A country that signs a climate pact can still decide to leave it. But what if the pact’s design made departure costly—not through an international court or a global police force, but by recognizing the disruption a departing member imposes on everyone else? In the model presented by Michel Grabisch, Elena Parilina, Agnieszka Rusinowska and Georges Zaccour, that cost can make a larger environmental coalition stable.
That is a striking possibility in a field often caught between two unsatisfying outcomes. In standard noncooperative models, countries have strong reasons to free-ride on others’ emissions cuts, so agreements tend to stay small. In cooperative models, countries can achieve a better collective result—but the model often assumes that everyone joins, even though no world government can compel them. This paper tries to bridge the gap: it models countries joining and leaving sequentially, then divides the coalition’s gains in a way that reflects the route taken to form it.
The proposal is not a blueprint for a treaty, and the supplied paper text does not include its numerical example or the full transition rules. It is a framework for asking a more realistic question: not only what countries would gain from cooperation, but how they might get there, who moves, and whether the resulting arrangement is acceptable to each participant. The distinction matters. The world’s climate agreements are made by sovereign governments, not by a planner able to impose the collectively best outcome.
The Science
The paper begins with a familiar climate-economics problem. A country gets economic benefits from activities that produce emissions, while the resulting pollution damages all countries. Emissions are therefore a negative externality: part of their cost falls on people beyond the country making the decision. Each government has an incentive to keep the benefits of production while hoping others will bear more of the cost of reducing pollution.
The authors represent country ’s payoff as a benefit from its own emissions minus damage from total emissions: . The benefit function is assumed to rise at a diminishing rate, while environmental damages increase with total emissions. A country acting alone chooses emissions to maximize its own payoff. Members of an agreement instead coordinate their emissions decisions to maximize the coalition’s combined payoff; countries outside it continue to make individual choices.
This setup gives a value to every possible coalition, not only to a grand agreement including all countries. The authors use what the literature calls the -characteristic function. In plain terms, it asks: if this particular set of countries cooperates while all the others act independently, what total payoff do the cooperating members receive? That makes it possible to compare the outcomes associated with different memberships.
The central innovation is to treat coalition formation as a visible sequence of individual decisions. At each step, one country can consider joining or leaving. A Markov process describes the possible transitions: the probability of the next coalition depends on the current coalition, rather than on the entire history of how it formed. The process continues until it reaches an absorbing state—a coalition that, under the model’s transition rules, no longer changes. The authors identify absorbing states as stable agreements, while also noting that the process can have periodic absorbing classes.
The word “Markov” can sound abstract, but the intuition is straightforward: the current membership is the state of the process. From that state, the model tracks which country has an incentive to alter membership and what state could follow. Different sequences can lead to the same final coalition, and different sequences can end in different coalitions. The formation path is part of the analysis rather than a detail discarded once the endpoint is known.
The second innovation concerns how to divide the coalition’s collective payoff. The Shapley value, a well-known cooperative-game solution, assigns each player an average marginal contribution across possible orders in which players could join a grand coalition. But the climate problem may not end in a grand coalition. The authors instead use a Markovian process value, a generalization that averages a country’s marginal contribution across possible coalition-formation scenarios, weighted by the probability of each scenario.
In a simple scenario, only one country changes membership at a time. The contribution assigned to that country is the change in coalition value associated with that step. The process value then averages these scenario-based contributions. If the formation process is constructed so that countries join in random order until everyone is in, this value reduces to the Shapley value. More generally, it can allocate value when the endpoint is a partial agreement and the route to it is uncertain.
There is a further safeguard: the authors retain only scenarios whose allocation is an imputation. An imputation is an allocation that gives each participant at least what it could obtain by acting alone. In the authors’ interpretation, this incorporates a form of farsightedness: a government does not evaluate a move only by its immediate effect, but considers whether the resulting path offers it an acceptable payoff.
What They Found
The paper’s headline result, as stated in the provided text, is that adding a cost to leaving an agreement can lead to a larger stable coalition. The authors motivate such a cost as compensation for harm caused to remaining members—for instance, because the agreement’s operating costs must be supported by fewer countries after one exits. They also suggest a related interpretation: remaining in the pact may encourage other countries to join, and an exit cost could reflect the value of that incentive.
This is a model result, not a claim that real-world exit penalties have already enlarged climate agreements. The supplied text gives no estimate of an optimal penalty, no comparison in tonnes of emissions or degrees of warming, and no numerical coalition sizes. It does not include the detailed example promised in the introduction, which compares countries that are more and less vulnerable to pollution. Those limits matter: the direction of an effect in a formal model is not yet evidence of its size or practical feasibility.
The model’s broader finding is institutional rather than numerical. It makes coalition stability and payoff allocation parts of one process. In the classic noncooperative approach, stability is usually checked using internal and external conditions: a member should not prefer to leave, and an outsider should not prefer to join. In cooperative approaches, by contrast, a joint outcome is chosen first and a rule such as the Shapley value or the core is used to share its rewards. The paper’s approach brings these concerns together: countries move through coalition states, and the eventual division of value depends on the scenarios that can lead there.
That matters because the identity of participants can matter as much as coalition size. A pact of ten countries is not necessarily equivalent to every other pact of ten. Their emissions, economic benefits, exposure to environmental damage, and gains from cooperation may differ. By making membership transitions individual and the allocation path-sensitive, the framework can represent asymmetries that a simple count of signatories misses.
The authors also make a structural claim about the process: absorbing classes in their Markov coalition model must be either absorbing states or periodic classes. In practical language, the process can settle on one stable membership pattern, or it can cycle among a set of patterns rather than converge to a single fixed coalition. This is a useful distinction for analysts: “stable” need not always mean that membership stops changing at one endpoint.
The supplied excerpt also situates the approach against decades of climate-coalition research. Standard models have often produced small agreements because countries can benefit from other members’ costly abatement without contributing themselves. Cooperative models can yield greater combined welfare, but their grand-coalition assumption needs some account of why a government would stay in. The new framework does not erase the free-rider problem. Instead, it asks whether sequential movement and a process-based allocation can make some partial agreements more sustainable.
Why This Changes Things
The real-world importance lies in the gap between a climate deal that is collectively desirable and one governments will actually join and honor. A global climate agreement is not simply an optimization problem. Its participants bargain under unequal risks, unequal capacity, domestic political constraints, and the knowledge that there is no sovereign authority above states with power to enforce every promise.
The paper’s emphasis on the path to agreement speaks to that political reality. Countries do not necessarily negotiate from a blank slate and instantly arrive at a final membership list. A government may join after others demonstrate commitment; another may stay out until the terms change. An exit can impose costs on partners and undermine confidence. A model that represents these steps can, in principle, help researchers ask which sequences are plausible and which allocations make a particular sequence acceptable.
The departure-cost result is especially provocative because it reframes a familiar policy tension. A penalty for leaving can look like a restriction on national sovereignty. But the authors’ rationale is that an exit may impose costs on the remaining members, who have to carry a larger share of the agreement’s burden. In that framing, an exit rule is not merely punishment: it is a way to account for spillovers created by withdrawal. Whether that argument justifies any particular treaty provision is a separate political and legal question.
The framework also offers a way to think about fairness without pretending that fairness is separable from institutions. A country’s allocation reflects not just an abstract average of its contribution across hypothetical permutations, but its marginal role along plausible formation paths. This could be valuable when coalition formation is constrained or when a partial pact is more realistic than universal participation. The paper’s process value can accommodate those cases; the classical Shapley value appears within it as a special case.
Still, a mathematical mechanism is not a negotiating strategy by itself. The model treats countries as players with payoff functions and assumes that emissions and coalition outcomes can be calculated. Actual agreements involve scientific uncertainty, changing governments, multiple policy instruments, trust, finance, technology transfer, and disputes about responsibility. A framework that focuses on membership and payoff allocation isolates one important layer; it does not settle all the others.
Nor does a larger coalition automatically mean a more effective one. The literature reviewed by the authors includes cases where adding members does not yield meaningfully lower emissions, particularly if the countries most willing to join contribute little to the environmental public good. To assess climate impact, coalition size must be connected to abatement, not treated as a success metric on its own. The excerpt provided here does not give emissions outcomes for the authors’ illustration, so no such conclusion can be drawn from it.
What’s Next
The next step is to inspect the full transition mechanism and worked example. The provided paper text ends as Section 4 begins setting out the rules for countries to join or leave, and it does not contain the later sections or figures. That means we cannot assess how the authors assign transition probabilities, what payoff parameters distinguish vulnerable from invulnerable countries, how large the departure cost must be, or which coalition emerges in their illustration. Those details would determine how much of the headline result depends on modeling choices.
For policy relevance, the model would need to be tested against a wider range of realistic conditions. How do outcomes change when countries differ in emissions, income, climate vulnerability, or capacity to pay? What if a country’s government changes, the value of cooperation shifts, or climate damages are uncertain? And how robust is a stable coalition when the cost of leaving is difficult to calculate or enforce? These are not objections unique to this paper; they are the next questions that any formal proposal for treaty design must face.
Empirical work could also help. Researchers might compare the model’s predicted entry and exit incentives with observed behavior in environmental agreements, while being careful not to treat participation alone as proof of effective action. They could then connect predicted membership paths to actual emissions commitments and implementation. Such evidence would show whether a process-based allocation captures a feature of diplomacy that static coalition models miss.
The paper offers a useful shift in perspective: climate cooperation is not only about finding the best collective outcome, or proving that a coalition is stable once formed. It is about designing a path from disagreement to a pact that countries have reason to join, stay in, and finance. If that path-sensitive view can be made robust to real-world asymmetries and political change, it could help turn the abstract promise of cooperation into agreements that are both broader and more durable.