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Formation of international environmental agreements and payoff allocation

Michel Grabisch, Elena Parilina, Agnieszka Rusinowska, Georges Zaccour

arXiv:2610.35443v1physics.soc-phcs.GT

TL;DR

Transboundary pollution and free riding make it difficult to secure countries’ emissions-reduction efforts without supranational enforcement. This paper combines cooperative and noncooperative approaches in a Markovian coalition-formation process with process-based payoff allocation. Adding a cost of leaving can yield larger stable coalitions, potentially including the grand coalition.

  • Problem

    Transboundary pollution makes countries’ payoffs interdependent, while the absence of a supranational authority makes instruments to internalize emissions costs nearly impossible to implement in practice.

    Source: §1 · pp. 1–2

  • Method

    The paper combines internal and external stability with a Markovian process of sequential coalition formation and allocates benefits using process-values across scenarios.

    Source: §1.2 · p. 5

  • Results

    Introducing a cost of leaving can yield stable coalitions with more than two members and may yield the grand coalition.

    Source: §6 · p. 25

  • Takeaways & Limitations

    The process-value accounts for all possible formation scenarios whether the final IEA is grand or partial.

    Source: §1.2 · p. 5

    Departures from a singleton coalition carry no cost, and a country may re-enter a scenario, so the first result’s process-value sign is not necessarily negative.

    Source: §4.1 · p. 8, §4.2 · p. 12

Abstract

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Dealing with climate change requires that all countries engage in costly efforts to reduce their emissions. Reaching this objective has so far been elusive because it is in the best interest of each country to let the others do the effort and benefit itself from a better environment. The presence of negative externalities and strategic behavior have made game theory a natural paradigm to design an international environmental agreement (IEA) that codifies what countries should do. Considering that countries are sovereign and no supranational entity can impose an agreement, a stream of literature adopted a noncooperative mode of play to the formation of an environmental coalition. On the other hand, as joint optimization of all countries' payoff leads to the best outcome, cooperative games approach has also been proposed to share the cost of climate change. Both approaches have their pros and cons.

1 Introduction

Because countries’ emissions impose transboundary costs and no supranational authority can readily enforce corrective measures, the paper combines cooperative and noncooperative game approaches to IEA formation. It proposes sequential coalition formation and a process-based benefit allocation, including a role for costs of leaving an agreement.

1.1 Review of the literature

  • Without added incentives, noncooperative IEA models typically produce only 2–3 signatories, while a cooperative outcome can be collectively optimal yet not self-enforcing.Free riding motivates the small-coalition result; a country may gain by deviating from the grand coalition.Source: §1.1 · pp. 2, 4
  • Earlier approaches span myopic two-stage stability tests, farsighted membership decisions, and dynamic models in which coalition size can change over time.Some dynamic models produce a range of stable coalitions, including the grand coalition, depending on parameter values.Source: §1.1 · p. 3

1.2 Contribution of our paper

  • The paper combines cooperative and noncooperative approaches by using internal and external stability conditions and a cooperative solution to allocate coalition outcomes.The authors distinguish this from approaches that coordinate signatories’ emissions but do not apply a cooperative-game solution to the resulting coalition.Source: §1.2 · p. 5
  • Coalitions form through sequential, visible join-or-leave decisions, and an absorbing state of the resulting Markov chain is a stable IEA.The authors also establish that absorbing classes must be absorbing states or periodic classes.Source: §1.2 · p. 5
  • After formation, the Markovian process value allocates benefits by averaging scenario-values across possible paths, whether the final IEA is grand or partial.The paper filters for scenarios whose scenario-value is an imputation, so entering countries receive at least their payoff from acting alone.Source: §1.2 · p. 5
  • Introducing a cost of leaving can produce larger stable coalitions, including a coalition with more than two members or the grand coalition.The paper considers this cost as a possible response to the harm departure causes remaining signatories, or as a price that incentivizes others to join.Source: §1.2 · p. 6

2 The Markovian process value4

The section defines a Markovian process value for coalition formation: it averages player contributions across possible coalition paths. The Shapley value is a special case under a process that gives every ordering equal probability.

2 The Markovian process value4

  • A time-homogeneous Markov process uses all possible coalitions as states, with transition probabilities that do not depend on history.The transition matrix is row-stochastic.Source: §2 · p. 6
  • A scenario is a possible sequence of coalitions from a starting coalition, and its probability depends on the start and the transitions along the path.A scenario may revisit coalitions, need not end at the grand coalition, and terminates when it reaches a terminal state.Source: §2 · p. 6
  • For simple scenarios, each transition changes at most one player’s membership, and the scenario-value records players’ contributions along that path.The process-value takes the expected scenario-value over possible scenarios.Source: §2 · pp. 6–7
  • The process starts from a specified coalition with probability 1, chosen so that at least one transition to a nonempty coalition is possible.The authors note that multiple starting coalitions can instead be used with assigned probabilities.Source: §2 · p. 7
  • The Shapley value is recovered when all n! coalition-building orders from the empty coalition to the grand coalition are equally likely.These orders are maximal chains in the Boolean lattice.Source: §2 · p. 7

3 The environmental game model

The environmental model links each country’s emissions benefits to damages from total emissions, making payoffs interdependent. It evaluates cooperative outcomes and allocates coalition payoffs using the Markovian process value.

3 The environmental game model

  • Each country’s payoff combines a concave, increasing benefit from its own emissions with a convex, increasing damage function of total emissions.Because damage depends on total emissions, a country’s payoff depends on other countries’ decisions.Source: §3 · p. 7
  • The cooperative IEA outcome maximizes the sum of countries’ payoffs, then allocates that outcome using the Markovian process value rather than a classical solution.The paper names the Shapley value, core, and nucleolus as alternatives it does not use.Source: §3 · p. 7
  • Coalition values use the γ-characteristic function: members act as one player, while non-members choose individually in a Nash equilibrium.The empty coalition is assigned value zero by convention.Source: §3 · p. 8

4 The IEA formation process

The process models coalition formation through one-country-at-a-time Markov transitions driven by payoff incentives and exit costs. Its absorbing outcomes and scenario feasibility depend on transition cycles, farsighted horizons, and whether countries remain in a coalition.

4.1 The transition mechanism

  • A country considers leaving when its payoff gain exceeds the exit cost; the model sets that cost to zero when a singleton coalition dissolves.The cost can represent money or reputational damage.Source: §4.1 · p. 8
  • Countries with an incentive to leave or join move with probabilities proportional to their incentives; otherwise, the process stays in the current state.Transitions change coalition membership one country at a time.Source: §4.1 · pp. 8–9
  • The transition matrix determines the possible coalition-formation scenarios and their probabilities, from which scenario-values and the process-value can be computed.Source: §4.1 · p. 9
  • Absorbing classes are either single absorbing states or periodic classes; the transition graph has no odd cycles beyond loops, but four-cycle absorbing classes can occur.Source: §4.1 · pp. 9–10

4.2 Myopic and farsighted behavior

  • Myopic countries decide using immediate entry and exit incentives, while farsighted feasibility requires a scenario-value at horizon τ at least equal to the country's singleton value.At τ = 1, farsightedness is equivalent to myopic behavior.Source: §4.2 · p. 10
  • In the three-country example, under superadditivity all scenarios are feasible for countries 2 and 3, while country 1 has additional feasibility constraints; at τ = 2, conditions are trivial or require v(ij) ≥ v(i) + v(j).Source: §4.2 · p. 11
  • With superadditive v and only singleton absorbing classes, a nonabsorbed country with v({i}) > 0 cannot feasibly follow some scenarios starting ∅→{i}; a country that never leaves a scenario ending in S is feasible.The first result needs care: a country may re-enter and later leave, and its process-value is not necessarily negative.Source: §4.2 · p. 12
  • Every country that enters a scenario and never leaves receives at least its individual payoff, and feasibility concerns the final coalition, which need not be the grand coalition.Source: §4.2 · pp. 10, 12

5 An example of formation of an IEA

The example examines IEA coalition formation among pollution-vulnerable and invulnerable countries, deriving equilibrium emissions and payoffs and considering incentives to join or leave. It explores how farsighted transitions, an allocation rule, and different costs of leaving shape coalition outcomes.

5.1 The setting

  • Equilibrium total emissions fall as coalition size, its number of vulnerable members, or the total number of vulnerable countries rises; invulnerable outsiders emit most and coalition members least.Vulnerable outsiders’ emissions lie between those two groups, and for a fixed coalition both player types benefit from being singletons, though joining or leaving changes coalition size.Source: §5.1 · p. 14
  • An invulnerable outsider never has an incentive to join a nonempty coalition, whereas vulnerable players’ joining or leaving incentives can depend on game parameters.When c = 0, vulnerable players are symmetric in these decisions; after the first transition, only vulnerable players join, and a third vulnerable player never joins a coalition of two.Source: §5.1 · pp. 15–17
  • With c = 0, an absorbing coalition can contain two vulnerable players but never more, while superadditivity makes scenarios starting with an invulnerable country infeasible.The possible two-player absorbing coalition also depends on β and k.Source: §5.1 · p. 17

5.2 Numerical example

  • For the four-country example, the game is superadditive at the chosen parameters, but at β = 0.44 it is not: v({1}) = −1.07, v({2}) = −6.57, and v({1, 2}) = −7.92.The example has three vulnerable countries and one invulnerable country; a higher α represents a more efficient economy.Source: §5.2 · p. 17

5.2.1 Without cost of leaving (c = 0)

  • Assuming farsighted countries, the superadditive game v has singleton absorbing classes 12, 13, and 23; scenarios S7–S36, where country 4 enters first, are infeasible, while the six remaining scenarios are feasible.To compute the process-value, probabilities of scenarios 1–6 are renormalized because scenarios 7–36 will not realize.Source: §5.2.1 · p. 21
  • The process-value allocation is φ1 = 9.3468, φ2 = 4.1351, φ3 = 1.0407, and φ4 = 0.For a realized scenario, its scenario-specific allocation is used; inactive players receive their individual payoff under the terminal coalition.Source: §5.2.1 · p. 21
  • When the final absorbing coalition is observed but the scenario is unknown, expected payoff is obtained by averaging over scenarios that lead to that coalition, including coalitions 12, 13, and 23.For coalition 12, scenarios S1 and S3 have normalized probabilities 0.656 and 0.344; for 13, S2 and S5 have 0.881 and 0.119; for 23, S4 and S6 have 0.795 and 0.205.Source: §5.2.1 · pp. 21–22
  • The process-value is unsuitable as a sharing rule because no final coalition would be observed, it lacks efficiency, and its payoff is not an imputation.The passage also notes that countries can be active in some scenarios and inactive in others.Source: §5.2.1 · p. 22
  • Without a leaving cost, the method’s payoffs are imputations for active players, and country 4 receives 4.500, its individual payoff; the results differ radically from Shapley-based values.The paper attributes the difference to infeasible scenarios and scenario probabilities reflecting externalities, rather than treating every ordering as possible and equiprobable.Source: §5.2.1 · p. 22

5.2.2 With a cost of leaving (c > 0)

  • At c ≥ 1.424, no backward transitions remain; 1234 becomes an absorbing coalition, while 123 is absorbing but disconnected and unreachable.The threshold is the minimum cost that eliminates all backward arrows in the transition graph.Source: §5.2.2 · p. 23, Figure 2 · p. 23
  • When c ≥ 1.424, scenarios S1–S6 retain their probabilities, while scenarios S7–S36 are replaced after backward transitions disappear.Table 4 presents the recomputed transition probabilities.Source: §5.2.2 · pp. 22–23, Table 4 · p. 24
  • The new scenarios S′7–S′12 are all feasible for every country, so their expected scenario value is an imputation for all countries.The six new scenarios each have probability of occurrence of about 0.03167.Source: §5.2.2 · p. 24
  • Introducing a cost of leaving makes all countries better off, strictly benefiting countries 1, 2, and 3; the cost must cover country 4’s gain from leaving the grand coalition.The paper interprets the cost as possible reputational damage.Source: §5.2.2 · p. 25

6 Concluding remarks

The paper models international environmental agreement formation with a Markov process, treating absorbing states as stable agreements and showing that a cost of leaving can permit larger coalitions, including the grand coalition. It identifies dynamic network formation as a possible extension for analyzing local interactions and allocating value among participants.

6 Concluding remarks

  • A cost of leaving can make the final coalition larger than two members, potentially reaching the grand coalition.Any absorbing state of the Markov chain represents a stable IEA.Source: §6 · p. 25
  • Network structure can enrich IEA analysis because local pollution interactions may affect whether stable agreements exist.Prior work cited in the paper finds stable IEAs for balanced networks, while highly asymmetric networks may lack them.Source: §6 · p. 25
  • A dynamic network formation process is proposed as a follow-up for modeling changing links and active players in IEAs.The paper suggests using a link-based scenario allocation rule to share value among participants.Source: §6 · p. 25

Statements and declarations

The authors report no relevant competing interests. The paper acknowledges financial support from the European Union and NSERC, Canada.

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