Climate change represents one of the most pressing global challenges of our time, necessitating unprecedented cooperation among nations to mitigate its escalating impacts. As global temperatures continue to rise, the international community has sought to foster collective action through mechanisms such as the Paris Agreement. While this landmark accord has been criticized for its insufficient stringency to achieve the targeted temperature goals of 1.5°C or 2°C, it remains the largest coalition of its kind, with 194 signatory states as of 2023 1 . The challenge of facilitating large-scale climate cooperation has been a central focus in the study of international environmental agreements (IEAs). A persistent obstacle, termed "the paradox of cooperation" by Barrett (1994), arises from the inherent tension between individual and collective rationality: the greater the potential benefits of cooperation, the more difficult it becomes to sustain a large coalition. This phenomenon is theoretically substantiated by Finus et al. (2024), who demonstrate that countries are incentivized to free-ride, opting to remain outside climate coalitions despite the collective benefits of participation. To address the free-rider problem in international climate negotiations, scholars have proposed the implementation of sanctions against non-participants (Bahn et al., 2009;Breton et al., 2010). Nordhaus (2015) applies club theory to advocate for a uniform tariff mechanism to penalize non-member countries, arguing that such punitive measures can enhance cooperation levels and overall welfare without significantly disrupting trade efficiency. This approach has spurred a growing body of research exploring the role of trade sanctions in climate cooperation. For instance, Clausing and Wolfram (2023) highlight the potential of market access and border carbon adjustments (BCAs) to strengthen international climate mitigation efforts. In 2023, the European Union operationalized this concept through the Carbon Border Adjustment Mechanism (CBAM), which imposes carbon tariffs on carbon-intensive imports 2 . However, the use of sanctions remains contentious. Critics argue that such measures may destabilize the existing international trade order. Ernst et al. (2023) employ a dynamic three-region multi-sector general equilibrium model to assess the welfare implications of carbon pricing and border adjustment taxes. Their findings suggest that while BCAs mitigate carbon leakage, they do not eliminate it, nor do they provide sufficient incentives for non-participating regions to join climate coalitions. Similarly, Hagen and 1 https://unfccc.int/process-and-meetings/the-paris-agreement 2 https://taxation-customs.ec.europa.eu/carbon-border-adjustment-mechanism Schopf (2024) examine the influence of industrial lobbying on coalition formation under BCAs, concluding that lobbying can exacerbate emissions and reduce welfare unless carbon tax revenues are redistributed to firms. In light of these challenges, alternative mechanisms to promote climate cooperation have been proposed, including intracoalition payment transfers (Eyckmans and Tulkens, 2003;Bréchet et al., 2011;Fanokoa et al., 2011), solar geoengineering (McEvoy et al., 2024), and technology-sharing (Barrett, 2013;Paroussos et al., 2019;Acemoglu et al., 2023). Nordhaus (2021) further contend that a dual approach-combining trade sanctions (such as tariffs) against non-members with low-carbon technology-sharing within the climate coalition-is essential to achieve cost-effective deep decarbonization. Despite their potential, these mechanisms have yet to be fully realized within the framework of the Paris Agreement, leaving an open question for further investigation. The urgency of climate cooperation is underscored by the risk of triggering abrupt and irreversible climate tipping events, which could precipitate catastrophic ecological and economic consequences (e.g., Kriegler et al., 2009;Solomon et al., 2009;Cai et al. 2016;Cai et al., 2017;Cai and Lontzek, 2019). According to the IPCC's Fifth Assessment Report, surpassing a global temperature increase of 2°C (IPCC, 2014)-or even 1.5°C (IPCC, 2022)-above pre-industrial levels could lead to devastating outcomes, including glacial melt, sea-level rise, and biodiversity loss. While the literature extensively discusses tipping events (e.g., Lenton et al., 2008;Solomon et al., 2009;Dietz et al., 2021), fewer studies have quantified their economic impacts (e.g., Alley et al., 2003;Lenton, 2011;Weitzman, 2012;Berger et al., 2016;Heutel et al., 2016;Daniel et al., 2019). The timing and likelihood of these tipping events depend critically on the effectiveness of global emission regulation and temperature control. The threat of tipping events may influence international cooperation, as regions adjust their emission strategies in response to potential catastrophes. Scholars have integrated tipping thresholds into integrated assessment models (IAMs) to explore optimal climate policies, such as carbon taxes, under the threat of tipping events (e.g., de Zeeuw and Zemel, 2012;Lontzek et al., 2015;Cai et al., 2016;Nkuiya and Costello, 2016;van der Ploeg and de Zeeuw, 2018;Cai et al., 2017;Cai and Lontzek, 2019). However, much of this research focuses on extreme cases of full cooperation or non-cooperation, with limited attention to partially cooperative arrangements and coalition stability (Barrett, 2013;Nkuiya, 2015;van der Ploeg and de Zeeuw, 2016). In this paper, we develop a dynamic game-theoretic model to investigate the formation and stability of coalitions among heterogeneous regions in the context of climate tipping events. The model builds on the literature on international environmental agreements (IEAs), which has explored various assumptions, including homogeneity versus heterogeneity among regions (e.g., Fuentes-Albero and Rubio, 2010;Kolstad, 2010), static versus dynamic game structures (e.g., Barrett, 1999;Rubio and Ulph, 2007), and deterministic versus stochastic frameworks (e.g., Kolstad, 2007;Finus and Pintassilgo, 2011). Additionally, the robustness of these models has been examined through both non-parametric and parametric approaches (e.g., Karp and Simon, 2013;Lessmann et al., 2015). By deriving Markov strategies for emissions among member and non-member regions, our heterogeneous dynamic model provides a more nuanced understanding of how coalitions evolve over time and in response to rising temperatures, offering both theoretical and practical insights. Our model extends the frontier of IEA research in several aspects: First, we focus on the conditions necessary to establish and maintain a grand coalition among heterogeneous regions in a dynamic setting. To cope with free-riding behavior, we introduce a technology-sharing mechanism that enhances the productivity of member regions, rather than relying on sanctions against non-members. This approach draws on the proposal by Paroussos et al. (2019) Second, in line with the overarching goal of climate cooperation to prevent tipping events, we model climate tipping events as stochastic processes dependent on temperature rise.
Actually, the impacts of climate shocks have been intensively studied, including using a Poisson distribution to uncover the influence of climate shocks on permanent damage to capital stock (Pindyck and Wang, 2013); introducing tipping events through an indicator function that triggers sudden increases in climate losses when temperature thresholds are exceeded (Barnett, 2023); and using a discrete-time Markov process to characterize the effect of tipping events on damages to production (Zhao et al., 2023). Drawing on these frameworks, we adopt a hazard rate approach (following Cai et al., 2016, Cai et al., 2017, and Cai and Lontzek, 2019) to incorporate the threat of tipping into regions' benefit functions dynamically. Our model also considers the possibility of multiple, sequentially occurring tipping events. Third, given the complexity of the model, we propose a nested numerical and analytical solution method. Nordhaus (2021) recognized that estimating coalition equilibria in a dy-namic framework is computationally prohibitive-if not intractable-and thus proposed an alternative approach to analyze how supportive policies influence individual countries' decisions. In contrast, we numerically solve this problem with endogenous emission control rates under non-cooperative dynamic Nash equilibrium. At each time period and temperature level, we distinguish between states where tipping has occurred and those where it has not. After a tipping event, the model is solved analytically with the derived Hamilton-Jacobi-Bellman (HJB) equations for member and non-member regions. We prove the existence and uniqueness of these analytical solutions. For periods before a tipping event, we employ Chebyshev polynomial approximation to solve the model numerically. Coalition stability is determined by internal and external stability conditions. By simplifying the model to a homogeneous framework and examining extreme cases, we theoretically demonstrate the tendency of coalitions to shrink as temperatures rise, and we also establish the equivalence between technology-sharing and sanctions in promoting coalition participation when the mechanisms are applied at consistent levels. Using parameters calibrated from the RICE model (Nordhaus, 2010), our numerical results yield:

Coalition shrinks over time with rising temperatures: Our findings confirm "the paradox of cooperation" from a dynamic perspective, revealing that coalitions tend to shrink as global warming rise. Even when a large coalition forms initially, its stability diminishes over rising temperature. This occurs because the benefits of a grand coalition increase, but the gains from free-riding for non-member regions grow even more rapidly, rendering the coalition increasingly shrink. The role of climate tipping events: The threat of climate tipping events reduces the size of stable coalitions compared to scenarios where tipping is not considered. However, coalitions formed after a tipping event are larger than those formed in its absence. This suggests that tipping events act as shocks, temporarily incentivizing cooperation. Nevertheless, this cooperative response is short-lived; if regions fail to implement effective temperature control measures, the coalition will continue to erode as temperatures rise. Overcoming the "Paradox of Cooperation": While technology-sharing and sanctions are equally effective in promoting coalition formation in theory, coalitions supported by technology-sharing are consistently larger than those relying on sanctions in numerical analysis, especially involving multiple tipping events. Given that technology-sharing generates greater overall benefits compared to sanctions, we propose a dynamic technologysharing path. This path represents the minimum rate of technology transfer required to maintain a stable grand coalition, reflecting the coalition's resistance to free-riding while ensuring that global temperature rise remains within the lowest possible bounds. The remainder of the paper is organized as follows. Section 2 presents the model and theoretical proposition results, Section 3 describes the numerical methods and parametriza-tion for cases, and Section 4 discusses the results from the aspects of dynamics of coalition formation, technology-sharing v.s. sanction, minimum technology-sharing path, and possible sequence of joining a coalition. Section 5 concludes this study, and the Appendix provides detailed derivations and solution processes. To analyze the long-term dynamics of climate coalition formation, we adopt a continuoustime model with n heterogeneous regions. Regions are divided into coalition members (i) and non-members (j), and the coalition among member regions is expressed as S = {1...i...m} when there is m members within it. The model incorporates the following assumptions:

Emissions and Production Benefits: Each region derives economic benefits from production, which is linked to its emissions level q (t) at time t. The benefit function is assumed to be quadratic in emissions:

where α, β and ϵ are region-specific benefit parameters. For simplicity, we exclude benefits from non-emissions growth (e.g., clean energy development or decoupling of GDP from CO 2 emissions), as the primary focus of climate cooperation is to address the negative externalities associated with greenhouse gas emissions. Non-emissions growth is treated as exogenous and does not influence coalition dynamics. Climate Damages: Global average temperature anomaly T (t) above preindustrial levels imposes two types of costs on regions:

Continuous Losses: These are increasing and convex in temperature, reflecting the escalating economic and ecological impacts of climate change. The continuous loss function is specified as (e.g., McGinty, 2007;Nordhaus, 2014;Barnett, 2023):

where ρ and η are region-specific loss parameters. Tipping Event Losses: If temperature exceeds a critical threshold, regions incur an additional, fixed loss L. The loss function in the presence of a tipping event becomes:

Temperature-Emissions Relationship: Temperature rise is modeled as a linear function of cumulative global emissions, following the transient climate response to emissions (TCRE) framework (Matthews et al., 2009):

where λ is the TCRE parameter, T 0 is set as an initial temperature level. The rate of temperature change is thus:

Hazard Rate of Tipping Events: The likelihood of a climate tipping event is modeled as a hazard rate H (T ), which increases with temperature rise. Following Cai et al. (2016), Cai et al. (2017), andCai andLontzek (2019), we specify:

where χ is the hazard rate parameter and T is the temperature threshold below which the hazard rate is zero. This hazard rate is incorporated into the Hamilton-Jacobi-Bellman (HJB) equations to capture the risk of tipping in regions' value functions.
Coalition Formation and technology-sharing: Member regions collaborate to maximize the collective net benefits of the coalition, while non-members optimize their individual net benefits. At any time t, regions decide whether to join, leave, or remain inside/outside the coalition. For n regions, there are 2 n -n-1 possible coalition structures, as a nonempty coalition must include at least two regions. To reflect the benefits of cooperation, we introduce a technology-sharing effect within the coalition. Drawing on Paroussos et al. (2019), we assume that coalition membership enhances production efficiency through technology diffusion and low-cost climate financing. This effect is captured by a technology-sharing parameter τ , which scales the benefits of member regions:

where τ is the technology-sharing rate achievable under a grand coalition, and t 0 is a reference year used to calculate the initial benefits. Consequently, regions decide their emissions to balance the benefits of emissionsassociated production against the costss of climate-related damages. The net emissionsassociated benefits for a member region i before and after a tipping event are:

respectively. For a non-member region j, the corresponding net benefits are:

θ j (t) -D j (t) and θ j (t) -D j (t) -L j . In this subsection, we analyze the strategic decisions of regions under the threat of a single climate tipping event. At any time t, the climate system can be in one of two states:

State 1 : The tipping event has occurred. State 2 : The tipping event has not yet occurred. For a given coalition S, member regions choose emissions i∈S q i (t), while non-member region j choose its emissions q j (t) individually. We examine the optimal emissions strategies and value functions for regions in each state separately. In State 1, where the tipping event has already occurred, member and non-member regions simultaneously solve their respective optimization problems to maximize their net emissions-associated benefits. The collective net benefits for member regions within the coalition S are given by:

while the net benefits for each non-member region j ̸ ∈ S are:

where, r denotes the discount rate, V 1 and W 1 j represent the value functions of the coalition and each non-member region j, respectively. Note that these value functions implicitly depend on the coalition S, though we omit this dependence for notational simplicity. The following proposition characterizes the value functions and optimal strategies for both coalition members and non-member regions in State 1, establishing key analytical properties of the equilibrium solution. Proposition 1. The value functions V 1 and W 1 j are quadratic in T , and the optimal emissions strategies q 1 * i and q 1 * j are linear in T . Proof. The proof of Proposition 1 is provided in Appendix A.1. Appendix A.1 provides a detailed derivation of the coefficients of these quadratic value functions for both member and non-member regions. Besides, we prove the existence and uniqueness of solutions for the quadratic coefficients in the special case (single nonmember region) and apply numerical simulations to confirm the unique real solution exists which corresponds to positive discriminant case for general cases in Appendix A.2. In State 2, the tipping event has not yet occurred, and the timing of the tipping event, denoted as t 1 , is stochastic. To account for this uncertainty, we model the decisionmaking process of member and non-member regions using a stochastic dynamic game framework. Regions maximize their expected collective net emissions-associated benefits, conditional on the stochastic nature of the tipping event. Let V 2 and W 2 j denote the value functions for member and non-member regions, respectively. The optimization problems for member regions (i ∈ S) and non-member regions (j / ∈ S) are given by:

To analyze the game, we adopt the Markov strategy, as the Markov perfect equilibrium ensures dynamic consistency and sub-game perfection (Zhang and Hennlock, 2018). Proposition 2 formalizes the equilibrium properties of value functions and optimal strategies for coalition members and non-member regions in State 2. Proposition 2. Before the occurrence of tipping, when T ≤ T , the value functions V 2 and W 2 j are quadratic in T , and the optimal emissions strategies q 2 * i and q 2 * j are linear in T . When T > T , the value functions V 2 and W 2 j are quadratic or cubic in T , and the optimal emissions strategies q 2 * i and q 2 * j are quadratic or linear in T . Proof. The proof of Proposition 2 is provided in Appendix A.3. Consequently, closed-form solutions cannot be derived analytically. In Section 3, we present a numerical solution process to determine the equilibrium strategies for member and non-member regions prior to the climate tipping event. This approach employs Chebyshev polynomial approximation methods to solve the system of HJB equations (See the Appendix A.4), ensuring computational tractability while maintaining the rigor of the theoretical framework. Two Sequential tipping events: We extend our analysis to incorporate two sequential climate tipping events, where the occurrence of the second tipping event is conditional on the first. Let T 1 and T 2 denote the respective temperature thresholds at which the hazard rates for the first and second tipping events become non-zero. The hazard functions are specified as:

where χ 1 and χ 2 represent the hazard rate parameters for each tipping event. The associated climate damages are denoted by L 1 and L 2 for the first and second tipping events respectively.
State Characterization: At any time t, the system may occupy one of three distinct states:

State 1 : Both tipping events have occurred (t ≥ t 2 )

State 2 : First tipping has occurred but not the second (t 1 ≤ t < t 2 ) State 3 : No tipping events have occurred (t < t 1 ) where t 1 and t 2 represent the stochastic realization times of the first and second tipping events, with t 1 < t 2 . Analytical and Numerical Solution Approach: For State 1 (both tipping events realized), the value functions maintain quadratic forms in temperature T , analogous to the single tipping event case. The solution methodology presented in Appendix A.1 remains applicable, with the complete analytical derivation provided in Supplement S2.1. For States 2 and 3 (representing intermediate stages of tipping), the value functions lose their quadratic structure, we adapt our numerical framework from the single tipping event case through recursive solution methods that account for state transitions. The complete numerical algorithm, including convergence criteria and implementation details, appears in Supplements S2.2 and S2.3. This approach maintains theoretical consistency while providing computational tractability. The framework naturally generalizes to cases with more than two tipping events through additional state variables tracking tipping event occurrences. However, we focus on the two-tipping-point case since it captures the essential economic dynamics of sequential climate shocks. Member regions within coalition S act collectively to maximize their collective net emissions-associated benefits. For a coalition comprising m heterogeneous regions (m ≤ n, where m = |S|), we first derive the optimal emission strategy q * i (T ) for each member region i ∈ S. The individual net benefits are then determined through intra-coalition payment transfers, for which we analyze two distinct allocation mechanisms: γ-core allocation: Building on cooperative game theory concepts (Chander, 2007), the γ-core allocation satisfies:

where

denotes region i's value after leaving coalition S; and V S\{i} is the coalition value without region i. This mechanism ensures each member receives at least its outside option while proportionally sharing the coalition's surplus. Shapley value allocation (Shapley, 1953): The Shapley value allocation distributes benefits according to:

where S(i) comprises all subsets containing region i of S, and |S s | denotes coalition size. Following the standard approach in IEA literature, we define stability through two conditions:

, ∀j / ∈ S These conditions ensure that: No member has incentive to leave (internal stability);

No non-member has incentive to join (external stability). Our framework captures the endogenous evolution of stable coalitions through temperaturedependent value functions, state-contingent payoff structures, and continuous re-evaluation of stability conditions. We compute all possible stable coalitions at each temperature level to track coalition transitions across climate states. While some studies suggest multiple agreements may enhance cooperation (e.g., Asheim et al., 2006;Hannam et al., 2017), we maintain the standard single-coalition framework due to empirical prevalence of unitary agreements (e.g., Paris Accord), and theoretical ambiguities in multiple-equilibrium settings (Hagen and Eisenack, 2019). The challenge of sustaining full participation in climate coalitions has prompted extensive debate between two policy approaches: 1) External sanctions (Nordhaus, 2015), which advocates imposing trade penalties on non-participants to create negative incentives for free-riding, however, it raises risk in trade distortions and retaliation. 2) Internal incentives (Paroussos et al., 2019), which advocates technology-sharing arrangements to create positive inducements for participation, thus enhances coalition productivity. Our theoretical analysis on homogeneous-region (Appendix B) reveals an equivalence: when technology-sharing (τ ) and sanction levels are calibrated to provide equivalent marginal benefits, their effects on coalition participation are identical. Moreover, we will numerically show that, technology-sharing consistently supports larger coalitions than sanctions. Meanwhile, also through homogeneous-region analysis (Appendix C), we identify two countervailing effects: 1) Temperature Rise Effect, which reduces participation incentives since free-riding benefits grow faster than coalition benefits; and 2) Tipping Event Effect, which temporarily increases cooperation and acts as coordination-facilitating shock. The net effect shows coalition fragility increasing with temperature, numerical simulations with heterogeneous regions confirm these dynamics. For a grand coalition S with n members, stability requires:

where the latter represents the aggregate free-riding temptation. We define the stability gap as: Ψ (τ , T ) = V S (τ , T ) -n i=1 W i, S\{i} (T ), if Ψ(0, T ) ≥ 0, then τ = 0 (no technology-sharing is required). Otherwise, there can find a minimum technology-sharing rate τ that stabilizes the grand coalition at temperature T , satisfying (Ψ(τ , T ) = 0). In our numerical illustration under γ-core allocation, we calculate τ (T ) iteratively using a sufficiently small step size at discrete temperature intervals. This approach identifies a dynamic technology-sharing pathway that evolves over time with rising temperatures. The model's forward-looking dynamics are closed through the temperature-emission feedback loop. At each time t, we compute optimal emissions (q * i,t , q * j,t ) and values (V t , W j,t ), and determines endogenous coalition formation (S t ). Then we update temperature trajectories T t+1 via climate dynamics and repeat the above calculation until a given terminal time is reached (t ≥ t terminal ). 3 Numerical solution and parametrization

For single tipping event cases, the value functions V 1 (T ) and W 1 j (T ) at climate tipping state 1 (post-tipping) admit analytical solutions (see Appendix A.1). These incorporate tipping damages. The value functions V 2 (T ) and W 2 j (T ) at state 2 (pre-tipping) require numerical treatment, when T > T . For T ≤ T , the hazard rate H(T ) = 0, allowing analytical solutions (Appendix A.1). These mirror the derivation for state 1 but exclude tipping damages. For T > T , the HJB equations are solved numerically via Chebyshev polynomial approximation (Appendix A.4). The solution for two tipping events case follows the same methodology, with additional states accounted for in the approximation. The Chebyshev regression algorithm is also applied in Supplements S2.2 and S2.3.
For foundational references on the method, see Judd (1998) and Cai and Judd (2013). Our model calibration draws on the regional economic and climate projections from the RICE-2010 model (Nordhaus, 2010), which comprises 12 aggregated regions (detailed in Supplement S4). Regional GDP (converted to trillion PPP USD) and emissions (gigatons CO 2 ) form the basis for estimating our benefit function parameters. We employ ordinary least squares to calibrate the continuous climate damage function coefficients, with technology-sharing effects benchmarked to 2005 values. The benefit function parameters (α, β) are derived through a three-step procedure: 1) establishing an α-β relationship using 2015 GDP-emissions data; 2) estimating β via firstorder conditions assuming cooperative optimization in RICE; 3) estimating α through least squares regression. This approach ensures consistency between our emissionsassociated benefits and RICE's marginal damage valuations. Key parameters include a 2.5% discount rate (r), 0.0035 hazard rate (χ), and 0.0021 TCRE coefficient (λ) (Matthews et al., 2009;Nordhaus, 2017;Cai et al., 2025). Tipping losses are modeled as percentage output reductions in 2100 RICE projections. Following the literature, we consider fixed membership (Petrosjan and Zaccour, 2003) coalition instead of flexible membership (Rubio and Ulph, 2007) since the model is dynamically forward updating through the temperature-emission feedback loop along with time. The cases setting is presented in Table 1. To determine stable coalition compositions, we integrate regional value functions with internal and external stability conditions. Since multiple stable coalitions may emerge within a state during numerical calculations, we select the configuration that maximizes overall benefits among regions, having verified through robustness checks (including random selection of the largest viable coalition) that alternative selection criteria do not materially affect our core conclusions. Our analysis tracks stable coalition formations alongside temperature increases from 2020 to 2100 across all cases, revealing dynamic patterns of coalition emergence and dissolution as temperatures rise. Prior to presenting these numerical results, we evaluate the two payment transfer mechanisms (detailed in Supplement S3), finding that while both Shapley and γ-core allocations are theoretically valid, the γ-core approach proves more effective at sustaining cooperation in practice. Consequently, we focus our reported results exclusively on γ-core implementations. To ensure robustness and demonstrate that our findings extend beyond single-tipping-point cases, we have systematically examined cases involving two sequential tipping events, which confirm the generalizability of our core conclusions. Consistent with prior findings, regional emissions-associated benefits decline monotonically with rising temperatures (Figure 2b), highlighting the enduring conflict between collective action and free-riding even under escalating climate risks. Our comparative analysis reveals that the presence of tipping events influences regional participation decisions in climate coalitions (Figure 3). When accounting for tipping threats, stable coalitions formed prior to tipping events are consistently the same size or smaller than those in cases without tipping considerations-a trend that intensifies with multiple potential tipping events. Conversely, coalitions established after tipping events are notably larger than their non-tipping counterparts. These findings align with Barrett (2013), who argue that when tipping thresholds are uncertain, regions face heightened incentives to free-ride, thereby impeding the formation of large-scale cooperation. The results underscore how the mere risk of catastrophic climate damages-even before materialization-can destabilize collective action by reinforcing free-riding over collective benefits. Our analysis of net emissions-associated benefits reveals a persistent benefit gap between full cooperation and non-cooperation cases, with cooperative actions consistently yielding higher collective benefits across all tested parameters (technology-sharing rates, tipping event frequencies, and tipping losses). While tipping events temporarily attenuate this differential (Figures 4), the cooperative advantage rebounds at post-tipping period and amplifies with rising temperatures-a pattern that starkly contrasts with the observed coalition shrink dynamics. This divergence underscores the fundamental conflict between individual and collective rationality: regions recognize the superior collective payoff from full cooperation yet increasingly defect to pursue larger individual gains through free-riding as climatic conditions deteriorate. Our findings dynamically validate Barrett (1994)'s "paradox of cooperation," which posits that the greater the benefit from cooperation, the more difficult it becomes to form a large coalition. Figure 4: Difference in net emissions-associated benefits between full cooperation and non-cooperation under various technology-sharing rates and tipping loss rates Note: a. Single tipping event cases; b. Two tipping events cases. Red vertical lines mark tipping event occurrences. While scholars have advocated for sanctions as a mechanism to overcome the cooperation paradox (Bahn et al., 2009;Breton et al., 2010;Nordhaus, 2015), our analysis of heterogeneous regions reveals the superior efficacy of technology-sharing. Our theoretical model demonstrates equivalence between sanctions and technology-sharing under homogeneous assumptions, and empirical results show advantages for technology-sharing in heterogeneous settings. With a single tipping event (Figure 5a), technology-sharing sustains grand coalitions longer than equivalent-level sanctions while generating higher net emissions-associated benefits (Figure 5b). This pattern persists in two-tipping case, where technology-sharing maintains larger stable coalitions prior to the second tipping event and delivers greater collective benefits (Figure 6a andb). The inherent advantages of technology-sharing stem from its dual capacity to: (1) enhance internal coalition stability through productive gains rather than punitive measures, and (2) avoid the trade conflicts associated with sanctions (Ernst et al., 2023;Hagen and Schopf, 2024). By creating positive-sum incentives rather than zero-sum penalties, technology-sharing emerges as both a more rational policy tool for coalition expansion and a more efficient mechanism for maximizing collective benefits. These findings suggest that an international climate agreements should prioritize cooperative technology frameworks over punitive trade measures to achieve lasting, large-scale participation. Our analysis proposes a novel solution to "the cooperation paradox" by identifying a dynamic path of minimum technology-sharing rates (τ ) required to sustain a stable grand coalition. Figure 7 demonstrates that while higher fixed technology-sharing rates slow global temperature rise, our flexible τ path achieves preferred results by adapting to changing climate conditions, thus it maintains the slowest temperature increase. The minimum technology-sharing rate drops sharply upon a tipping event because the elimination of associated tipping risks narrows the stability gap for the grand coalition, reducing the free-riding incentives for non-members.
The τ path thus serves both as a policy tool and a diagnostic metric, quantifying the growing challenge of preserving cooperation as climate impacts intensify. Figure 7: Minimum technology-sharing rates (τ ) for grand coalition stability and corresponding temperature trajectories Note: a. Single tipping event cases (4% loss rate in 2050); b. Two tipping events cases (2% and 4% loss rates in 2030 and 2050, respectively). Red vertical lines mark tipping event occurrences. Our analysis of coalition formation across all cases reveals consistent patterns in regional participation. By aggregating coalition membership at each possible coalition size (across various temperature levels, tipping losses, and technology-sharing rates), we identify a statistical hierarchy in regional participation likelihoods (Figure 8). The results demonstrate that: 1) dyadic coalitions invariably form between the US and China; 2) triadic coalitions most frequently include these two nations plus OHI (Other High Income countries); and 3) subsequent expansion follows a predictable sequence: EU → Japan → Eurasia → Russia → LatAm → MidEast → India → Africa → OthAsia. This participation ordering persists regardless of exogenous parameter variations, suggesting that coalition formation depends primarily on intrinsic regional characteristics rather than external conditions. These empirically derived participation patterns provide valuable insights for predicting probable coalition formation pathways during international bargaining processes. We propose technology-sharing as a superior alternative to sanction measures for sustaining grand coalitions. By calculating the minimum required technology-sharing rates, we identify a dynamic solution where τ increases with temperature to offset growing freeriding incentives, yet discontinuously decreases at post-tipping period due to renewed cooperation impulses. This pathway serves dual purposes: quantitatively measuring the free-rider challenge while providing actionable policy guidance. Unlike sanctions that reduce overall benefits (Ernst et al., 2023), technology-sharing expands the cooperative benefit pool through productivity gains, creating positive-sum incentives that align individual and collective rationality. Our regional participation patterns -remarkably consistent across cases -further suggest that coalition building should prioritize nations with the highest inherent cooperation propensity (e.g., US and China initially), leveraging their participation as a catalyst to stimulate broader involvement. The persistent gap between theoretical cooperation potential and actual negotiation outcomes -evident in the Paris Agreement's struggles -underscores the need for mechanisms that structurally reward participation. Our results suggest that technology-shared coalitions could break this impasse by creating tangible economic benefits for members that outweigh free-riding gains, and establishing predictable expansion pathways based on empirical participation hierarchies. While challenges remain in implementing such systems, particularly in intellectual property rights and technology transfer governance, this approach offers a scientifically-grounded framework for achieving the global climate governance scale required to avoid catastrophic warming trajectories. Future research should explore institutional designs that operationalize these insights within real-world negotiation contexts. A.1 Proof of Proposition 1

We derive the optimal strategies for regions within coalition S by maximizing their collective net emissions-associated benefit:

Where temperature evolution follows:

With derived Hamilton-Jacobi-Bellman (HJB) equation through standard dynamic programming method, the value function satisfies:

where dV 1 (T ) dT is the first-order derivative of the value function V 1 (T ) with respect to the state variable T . The first-order conditions yield optimal emissions for coalition members:

The derivation of the Markov strategy of region j outside the coalition is similar. By substituting the optimal strategies for member and non-member regions into the Hamilton-Jacobi-Bellman (HJB) equations, we obtain:

for coalition members, and for each non-member region j / ∈ S:

dT is the first derivative of U 1 (T ) with respect to T . Given the linear-quadratic structure of the game, we postulate quadratic forms for the value functions:

where v 0 , v 1 , v 2 , w j,0 , w j,1 , and w j,2 are undetermined coefficients3 . Substituting these into the HJB equations yields three coupled systems for the coefficients:

For coalition members:

For non-members (∀j / ∈ S):

The solved coefficients are presented in Table A.1. Member regions Non-member regions

, and u 2 is the solution to the following equation:

For a coalition S with m = |S| member regions, by substituting the solved coefficients into the optimal emission strategies for both member and non-member regions, we obtain the complete equilibrium solutions, including the explicit form of the value functions. We establish the existence and uniqueness of solutions for the quadratic coefficients v 2 and w j,2 through the following proof:

Special Case Analysis (Single Non-Member Region):

For the simplified case with one non-member region j, the system reduces to six equations. The quadratic coefficients satisfy:

where

Existence of Negative Roots: For F (v 2 ), F (0) =

12)

It has at most three real roots (cubic equation), and always non-negative for v 2 ≤ 0 (analysis of coefficients). Therefore F (v 2 ) is strictly increasing on (-∞, 0]. Combined with F (0) > 0, guarantees unique negative root. The uniqueness proof of w j,2 is similar. General Case (Multiple Non-Members):

For systems with multiple non-members, analytical proof becomes intractable. The

Consider member regions maximizing their expected net emissions-associated benefit collectively:

The state variable evolves as dT dt = λ i∈S q i + j / ∈S q j , T (0) = T 0 . And with hazard rate H(T ) = χ max(0, T -T ), The value function V 2 (T ) satisfies:

The first-order condition yields:

The derivation of the Markov strategy of region j outside the coalition is similar. From the definition of H(T ), when T < T , there is H(T ) = 0. The HJB equations in state 1 will be identical to HJB equations in state 2 in form. Thus, the value functions V 2 and W 2 j can be conform to a quadratic form.
For temperature levels T ≥ T , substituting the optimal emission strategies q 2 * i and q 2 * j into the HJB equations, we can obtain:

where

It is easy to verify that when the value functions V 2 and W 2 j are quadratic polynomials or cubic polynomials, the highest terms on both sides of these two equations are the same. This means that the value functions V 2 and W 2 j can be quadratic polynomials or cubic polynomials. We employ Chebyshev polynomial approximation on T , T :

Chebyshev polynomials adapted to T , T for k = 0, 1, 2, ... with degree-K.

We

, and the coefficients a k can be computed:

See Judd (1998) and Cai and Judd (2013) for more details. We then assume:

where ϕ k (T ) are Chebyshev basis functions with coefficients a k and b k,j for V 2 (T ) and W 2 j (T ), respectively. We solve

For each non-member region (j / ∈ S) and Chebyshev node (d = 1, ..., K + 1). We use the analytical solutions V 2 (T ) and W 2 j (T ) at T = T as boundary conditions, where ϕ k T d denotes the first derivative of ϕ k (T d ) at T d . The resulting system comprises (m + 1)(1+n-m) unknown coefficients (a k and b k,j ) and (K + 1) (1 + n -m) equations, which we solve numerically using MATLAB's fsolve. Regarding the selection of nodes for the solution, we compared both K = 4 and K = 6 options and found virtually no difference in computational efficiency between the two. Therefore, we opted for K = 4. For details, please refer to the Supplement S1. For homogeneous regions with technology-sharing rate τ , we analyze the benefit differential between full cooperation and non-cooperation after the tipping event:

Member regions:

Non member regions: w 0 + w 1 T + 1 2 w 2 T 2 . let B ≡ nα β , we have:

where

, and u 2 is determined by the following equation:

Considering asymptotic behavior (n → ∞), it is easy to deduce that λ

n, then we have:

Non member value:

And participation incentive:

For sanction rate τ 2 applied to non-members, member regions (with size m) and nonmember regions (n -m) make emissions decisions after the tipping has occurred, and solve the following collective net emissions-associated benefit functions, respectively:

Member regions:

Non member regions:

The difference in benefit with sanctions are:

Member regions:

Non-member regions: w 3 + w 4 T + 1 2 w 5 T 2 Let B ≡ nα β , we have

, and u 4 is determined by the following equation:

Considering asymptotic behavior (m → n → ∞), it is easy to deduce that λ 2 u 4 -1 2 r = -mλ (1-τ 2 )α 2 2β

+ η ρ -L r ; And participation incentive:

with rising temperatures. Changes in emissions for a region inside versus outside a coalition: Assume τ = 0. Let q (T, m) and q (T, m -1) denote emissions for a region inside and outside a coalition of size m, respectively. The difference is:

Taking derivative with respect to T : ,m)   dT > 0, emission reduction pressure grows faster for outsiders, incentivizing joining. If dq ψ (T,m) dT < 0, pressure grows faster for insiders, incentivizing leaving. Let σ m = λ 2 u 2 (m) -1 2 r. Then:

< 0, we conclude dq ψ (T,m) dT < 0. Thus, as temperatures rise, the reduction pressure on coalition members increases more rapidly, making regions more likely to leave. We analyze how the occurrence of tipping events affects the incentive for regions to join a coalition. Consider a coalition of size m in a homogeneous setting. Define the net coalition benefit difference before and after tipping as:

Post-tipping gain (A.43) If ψ tp > 0, the net benefit of joining is higher before tipping. Regions are more inclined to join before tipping. If ψ tp < 0, it indicates that the region's benefit within the coalition is smaller before tipping, making it less inclined to join the coalition before tipping. Due to analytical complexity, we illustrate this numerically using homogeneous parameters, in which tipping losses L 1 = 2%, L 2 = 4%,technology-sharing rate τ = 5%. It can be observed that, for small coalitions (m = 2, 3), ψ tp > 0, regions prefer joining before tipping. For larger coalitions (m ≥ 4) and across temperature rises, ψ tp < 0, regions are more likely to join after tipping. Thus except for very small coalitions, the occurrence of tipping events incentivizes coalition expansion, as regions gain greater benefits from joining post-tipping. After both tipping events have occurred (State 1), coalition members and non-members solve their respective optimization problems:

for each j / ∈ S. The HJB equations yield optimal strategies:

3)

The quadratic value functions take the form which is similar to that of after the climate tipping with only one tipping (Appendix A1), with coefficients determined in Table S.1. Member regions Non-member regions

, and u 2 is the solution to equation(for each region j / ∈ S, which is not in the coalition):

This solution extends the single-tipping case while incorporating tipping damages L 1 and L 2 . For State 2 (first tipping occurred, second pending), regions solve:

The value functions satisfy:

where dV 2 (T ) dT and dW 2 j (T ) dT are the first-order derivatives of value functions V 2 (T ) and W 2 j (T ) on T , respectively.
Yielding optimal strategies:

Using Chebyshev approximation on T ∈ [T , T ]:

(S.9)

We solve at Chebyshev nodes T d :

In State 3 (no tipping events occurred), regions solve sequential optimization problems accounting for potential future tipping events:

(S.12)

for each j ̸ ∈ S.

The value functions satisfy:

rV 3 (T ) = max

Taking the optimal emission strategies of member regions and non-member regions into HJB functions, and using Chebyshev approximation, We solve: β j . d = 1, ..., K + 1, V 3 (T ) and W 3 j (T ) are given by the analytical solution for T = T . V 2 (T ) and W 2 j (T ) can be obtained from the numerical solutions in Supplement S.2.2. We evaluate two payment transfer mechanisms to sustain cooperation: Shapley Allocation, benefits are distributed based on each member's marginal contribution to the coalition; γ-core allocation, benefits are allocated according to the proportion of benefits each member would retain if they leave the coalition. While both mechanisms are logically justified, our numerical simulations demonstrate that the γ-core allocation is more effective in fostering cooperation and maximizing collective benefits. As a result, the main text focuses on the γ-core allocation for comparative analysis. To simplify our discussion, these undetermined coefficients will be used repeatedly in the following individual sections