferences at both stages of the game and argues that governments interested in re-election must take (median) voters’ fairness preferences into account in national policy and international negotiations. Second, the literature incorporating reciprocal fairness (Grüning and Peters 2010; Nyborg 2018) also considers social preferences at both stages of the game.31 Nyborg (2018,p. 707) argues that although groups may act differently than individuals, policy makers and treaty negotiators tend to be reciprocal when the general population is reciprocal, and tend to act reciprocally when the median voter is reciprocal. Third, the literature incorporating Kantian ethics (Eichner and Pethig 2022; Ulph and Ulph 2023) allows for different moral behaviour at the two stages of the game. Nevertheless, Eichner and Pethig (2022,pp. 18–19) argue that moral behaviour at just one stage of the game or different moral behaviour at the two stages of the game appears to be implausible, and Ulph and Ulph (2023,p. 12) “recognise that there is a strong argument that an agent should take the same moral stance to all decisions.” However, the latter argue that governments decide on coalition membership, while both governments and individuals decide on domestic emissions through public policy and private behaviour, respectively, such that the “decisions involve somewhat different agents”, which might explain different moral behaviour at the two stages of the game. Finally, van der Pol etal. (2012,p. 114) argue that “agents may hold different preferences when acting in different social situations, for example as consumers or as citizens.” They then distinguish between the decision about the technology employed and the domestic regulations adopted of a homo economicus with “personal well-being functions” (Nyborg 2000,p. 305), and the decision about the membership in the coalition of a homo politicus with “subjective social welfare functions” (Nyborg 2000,p. 305). To sum up, most of the literature assumes that individual social preferences affect both stages of the game via the median voter (or because policy makers and treaty negotiators tend to have the same social preferences as the general population). On the other hand, taken together, vander Pol etal. (2012) and Ulph and Ulph (2023) provide good arguments that social preferences could be different at the two stages of the game: If there is a distinction between homo economicus and homo politicus and if homo economicus can influence domestic emissions, then social preferences might be more pronounced at the membership stage than at the policy stage. However, note that the cited literature (and the present paper) abstracts from individual decisions and assumes that countries or governments decide on both coalition membership and domestic emissions. In this case, all decisions are made by citizens rather than by consumers, and there should be no qualitative difference between social preferences at the two stages of the game. 30 We owe this section to an anonymous reviewer who suggested to discuss which realms of decision making might be influenced by altruistic preferences. 31 InBuchholz etal. (2018) countries are reciprocal when they decide on their membership in the coalition, but the emissions policy inside and outside the coalition is exogenously given. 2326 M.Schopf 1 3 However, strategic delegation could be another argument for different social preferences at different game stages. The literature on strategic delegation shows that strategic voters elect policy makers who care less about global public goods when policy makers bargain over their provision. Thereby, Buchholz etal. (2005) and Loeper (2017) assume that policy makers decide on the cooperative bargaining outcome and on the non-cooperative bargaining default (strong delegation), while Segendorff (1998) and Graziosi (2009) also consider policy makers deciding on the cooperative bargaining outcome but median voters deciding on the non-cooperative bargaining default (weak delegation), which leads to different preferences at different game stages.32 Spycher and Winkler (2022) introduce strategic delegation into the standard two-stage game of self-enforcing international environmental agreements. They distinguish between weak delegation, i.e. elected policy makers decide on coalition membership but median voters decide on emissions policy, and strong delegation, i.e. elected policy makers decide at both stages of the game. While weak delegation does not increase coalition size but does increase global emissions, strong delegation can stabilise the grand coalition and thereby bring about the social optimum. These results suggest that strategic delegation can pay off. In this context, Lange and Schwirplies (2017) argue that there is indeed strategic delegation in climate policy because the social preferences of climate negotiators and the general population differ in that the former are more likely to support burden-sharing rules with low economic costs for their regions than the latter. 6 Extensions This section performs two model extensions. The first subsection shows that a small degree of community altruism can stabilise the grand coalition. The second subsection reveals that altruism can stabilise coalitions of two countries but tends to destabilise coalitions of three or more countries with linear climate damages and asymmetric countries.33 6.1 Community Altruism In this subsection, we consider community altruism. In particular, we distinguish between out-group altruism 𝛼 and in-group altruism 𝛽>𝛼 . The psychological literature has developed and tested two theories in particular for the preference of in-group members over out-group members (Balliet etal. 2014): The social identity theory assumes that individuals identify themselves with their memberships in social groups (Tajfel etal. 1979), while the theory of bounded generalized reciprocity assumes that groups contain individuals with cooperative reputations, which induces indirect reciprocity (Yamagishi and Kiyonari 1999). Cheikbossian (2021a) shows that in-group altruism is evolutionary stable, Cheikbossian (2021b) finds that a combination of in-group altruism and out-group altruism can be evolutionary stable, and Balliet etal. (2014) summarize the significant evidence for ingroup altruism (and against out-group spite) in psychological experiments. 32 While Buchholz et al. (2005) and Graziosi (2009) find that strategic voting cancels the gains of international cooperation, Segendorff (1998) finds that weak [strong] delegation increases [decreases] the gains of international cooperation, and Loeper (2017) finds that the results depend on the type of global public goods. 33 We owe this section to an anonymous reviewer who suggested to extend the analysis to community altruism and asymmetric countries. 2327 Self‑Enforcing International Environmental Agreements and… 1 3 In the following, we first analyse community altruism in the Nash game then report the effects of community altruism in the Stackelberg game. In the Nash game of coalition formation, each fringe country does not belong to a group, such that its moral welfare (1) and its first-order condition (4) do not change. By contrast, each coalition country belongs to the climate coalition, such that its moral welfare becomes Thus, community altruism ceteris paribus increases the weight of the coalition’s material welfare. Rearranging the corresponding first-order condition yields34 where the numerator reflects the altruistic preferences of the respective coalition country for all other countries’ climate damages, and the denominator reflects the altruistic preferences of all other coalition countries for the respective coalition country’s consumption benefits. Community altruism reduces the relative importance of the fringe countries’ climate damages, which ceteris paribus increases the coalition’s emissions. However, from (4) and (17), we infer Consequently, each fringe country’s emissions are still greater than each coalition country’s emissions. In Online Appendix B.1.1, we prove that the first bullet and the third bullet of Proposition1 also hold with community altruism: Each coalition country consumes less than at BAU, which reduces global emissions and, thus, marginal climate damages, such that each fringe country consumes more than at BAU. Consequently, the material welfare of each fringe country is greater than that of each coalition country and greater than that at BAU. However, and in contrast to the case without community altruism, the moral welfare of each coalition country can be greater than that of each fringe country if the material welfare of each coalition country is positive, because then community altruism ceteris paribus increases the former but does not affect the latter. In particular, we have Vf−Vc=(1−𝛼)(Wf−Wc)−(𝛽−𝛼)(m−1)Wc , which ceteris paribus decreases with the degree of community altruism and with the coalition size if Wc>0 . This welfare effect of community altruism can reduce the free-rider incentives. To analyse the policy effect of community altruism, we prove in Online Appendix B.1.135 Lemma 4 (Effects of community altruism on emissions and welfare) (16) ∑ i ∈M Vi=∑ i∈M [ Wi+𝛽∑ j∈M�i Wj+𝛼∑ j∈N�M Wj ] =[1−𝛼+(𝛽−𝛼)(m−1)] ∑ i∈M Wi+𝛼mW . (17) B �(ec)= 1+𝛼(n−m)+𝛽(m−1) 1+𝛽(m−1) mD�(e)≤nD�(e) , (18) B � (ec) B�(e f )=m 1+𝛼(n−1) 1+𝛼(n−m)+𝛽(m−1) 1+𝛽(m−1)∈(1, m] . 34 The second-order conditions are fulfilled. 35 Furthermore, we there prove that Lemma1 also holds with community altruism. 2328 M.Schopf 1 3 • de f d 𝛽 < 0 , de c d 𝛽 > 0 and de d𝛽 > 0 , • dW f d 𝛽 < 0 and dW c d𝛽 > 0 , • dV c d 𝛽 > 0 and d(V f −V c ) d𝛽 < 0 for Wc≥0 (sufficient). Ceteris paribus, community altruism does not affect each fringe country’s consumption, see (4), but increases each coalition country’s consumption, see (17). Thus, ec increases, which increases global emissions and, thus, marginal climate damages, such that ef decreases. Consequently, each fringe country’s material welfare decreases, while the increase in each coalition country’s consumption benefits outweighs the increase in the climate damages, such that its material welfare increases. This policy effect of community altruism raises each coalition country’s moral welfare and reduces the difference between the moral welfare of each fringe country and each coalition country if Wc ≥ 0 . Furthermore, it reduces each fringe country’s moral welfare Vf=(1−𝛼)Wf+𝛼W if 𝛼 is sufficiently small, such that the decrease in Wf outweighs the potential increase in W. Taken together, the sign of each coalition country’s material welfare plays an important role for the effects of community altruism on the free-rider incentives. This also becomes clear when we look at the (internal) stability condition of the grand coalition: First note that the grand coalition maximizes global material welfare from Vc(n)=[1+𝛽(n−1)]Wc(n) as in the case without community altruism. Now consider 𝛽=1 . Then, Vc(n) is maximized global material welfare W(n), and Vf(n−1) is non-maximized global material welfare W(n−1) minus the coalition’s material welfare weighted by 1−𝛼 . Thus, the grand coalition is stable if Wc(n−1)≥0 . By contrast, if Wc(n−1)<0 , the grand coalition need not be stable. For example, consider 𝛼=0 . Then, Vf(n−1) is equal to the fringe country’s material welfare Wf(n−1) , and this material welfare can exceed maximized global welfare W(n) if Wc(n−1)<0 . However, it can be shown that Wc(n−1)>0 holds with specification (9) when there is a sufficiently large number of countries ( n≥9 ). Then, the emissions policy of one fringe country is relatively unimportant for the material welfare of many coalition countries. In this case, the grand coalition is stable for 𝛽=1 . Note that Wc(n) is always non-negative because the grand coalition could choose ec =0 and, thus, B(0)≥0 and D(0)=0 . Consequently, in-group altruism increases each country’s moral welfare in the grand coalition. Furthermore, it reduces each fringe country’s material welfare, but raises each coalition country’s material welfare with any other coalition, such that the effect on each fringe country’s moral welfare is in general ambiguous. However, in Online Appendix B.1.1 we prove that this effect is definitely negative if there is only one fringe country. Then, in-group altruism increases the emissions of so many coalition countries that the corresponding increase in climate damages and decrease in the fringe country’s consumption benefits outweigh the increase in the coalition country’s consumption benefits. To sum up, in-group altruism increases the stability of the grand coalition. Finally, out-group altruism does not affect each country’s moral welfare in the grand coalition, but it reduces global emissions and increases global (19) V c (n)−V f (n−1) =(1−𝛽)Wc(n)+𝛽W(n)− [ (1−𝛼)Wf(n−1)+𝛼W(n−1) ] =W(n)−(1−𝛽)(n−1)W c (n)− [ W(n−1)−(1−𝛼)(n−1)W c (n−1) ]. 2329 Self‑Enforcing International Environmental Agreements and… 1 3 material welfare with any other coalition, see Lemma1. Thus, it also tends to increase Vf(n−1)=Wf(n−1)+𝛼(n−1)Wc(n−1) if Wc(n−1)>0 . Consequently, it can be shown that out-group altruism increases the fringe country’s moral welfare with specification (9) when there is a sufficiently large number of countries ( n≥10 ). In this case, outgroup altruism decreases the stability of the grand coalition. We prove our results in Online Appendix B.1.1 and summarize them in Proposition 7 (Stability of grand coalition with community altruism) Consider specification (9). • The internal stability condition of the grand coalition increases with 𝛽 , and it decreases with 𝛼 for n≥10 and d≤  d (sufficient). • The grand coalition is stable for 𝛽=1 , n≥6 and d≤  d (sufficient). Thus, community altruism can stabilise the grand coalition and thereby bring about the social optimum. Figure5 depicts thresholds of 𝛽−𝛼 for the grand coalition to be stable dependent on 𝛼 in the Nash game (left-hand side figure) for different values of the climate damage parameter d.36 When this parameter is small, emissions policy is determined mainly by marginal emissions benefits rather than marginal climate damages, so that the difference between the emissions and therefore the material welfare of each fringe country and each coalition country is small, which reduces free-rider incentives and stabilises the grand coalition. Figure5 shows that the grand coalition can be stable even with a small difference between in-group altruism and out-group altruism of less than 0.03.37 Note that community altruism does not change the grand coalition’s emissions policy, so its stabilizing effect relies on the worse emissions policy for the final fringe country (policy effect) and on the increased moral welfare of each coalition country for a given emissions policy (welfare effect).38 Finally, we report the effects of community altruism in the Stackelberg game. Since the results are simply a combination of those in the Stackelberg game without community altruism and those in the Nash game with community altruism, we delegate the full analysis to Online Appendix B.1.2. First, there is a unique threshold coalition m∈(1, n) where the Stackelberg equilibrium and BAU coincide in terms of emissions and material welfare. If WBAU i ≥ 0 , then Vc (m)≥V BAU i and Vf (m)≤V BAU i for m≤m , such that all coalition m≤m are externally unstable, and the coalition m=⌊m+1⌋≥2 is internally stable as in the case without community altruism.39 Second, community altruism increases each coalition country’s emissions and global emissions, and it decreases each fringe country’s 36 For n=100 and d∕b>3.75∕10,000 , each coalition country’s minimal emissions become negative for certain combinations of 𝛼 and 𝛽 . 37 Coincidentally, this is consistent with the numerical analysis of vander Pol etal. (2012), who find that the grand coalition is stable for 𝛼=0 and 𝛽≥0.03 or 𝛼≤0.024 and 𝛽=0.06 and unstable for 𝛼≥0.036 and 𝛽=0.06 . 38 Finally, we ran several examples to find other possible stable coalition sizes. There are economies in which no coalition is stable, only the coalition m=2 is stable, only the grand coalition is stable, or some coalition m≥2 and the grand coalition are stable. We did not find an example in which the coalition m=2 is stable in the case without community altruism and no coalition is stable in the case with community altruism. However, there are economies in which only the coalition m=2 is stable in both cases, and in which global moral welfare is smaller with than without community altruism due to the higher emissions damages. 39 By contrast, if WBAU i < 0 , t hen Vc (m)<V BAU i and Vf (m)=V BAU i , such that the coalition m=m−1 is externally stable. 2330 M.Schopf 1 3 emissions as in the Nash game. Consequently, m increases with 𝛽 , which can stabilise larger coalitions. Furthermore, community altruism decreases each fringe country’s moral welfare, and it increases each coalition country’s moral welfare if Wc≥0 as in the Nash game, which then stabilises larger coalitions. In particular, in-group [out-group] altruism increases [decreases] the internal stability condition of the grand coalition [when there is a sufficiently large number of countries] as in the Nash game. Furthermore, it can be shown that the condition for the grand coalition to be stable is laxer in the Stackelberg game than in the Nash game. However, Fig.5 shows that the thresholds of 𝛽−𝛼 for the grand coalition to be stable dependent on 𝛼 in the Stackelberg game (right-hand side figure) are close to those in the Nash game (left-hand side figure). 6.2 Asymmetric Countries In this subsection, we consider asymmetric countries in terms of consumption benefits Bi(ei) and climate damages Di(e) . With convex climate damages D′′ i>0 (or abatement benefits), the results of previous literature on coalition stability with asymmetric countries are based on numerical analyses (Barrett 1997; Botteon and Carraro 2001; McGinty 2007; Bakalova and Eyckmans 2019; McGinty 2020). In order to obtain analytical results, we thus rely on linear climate damages D�� i=0 in this subsection. This implies that each fringe country has a dominant strategy, such that the Nash game and the Stackelberg game coincide. Furthermore, we focus on the case without transfers. In such a framework with two types of countries, Fuentes-Albero and Rubio (2010) show that a coalition of at most three countries is stable if either climate damages or emissions benefits are symmetric. Pavlova and DeZeeuw (2013) confirm this result and show that the same holds if climate damages and emissions benefits are positively correlated. By contrast, if climate damages and abatement costs are negatively correlated, a Fig. 5 Thresholds of 𝛽−𝛼 for the grand coalition to be stable dependent on 𝛼 in the Nash game (lefthand side figure) and in the Stackelberg game (right-hand side figure) with n=100, a=100, b=1 and d=1∕10,000 (solid curve), d=2.375∕10,000 (dashed curve) and d=3.75∕10,000 (dotted curve) 2331 Self‑Enforcing International Environmental Agreements and… 1 3 coalition of all countries with low climate damages and two countries with high climate damages can be stable. Finus and McGinty (2019) extend this analysis by allowing for any type of countries. They show that the grand coalition can be stable if climate damages and abatement costs are negatively correlated. Following van der Pol etal. (2012), we consider a unique altruism parameter to derive the general effects of altruism on the formation of climate coalitions. Furthermore, we abstract from community altruism to check the robustness of our main model with linear climate damages and asymmetric countries. With a unique altruism parameter and without community altruism, each country’s moral welfare function is still given by (1), and the global moral welfare function is still given by (2). Consequently, the socially optimal emissions always maximize global material welfare, and the individually optimal emissions maximize global material welfare if and only if 𝛼=1 . In AppendixA.1, we prove that global emissions decrease and global material welfare increases with the altruism parameter in the individually optimal solution as in the main model. Consequently, the relative global emissions eBAU ∕eSO decrease and the relative global material and moral welfare WBAU ∕WSO =VBAU ∕VSO increase with the altruism parameter. In the coalition formation game, each fringe country’s moral welfare is still given by (1), and the sum of the coalition countries’ moral welfare is still given by (3). Rearranging the corresponding first-order conditions yields Consequently, considering two ex ante identical countries, the emissions of the country outside the coalition are greater than those of the country inside the coalition, which means that also the material and moral welfare of the country outside the coalition are greater than those of the country inside the coalition. In Online AppendixB.2.1, we prove40 Proposition 8 (Comparison of Nash equilibrium and BAU with asymmetric countries) • ei <e BAU i for all i∈M , ei =e BAU i for all i∉M and e<eBAU , • Vi >V BAU i for all i∉M and V>VBAU , • Wi >W BAU i for all i∉M and W>WBAU . (21) implies that the coalition countries are ceteris paribus more ambitious in the fight against climate change than at BAU. This results in smaller emissions inside the coalition, (20) B � i(ei)=(1−𝛼)D� i(e)+𝛼 ∑ j ∈ N D� j(e)≤ ∑ j ∈ N D� j(e),∀i∉M , (21) B � i(ei)= ( 1 − 𝛼 )∑ j∈MD � j ( e )+ 𝛼m ∑ j∈ND � j ( e ) 1+𝛼(m−1) =(1−𝛼)D� i(e)+𝛼� j∈N D� j(e) +(1−𝛼) ∑ j∈M�iD� j(e)+𝛼(m−1) ∑ j∈N�iD� j(e) 1+𝛼(m−1)≤ � j∈N D� j(e),∀i∈M . 40 For 𝛼=0 , the results of Proposition8 have been proven by Finus and McGinty (2019,p. 544) for the abatement game with linear benefits and quadratic costs. 2332 M.Schopf 1 3 while the fringe countries’ dominant strategies imply that the emissions outside the coalition do not change, such that global emissions are smaller than at BAU. Smaller global emissions and constant fringe country’s emissions imply Wi >W BAU i for all i∉M . The remaining results arise from the superadditivity of the game, i.e. ∑ j∈MVj≥ ∑ j∈MV BAU j : Superadditivity implies W>WBAU , which in turn implies Vi >V BAU i for all i∉M and V>VBAU . To prepare the analysis of the first stage of the Nash game, we prove in Online Appendix B.2.241 Lemma 5 (Effects of coalition size and altruism on emissions and welfare with asymmetric countries) • If another country joins the coalition, then each coalition country’s emissions decrease, each fringe country’s emissions do not change and global emissions decrease. Furthermore, each fringe country’s material and moral welfare increase, and global material and moral welfare increase. • de i d𝛼 < 0 for all i∈N and dW d𝛼 > 0 . The first bullet of the lemma reveals that the comparison between Nash equilibrium and BAU from Proposition8 can be transferred to the comparison between the equilibrium with some coalition M and the equilibrium with some smaller coalition M∖i : The larger the coalition, the more it fights against climate change, which leads to smaller emissions of the original coalition members, to smaller emissions of the new coalition member and, thus, to smaller global emissions. The material welfare of each fringe country increases, because its consumption benefits do not change, but its climate damages decrease. Superadditivity implies that global welfare increases, which in turn implies that the moral welfare of each fringe country and global moral welfare increase. The second bullet of the lemma reveals that each country’s emissions decrease and global material welfare increases with the altruism parameter. Consequently, the relative global emissions e∕eSO decrease and the relative global material and moral welfare W∕WSO =V∕VSO increase with the altruism parameter as in the main model with symmetric countries and convex climate damages. For the first stage of the game, we use the following specification In Online Appendix B.2.3, we then prove Proposition 9 (Stability of coalitions with asymmetric countries) Consider the specification (22). • Suppose bi=b and di=d for all i∈N . Then, any coalition with three members is stable for 𝛼=0 , and any coalition with two members is stable for 𝛼>0 . (22) Bi(ei)=aiei− b i 2 e2 i,Di(e)=die, with ai,bi,di> 0. 41 For 𝛼=0 , the results of Lemma 5 have been proven by Finus and McGinty (2019, p. 544) for the abatement game with linear benefits and quadratic costs. 2339 Self‑Enforcing International Environmental Agreements and… 1 3 Solving for ef , ec and e yields where Note that 𝜕 [ 𝜕2 ] e c Ω 1+𝛼(m−1) m> 0 . For 𝛼=0 , e c Ω 1+𝛼(m−1) is minimal at m = n+1 2 , and then e c Ω 1+𝛼(m−1) is non-negative if and only if d ≤  d∶= 4b (n−1) 2 , which is thus an upper bound for d. Using this upper bound in (A.34) yields For 𝛼>0 , e c Ω 1+𝛼(m−1) is minimal at and then e c Ω 1+𝛼(m−1) is non-negative if and only if Using (A.33), (A.34) and (A.35) yields (A.31) a −bec= 1+𝛼(n−1) 1+𝛼(m−1) mde , (A.32) e=(n−m)ef+mec. (A.33) e f= 1+𝛼(m−1)+m(m−1)(1−𝛼)[1+𝛼(n−1)] d b Ω a b > 0, (A.34) e c= 1+𝛼(m−1)−(n−m)(m−1)(1−𝛼)[1+𝛼(n−1)] d b Ω a b, (A.35) e = 1+𝛼(m−1) Ω na b > 0, Ω ∶= 1+𝛼(m−1)+[1+𝛼(n−1)]{[1+𝛼(m−1)]n+(1−𝛼)m(m−1)} d b > 0. e c= a 4b2Ω{{𝛼(m−1)[(n−1)(n−m)(4𝛼−3)+(m+3)(n−m)+(m−1)2] +(n+1−2m)2 } d+(n−1)2[1+𝛼(m−1)]( d−d) } >0⟸d≤ d∧𝛼≥3∕ 4. m =1+ √ 1+𝛼(n−1)−1 𝛼 =n+1 2−𝛼(n−1)2∕2 2 √ 1+𝛼(n−1)+2+𝛼(n−1) ∈ � 1, n+1 2 �, d ≤𝛼 2 b (1−𝛼){1+𝛼(n−1)− √ 1+𝛼(n−1)}2 . 2340 M.Schopf 1 3 The internal stability condition reads where (A.36) V f=[1+𝛼(n−m−1)] [ aef− b 2e2 f− d 2e2 ] +𝛼m [ aec− b 2e2 c− d 2e2 ] =a2 2bΩ2[1+𝛼(n−1)]{[1+𝛼(m−1)]2−[1+𝛼(m−1)]{[1+𝛼(m−1)][n2−2m 2 −2n+2m−2𝛼(n−1)(n−m)] − 2𝛼m2(n−m)}d b+(1−𝛼)2[1+𝛼(n−1)]m(m −1){m2+2n−m−𝛼(m−1)(n2−nm −2n+m)} ( d b) 2 }, (A.37) V c=[1+𝛼(n−m)] [ aef− b 2e2 f− d 2e2 ] +[1+𝛼(m−1)] [ aec− b 2e2 c− d 2e2 ] =Vf−a2 2bΩ 2n2(m−1)(1−𝛼)2[1+𝛼(n−1)]2[m+1+𝛼(m−1)] ( d b) 2 . (A.38) V c(m)−Vf(m−1)= a2n2(m−1)(1−𝛼)2[1+𝛼(n−1)]2 ( d b )2 2bΩ(m) 2 Ω(m−1) 2Φ(m) , (A.39) Φ( m) ∶= −[1+𝛼(m−1)][m−3+𝛼(m−2) 2 ]−[1+𝛼(n−1)]{2[(n−m)(m−1)+( m −3)3+6(m−3)2+11(m−3)+4]+2𝛼{(n−m)[2(m−2)2+5(m−2)+1] +(m−2)[(m−2)3+4(m−2)2+6(m−2)+2]} + 𝛼2(m−2)[2(n−m)(m2 +m−3)+(m−1)(m−2)] + 2𝛼3(n−m)(m−1)(m−2)2}d b−[1+𝛼(n−1)] 2 ⋅{[n(m+1)+m(m−1)2][n+m(m−3)] + 𝛼{(n−m)2[2(m−2)2+10(m−2) +3]+2(n−m)[(m−2)4+7(m−2)3+16(m−2)2+14(m−2)+2]+m2(m −2)2}+𝛼2(n−m)(m−2){(n−m)[(m−2)2+9(m−2)+6]+2(m−2)3+8 ⋅(m−2)2+12(m−2)+4}+𝛼3(n−m)(2n−2m+1)(m−1)(m−2)2}(d b)2 <0⟸m≥3, 2341 Self‑Enforcing International Environmental Agreements and… 1 3 such that all coalitions m≥3 are internally unstable, which proves the first bullet of the proposition. Furthermore, which proves the second bullet of the proposition. Finally, where such that m=2 is internally stable if d b , 𝛼 and n are sufficiently small. Since all coalitions m≥3 are internally unstable from (A.39), and the condition for m=2 to be stable (A.40) Φ( 2)�𝛼=0=1−2(n−4)d b−(n−2)(3n+2) � d b �2 ⋛ 0 ⟺ d b ⋚2 √ n2−3n+3−n+4 3n 2 −4n−4 , (A.41) = (n−1) 4 16b2 { (n−12) 4 +36(n−12) 3 +438(n−12) 2 +1860(n−12)+817 (n−1)4d 2 +2(n−3)2+16 (n− 1 ) 2( d−d)d+( d−d)2 } >0⟸n≥12 ∧d≤ d, (A.42) =− (n−1)4 16b2{ ( 11−n n−3 )4 +16 ( 11−n n−3 )3 +66 ( 11−n n−3 )2 +88 ( 11−n n−3 ) +5 32(n−1 n−3)4d 2 −2(n−3)2+16 (n− 1 ) 2( d−d)d−( d−d)2 } <0⟸n≤11 ∧d= d, (A.43) Φ(2) 1+ 𝛼 =1−2[n−4+𝛼(n−2)] (1+ 𝛼 )∕[1+ 𝛼 (n−1)] d b −(n−2)[3n+2+𝛼(3n−2)] (1+𝛼)∕[1+𝛼(n−1)] 2 ( d b)2 , (A.44) 𝜕( Φ(2) 1+𝛼 ) 𝜕 ( d b) <0⟸n≥ 4, (A.45) 𝜕( Φ(2) 1+𝛼 ) 𝜕𝛼 =− 2(n−2)[(n+1)(3n−4)+(2+𝛼)𝛼(n−1)(3n−2)] (1+𝛼)2∕[1+𝛼(n−1)] (d b) 2 −2(n−2)[n−3+(2+𝛼)𝛼(n−1)] ( 1 +𝛼) 2 d b <0⟸n≥3, (A.46) 𝜕( Φ(2) 1+𝛼 ) 𝜕n=− 2[3n−2+6𝛼(n2−n−1)+𝛼2(6n2−15n+8)] (1+𝛼)∕[1+𝛼(n−1)] (d b) 2 −2[1+2𝛼(n−2)+𝛼2(2n−3)] ( 1 +𝛼) d b <0⟸n≥2, 2342 M.Schopf 1 3 becomes stricter as 𝛼 increases from (A.45), the coalition size weakly decreases with 𝛼 , which proves the third bullet of the proposition. ◻ Effects ofAltruism ontheInternal Stability Condition From (11), the direct effect of altruism on the internal stability condition reads which is positive if ( m−1)dWc(m) dm +(n−m) dW f (m) dm > 0 . Using (4), (5) and (A.21) yields such that 𝜕[ Vc(m)−Vf(m−1) ] 𝜕𝛼 > 0 for m∈{2, n} . Using specification (9), it can be shown that 𝜕[ Vc(m)−Vf(m−1) ] 𝜕𝛼 > 0 for m∈[2, n] . The corresponding Maple file is available on request. From (12), the indirect effects of altruism on the internal stability condition read Using (4), (5) and (A.21) yields and (A.47) 𝜕[ Vc(m)−Vf(m−1) ] 𝜕𝛼 = [ (m−1)W c (m)+(n−m)W f (m) ] − [ (m−1)W c (m−1)+(n−m)W f (m−1) ], (A.48) ( m−1)dWc(m) dm+(n−m) dW f (m) dm =(m−1)[B�(ec)dec dm−D�de dm]+(n−m)[B�(ef) def dm−D�de dm] =D�{(m−1)[1+𝛼(n−1) 1+𝛼(m−1)mdec dm−de dm]+(n−m)[[1+𝛼(n−1)] def dm−de dm ]} = (m−1)(ef−ec)+(n−m){[2−m+𝛼(m−1)]def dm−[1+𝛼(n−1)]−1de dm} [ 1 + 𝛼 (n− 1 )] −1 [ 1 + 𝛼 (m− 1 )]∕D �, (A.49) ( 1−𝛼) dWc(m) d𝛼 +𝛼 dW(m) d𝛼 − [ (1−𝛼) dW f (m−1) d𝛼 +𝛼 dW(m−1) d𝛼 ]. (A.50) ( 1−𝛼) dW c (m) d𝛼+𝛼 dW(m) d𝛼 =(1−𝛼)[B�(ec(m)) dec(m) d𝛼−D�de(m) d𝛼] +𝛼[mB�(ec(m)) dec(m) d𝛼+(n−m)B�(ef(m)) def(m) d𝛼−nD�de(m) d𝛼 ] =[1+𝛼(n−1)]D�[m dec(m) d𝛼+𝛼(n−m) def(m) d𝛼−de(m) d𝛼] = −(1−𝛼)[1+𝛼(n−1)](n−m)D�def(m) d𝛼 >0 2343 Self‑Enforcing International Environmental Agreements and… 1 3 such that ( 1−𝛼) d Wc ( m ) d𝛼 +𝛼dW(m) d𝛼 > 0 and ( 1−𝛼)dWf(2−1) d𝛼 +𝛼dW(2−1) d𝛼 =[1+𝛼(n−1)] dW BAU i d𝛼 > 0 from Appendix A.1. Using specification (9), it can be shown that ( 1−𝛼) dW f (m−1) d𝛼 +𝛼dW(m−1) d𝛼 > 0 for m∈[2, n−2] . The corresponding Maple file is available on request. Proof ofProposition 3 Without altruistic preferences at the second stage of the game, the emissions for a given coalition size are given by substituting 𝛼=0 into (A.33), (A.34) and (A.35), and the material welfare levels for a given coalition size are given by substituting 𝛼=0 into (A.36) and (A.37). Using these results, the internal stability condition reads where where Ni∶= n−i and Mi∶= m−i . From (A.39) and (A.53), Φ(m)|𝛼=0=𝜑(m)|𝛼=0 , and from the proof of Proposition2, Φ(2)|𝛼=0>0 for n≥12 , which proves the second bullet of (A.51) ( 1−𝛼) dW f (m−1) d𝛼 +𝛼 dW(m−1) d𝛼 =(1−𝛼)[B�(ef(m−1)) def(m−1) d𝛼 −D�de(m−1) d𝛼] +𝛼[(m−1)B�(ec(m−1)) dec(m−1) d𝛼 +(n−m+1)B�(ef(m−1)) def(m−1) d𝛼 −nD�de(m−1) d𝛼 ] =[1+𝛼(n−1)]D�{[1+𝛼(n−m)] def(m−1) d𝛼 +𝛼(m−1)2 1+𝛼(m−2) dec(m−1) d𝛼 −de(m−1) d𝛼} =− (1−𝛼)[1+𝛼(n−1)]D� 1+𝛼( m −2) { [𝛼(m−2)(n−m)−1] def(m−1) d𝛼 +de(m−1) d𝛼 } , (A.52) V c(m)−Vf(m−1)=(1−𝛼)Wc(m)+𝛼W(m)− [ (1−𝛼)Wf(m−1)+𝛼W(m−1) ] = (1+2𝛼)n2(m−1)a2 2b(d b)2 [ 1+(m2+n−3m+2)d b] 2 [ 1+(m2+n−m)d b] 2𝜑(m), (A.53) 𝜑 (m) ∶= 𝛼 1+2𝛼 { 4N3+3+2[2N2 3+N3(2M2 2+4M2+13)+2M3 2+9M2 2+7M2+9] d b +[7N2 3+N3[4M3 2+22M2 2+22M2+34]+4M5 2+23M4 2+54M3 2+85M2 2 +50M2+27](d b)2}−{m−3+2[(n−m)(m−1)+(m−3)(m2+2)+4]d b +[n−m+m(m−2)][(n−m)(m+1)+m(m2−m+2)] ( d b ) 2 } , 2344 M.Schopf 1 3 the proposition. From (A.53), 𝜑(m) increases with 𝛼 1+2𝛼 and, thus, with 𝛼 , which proves the third bullet of the proposition. Furthermore, which proves the third bullet of the proposition. Furthermore, where Ψ>0 for n≥4 and m≥3 . The corresponding Maple file is available on request. 𝜑(m)≤0 for some m≥3 implies 𝜕𝜑(m) 𝜕m < 0 and, thus, 𝜑(m)<0 for all m≥m . Fur thermore, 𝜑(m)≥0 ⟺ Vc(m)−Vf(m−1)≥0 ⟺ Vf(m−1)−Vc(m)≤0 . Thus, an internally stable coalition m implies an externally unstable coalition m−1 . Consequently, there is at most one internally and externally stable coalition, which proves the first bullet of the proposition. Finally, note that (A.54) 𝜑 (n)=(n−1)4 16b2 { 50N 5 2+305N 4 2+720N 3 2+790N 2 2+358N2+17 (1+2𝛼)(n−1)4 ⋅[𝛼−4 7+ N3(5N2 3+24N3+8) 7(10N3 3+55N2 3+100N3+56)]d2 + 2(10N3 2+33N2 2+36N2+9) (1+2𝛼)(n−1)2 ⋅[𝛼−4 7+5n3+11n2−9n+33 7(10N3 2+33N2 2+36N2+9)]( d−d)d +2N2+1 1 + 2𝛼 [ 𝛼−4 7+n+9 7 ( 2 N2+ 1 )] ( d−d)2 } >0⟸𝛼≥4 7∧d≤ d , (A.55) =− n 2 (n−2)(2n 2 −3n+2) 1+2𝛼 [ 3 7−𝛼+n 2 +2n+8 7(2n2−3n+2) ]( d b )2 − 2(2N3 2+7N2 2+8N2+2) 1+2𝛼[3 7−𝛼+ N3 6+12N2 6+38N6+4 7(2N3 2+7N2 2+8N2+2)] d b −2N2+1 1 + 2𝛼 [ 3 7−𝛼+ N12 7 ( 2 N2+ 1 )] <0⟸𝛼≤3 7∧n≥12, (A.56) 𝜕𝜑 (m) 𝜕m= { 1+2(n−m+3m2−7m+3)d b+ [(n−m)2+(6m2−6m−2)(n−m)+5m4 −14m3+14m2−8m](d b)2}/{m−3+2[(m−1)(n−m)+(m−3)(m2+2) +4]d b+[n−m+m(m−2)][(m+1)(n−m)+m(m2−m+2)] ( d b ) 2 } 𝜑(m )−Ψ, (A.57) W ={b 2 −b[n(n−2)−2m(m−1)]d−m(m−1) 2 (n−m)d 2 }na 2 2b[b+(m 2 +n−m)d] 2 , 2345 Self‑Enforcing International Environmental Agreements and… 1 3 such that [ 1+𝛼(n−1)] 𝜕W 𝜕m = 𝜕V 𝜕m > 0 . ◻ Stackelberg Game Derivation of(13) The first-order condition of (3) reads Substituting (4) and rearranging yields (13). The second-order condition of (3) reads which is fulfilled if D′′′ ≤0, B′′′ ≥0 . Proof ofProposition4 From (4), (6) and (13), the equilibrium is characterised by First differentiating (A.61), (A.62) and (A.63) with respect to 𝜃 yields (A.58) 𝜕 W 𝜕 m=[(4m−2)(n−m)+(m−1) 2 ]n 2 ad 2 2[b+(m2+n−m)d]2ec+(m−1) 4 n 2 (n−2)a 2 d 3 2(n−1)2b[b+(m2+n−m)d] 3 ⋅ [ 2 ( n−m m−1) 3 +(4n−1) ( n−m m−1) 2 +(n−1) ( n−m m−1) +n2 n−2] >0, (A.59) [ 1+𝛼(m−1)] { B�(ec)− m[1+𝛼(n−1)] 1+𝛼(m−1)D� [ 1+ d(n−m)e f de c] +𝛼m(n−m) 1+𝛼(m−1)B�(ef) de f de c} = 0. (A.60) [ 1+𝛼(m−1)]{B��(ec)− m[1+𝛼(n−1)] 1+𝛼(m−1)D��[1+ d(n−m)ef dec] 2 +𝛼m(n−m) 1+𝛼(m−1)B��(ef)(def dec) 2 −m 1+𝛼(m−1){[1+𝛼(n−1)]D�−𝛼B�(ef)} (n−m)[1+𝛼(n−1)] {(n−m)[1+𝛼(n−1)]D�� −B��(ef)}2 ⋅ [ B��(ef)D��� [ 1+ d(n−m)ef de c] −D��B���(ef) def de c]} <0, (A.61) B � (e f )=[1+𝛼(n−1)]D �, (A.62) B�(ec)=[1+𝛼(n−1)]𝜃D�, (A.63) e=mec+(n−m)ef. (A.64) B ��(ef) de f d𝜃 =[1+𝛼(n−1)]D�� de d𝜃, (A.65) B ��(ec) de c d𝜃 =[1+𝛼(n−1)] [ 𝜃D�� de d𝜃 +D� ], 2346 M.Schopf 1 3 Solving for de d𝜃 , de f d𝜃 and dec d𝜃 yields Note that 𝜃 =1⟺e f =e c =e BAU i . Thus, 𝜃⋛ 1⟺e f⋛ e BAU i⋛ e c ∧e BAU ⋛e . Second differentiating Vi and Wi with respect to 𝜃 and using (4), (6), (13), (A.67), (A.68) and (A.69) yields (A.66) d e d𝜃 =m dec d𝜃 +(n−m) de f d𝜃 . (A.67) de f d𝜃=m[1+𝛼(n−1)] 2 D �� D � B��(e c )B��(e f )−[1+𝛼(n−1)][(n−m)B��(e c )+m𝜃B��(e f )]D�� > 0, (A.68) d ec d𝜃=− [1+𝛼(n−1)]{(n−m)[1+𝛼(n−1)]D �� −B �� (ef)}D � B��(e c )B��(e f )−[1+𝛼(n−1)][(n−m)B��(e c )+m𝜃B��(e f )]D�� < 0, (A.69) d e d 𝜃=m[1+𝛼(n−1)]B �� (ef)D � B��(e c )B��(e f )−[1+𝛼(n−1)][(n−m)B��(e c )+m𝜃B��(e f )]D�� < 0. (A.70) dV f d𝜃=[1+𝛼(n−m−1)]B�(ef) de f d𝜃+𝛼mB�(ec) dec d𝜃−[1+𝛼(n−1)]D�de d𝜃 =[1+𝛼(n−1)]D�{(1−𝛼) def d𝜃+𝛼[(n−m) def d𝜃+m𝜃 dec d𝜃]−de d𝜃} = 𝛼m[1+𝛼(n−1)]2{(n−m)[1+𝛼(n−1)]D�� −B��(ef)}(D�)2 B��(ec)B��(ef)−[1+𝛼(n−1)][(n−m)B��(ec)+m𝜃B��(ef)]D�� ⋅ { (1−𝛼){[1+𝛼(n−1)]D�� −B��(ef)} 𝛼{(n−m)[1+𝛼(n−1)]D�� −B��(e f )} +1−𝜃 } (A.71) = 𝛼m[1+𝛼(n−1)] 2 {(n−m)[1+𝛼(n−1)]D �� −B �� (ef)}(D � ) 2 B��(ec)B��(ef)−[1+𝛼(n−1)][(n−m)B��(ec)+m𝜃B��(ef)]D�� ⋅ { (1−𝛼){[1+𝛼(n−1)]2D�� −B��(ef)} 𝛼[1+𝛼(m−1)]{(n−m)[1+𝛼(n−1)]D�� −B��(ef)} + 𝜃−𝜃 }, (A.72) d Vc d𝜃=𝛼(n−m)B�(ef) de f d𝜃+[1+𝛼(m−1)]B�(ec)dec d𝜃−[1+𝛼(n−1)]D�de d𝜃 =[1+𝛼(n−1)]D�{(1−𝛼)𝜃dec d𝜃+𝛼[(n−m) def d𝜃+m𝜃dec d𝜃]−de d𝜃} = [1+𝛼(m−1)][1+𝛼(n−1)]2{(n−m)[1+𝛼(n−1)]D�� −B��(ef)}(D�)2( 𝜃−𝜃) B��(e c )B��(e f )−[1+𝛼(n−1)][(n−m)B��(e c )+m𝜃B��(e f )]D�� , 2347 Self‑Enforcing International Environmental Agreements and… 1 3 (A.70) [(A.73)] yields dV f d𝜃 > 0 [ dV d 𝜃 > 0 and d W d 𝜃 >0 ] for 𝜃≤1 , which implies Vf <V BAU i [ V<VBAU and W<WBAU ] for  𝜃<1 . (A.71) [(A.74)] yields dV f d𝜃 > 0 [ dV d𝜃> 0 and d W d𝜃>0 ] for 𝜃≤ 𝜃 , which implies Vf >V BAU i [ V>VBAU and W>WBAU ] for  𝜃>1 . (A.72) yields dV c d𝜃 ⋛ 0 for 𝜃⋚ 𝜃 , which implies Vc >V BAU i for  𝜃≠1 . (A.75) implies Wf⋛ W BAU i for  𝜃⋛1 . (A.76) yields dW c d𝜃 < 0 for 𝜃≥ 𝜃 , which implies Wc >W BAU i for  𝜃<1 . Finally, (A.77) implies Vf⋛ V c and Wf⋛ W c for  𝜃⋛1 . Third suppose e≤eSO . Then, the right-hand sides of (4) and (13) would be smaller than nD�(eSO) , such that the left-hand sides would have to be smaller than B� (e SO i) , implying ec ,e f <e SO i and contradicting e≤eSO . Thus, e>eSO . ◻ (A.73) d V d𝜃=[1+𝛼(n−1)] dW d𝜃=[1+𝛼(n−1)] { (n−m)B�(ef) de f d𝜃+mB�(ec) dec d𝜃−nD�de d𝜃 } =[1+𝛼(n−1)]D�{[1+𝛼(n−1)][(n−m) def d𝜃+m𝜃 dec d𝜃]−nde d𝜃} =m[1+𝛼(n−1)]3{(n−m)[1+𝛼(n−1)]D�� −B��(ef)}(D�)2 B��(ec)B��(ef)−[1+𝛼(n−1)][(n−m)B��(ec)+m𝜃B��(ef)]D�� ⋅{− (1−𝛼)(n−1)B��(ef) [1+𝛼(n−1)]{(n−m)[1+𝛼(n−1)]D�� −B��(e f )} +1−𝜃} (A.74) = m[1+𝛼(n−1)] 3 {(n−m)[1+𝛼(n−1)]D �� −B �� (ef)}(D � ) 2 B��(ec)B��(ef)−[1+𝛼(n−1)][(n−m)B��(ec)+m𝜃B��(ef)]D�� ⋅ { (1−𝛼)(n−m){[1+𝛼(n−1)]2D�� −B��(ef)} [1+𝛼(m−1)][1+𝛼(n−1)]{(n−m)[1+𝛼(n−1)]D�� −B��(ef)} + 𝜃−𝜃 }, (A.75) dW f d𝜃 =B�(ef) de f d𝜃 −D�de d𝜃 > 0, (A.76) dW c d𝜃=B�(ec) de c d𝜃−D�de d𝜃=D� { [1+𝛼(n−1)]𝜃 de c d𝜃−de d𝜃 } =− [1+𝛼(n−1)]2{(n−m)[1+𝛼(n−1)]D�� −B��(ef)}(D�)2 B��(ec)B��(ef)−[1+𝛼(n−1)][(n−m)B��(ec)+m𝜃B��(ef)]D�� ⋅ { 𝛼m(n−m){(n−m)[1+𝛼(n−1)]2D�� −B��(ef)} [1+𝛼(m−1)][1+𝛼(n−1)]{(n−m)[1+𝛼(n−1)]D�� −B��(ef)} +𝜃− 𝜃 }, (A.77) d(V f −V c ) d 𝜃 =(1−𝛼) d(W f −W c ) d 𝜃 =(1−𝛼) [ B�(ef) de f d 𝜃 −B�(ec) dec d 𝜃] > 0. 2348 M.Schopf 1 3 Proof ofProposition5 Totally differentiating (4), (13) and e=mec+(n−m)ef yields where Solving for def , dec and de yields where (A.78) B�� (e f )de f =[1+𝛼(n−1)]D �� de+(n−1)D � d𝛼 , (A.79) B�� (ec)dec=𝜆ede−𝜆e f def+𝜆mdm+𝜆𝛼d𝛼 , (A.80) de=mdec+(n−m)def+(ec−ef)dm, 𝜆e∶= 1+𝛼(n−1) 1+𝛼(m−1)mD�� 𝛼(n−m)[1+𝛼(n−1)]D �� −B �� (ef) (n−m)[1+𝛼(n−1)]D�� −B��(ef) +1+𝛼(n−1) 1+𝛼(m−1)mD�(1−𝛼)(n−m)[1+𝛼(n−1)]B��(ef) [(n−m)[1+𝛼(n−1)]D�� −B��(ef)]2D��� >0⟸D��� ≤0, 𝜆 ef∶= 1+𝛼(n−1) 1+𝛼(m−1)mD�(1−𝛼)(n−m)[1+𝛼(n−1)]D�� [(n−m)[1+𝛼(n−1)]D�� −B��(ef)]2B���(ef)⋛0⟺B��� ⋛0, 𝜆 m∶= 1+𝛼(n−1) [1+𝛼(m−1)]2D�1−𝛼 [(n−m)[1+𝛼(n−1)]D�� −B��(ef)]2{𝛼(n−m)2[1+𝛼(n−1)] 2 ⋅[D��]2−[n+𝛼(m2+n−2m)][1+𝛼(n−1)]D��B��(ef)+[B��(ef)]2]} >0, 𝜆𝛼∶= n−m [1+𝛼(m−1)]2mD�1 [(n−m)[1+𝛼(n−1)]D�� −B��(ef)]2{[1+𝛼(n−1)][1+𝛼( m −1)][1+2𝛼(n−1)+𝛼2(n−1)(m−1)][(n−m)[1+𝛼(n−1)]D�� −B��(ef)]D��(1 − 𝜃)∕(1−𝛼)+[1+𝛼(n−1)][1+𝛼(n−1)(m2−1)+𝛼(n−1)(m+2)]D��B��(ef) +[B��(e f )]2}>0⟸ 𝜃≤1. (A.81) 𝜆de f =[1+𝛼(n−1)]D��[m𝜆m−(ef−ec)B��(ec)]dm − {(n−1)[m𝜆 e −B��(e c )]D�−m[1+𝛼(n−1)]D��𝜆 𝛼 }d𝛼 , (A.82) 𝜆de c = −{[1+𝛼(n−1)]D��[(n−m)𝜆m+(ec−ef)𝜆ef]−B��(ef)[𝜆m+(ec−ef)𝜆e]}d m + {(n−1)[(n−m)𝜆e−𝜆e f ]D�− {(n−m)[1+𝛼(n−1)]D�� −B��(ef)}𝜆𝛼}d𝛼, (A.83) 𝜆de =B��(ef)[m𝜆m−(ef−ec)B��(ec)]dm − {(n−1)[m𝜆e f −(n−m)B��(ec)]D�−mB��(ef)𝜆𝛼}d𝛼 , 2355 Self‑Enforcing International Environmental Agreements and… 1 3 Thus,  m is minimal for 𝛼=0 and n=7 with  m=1.96 , and it is maximal for 𝛼=1 and n→∞ with  m=5 . m∈(m,m+2) and m≤  m then imply m∈{2, 3} for 𝛼=0 and m∈{2, 3, 4, 5, 6} for 𝛼>0 . This proves the second bullet of the proposition and the fourth bullet of the proposition, respectively. Furthermore, such that m=3 is internally unstable for 𝛼=0 , n≥26 and d≤   d . Finally, such that 𝜙(3) decreases with d/b, and increases with (d∕b)3 and (d∕b)4 , which implies that 𝜙(3) is positive if and only if d/b is greater than some unique threshold d∕b=[arg 𝜙(3)=0] . Figure6 shows that the derivative of this threshold with respect to 𝛼 is negative for n∈[7, 25] . This proves the third bullet of the proposition. ◻ (A.106) 𝜕 m 𝜕n =16𝛼(n−1) 2 −8𝛼(n−1)(n−2)−4n(n−2) [(n−2)2+4𝛼(n−1)]2⋛0 ⟺𝛼⋛n−2+ √ (n−2)(5n−2) 4(n−1) ∈[0.576, 0.809] . (A.107) 𝜙(3)| 𝛼=0 =−8d b+(n2+10n−23)(d b)2 +2(n−1)2(n−2)(d b)3 +(n−2)2(n−3)2(d b)4 =− d 16b4{(N6 26 +122N5 26 +5951N4 26 +145224N3 26 +1778248N2 26 +8867520N26 +2064240)d3+2n(n−4)(2N4 26 +174N3 26 +5587N2 26 +78148N26 +399316)(  d−d)d 2 +n2(n−4)2(5N2 26 +226N 26 +2519)(  d−d)2d+2n3(n−4)3(  d−d)3}, (A.108) 𝜙 (3)=−2𝛼−2[1+𝛼(n−1)][4+(2n−11)𝛼+(2n−6)𝛼2] d b+[1+𝛼(n−1)]2[n2+10 n −23 +(2n2−28n+68)𝛼−(3n2−24n+29)𝛼2](d b)2 +(n−2)[1+𝛼(n−1)]3 ⋅[2(n−1)2+(8n2−10n−20)𝛼+(n−3)(6n−26)𝛼2](d b)3 +(n−3)(n−2)2 ⋅[1+𝛼(n−1)]4[n−3+4(n−1)𝛼+(5n+7)𝛼2] ( d b) 4 , 2356 M.Schopf 1 3 Supplementary Information The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s1064002400885-8. Acknowledgements I would like to thank Marc Lenders, Julia Kommritz and participants of the EAERE 2023 and the VfS 2023 for helpful comments. Moreover, the comments from three anonymous reviewers significantly improved this article. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. We have not received any financial support for the conduct of the research and/or preparation of the article from any third party. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References Alger I, Weibull JW (2010) Kinship, incentives, and evolution. Am Econ Rev 100(4):1725–1758 Alger I, Weibull JW (2013) Homo Moralis-preference evolution under incomplete information and assortative matching. Econometrica 81(6):2269–2302 Fig. 6 Derivative of d∕b with respect to 𝛼 for n∈[7, 25] 2357 Self‑Enforcing International Environmental Agreements and… 1 3 Altamirano-Cabrera J-C, Finus M (2006) Permit trading and stability of international climate agreements. 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