# Analyzing Cooperative Environmental Agreements: Mechanism of Benefit Redistribution Among Signatory Nations

International Environmental Agreements (IEAs) provide a framework for countries to collectively address environmental challenges such as greenhouse gas (GHG) emissions reduction. A crucial aspect ensuring the stability and effectiveness of such coalitions is the equitable distribution of benefits derived from joint emission reduction efforts. Disparities among nations in terms of emissions levels, economic capacities, and cost-benefit analyses pose obstacles to fair benefit sharing.

In light of these challenges, establishing a mechanism for benefit redistribution that is perceived as fair and just by all member nations is imperative. Such a mechanism would not only incentivize nations to join and adhere to the coalition but also foster a sense of trust and mutual benefit among them. This study delves into one such proposed mechanism, diving into cooperative game theory to explore how benefits can be equitably redistributed among nations in a way that reflects their individual contributions and sustains the coalition’s integrity in the long term. Through the insights offered by McGinty (2020) [1], this discourse aims to illuminate the path towards more stable and effective International Environmental Agreements.

## Table of Contents

## Problem Statement

To ensure the long-term stability of a coalition of signatories to an environmental cooperation treaty, there must be a mechanism for transfer between countries that allows for the redistribution of benefits derived from their collective efforts to reduce GHG emissions.

By setting up this mechanism based on the distribution proposed by Shapley (1953b) [2], each country would receive a share proportional to its contribution to all possible sub-coalitions within the broader coalition.

## Mathematical Solution

It’s essential to define the expressions that describe the benefit calculation of a coalition in this cooperative game to compute its Shapley value. McGinty (2020) [1] characterized a country’s net benefit as the difference between the gross benefit from the total emissions reduction for that country, $B_{i}(Q)$ , and the individual cost of that country, $C_{i}(q_{i})$, in reducing an amount $q_{i}$ of GHG emissions.

\[\begin{equation} \pi_{i}(\alpha_{i}, c_{i}, q_{i}, Q) = b\alpha_{i}\left(aQ-\frac{Q^{2}}{2}\right) - \frac{c_{i}(q_{i})^{2}}{2} \label{eq:pi} \end{equation}\]Here, $b$ and $a$ are strictly greater than zero, and $\alpha_{i}$ represents country $i$’s share in the global benefit of emissions reduction, while $c_{i}$ denotes the marginal cost of reducing one unit of emissions for country $i$.

Furthermore, the profit value that a coalition $S$ obtains is the sum of the net benefits of the countries that are part of this coalition, which have collaborated to reduce emissions by an optimal amount $Q_{S}$.

Concurrently, the global emissions reduction $Q$, defined in equation \eqref{eq:q}, comprises both the optimal emissions reduction for the signatory members of the coalition $S$ and the reduction contributed by non-signatory members, represented as $T = N \setminus S$

\[\begin{equation} Q = \frac{ab\left(\sum_{j \in T}{\theta_{j}} + \sum_{i \in S}{\frac{1}{c_{i}}} \sum_{i \in S}{\alpha_{i}}\right)}{1+b\left(\sum_{j \in T}{\theta_{j}} + \sum_{i \in S}{\frac{1}{c_{i}}} \sum_{i \in S}{\alpha_{i}}\right)} \label{eq:q} \end{equation}\]Where $\theta_{j} = \frac{\alpha_{j}}{c_{j}}$ is the ratio between player $j$’s share $\alpha_{j}$ in the global benefit and their marginal cost of emissions reduction $c_{j}$.

## Experimental Example

To illustrate the above mathematical solutions in a practical scenario, consider the game where ten countries have shown active participation in global environmental agreements. These example include Australia, Brazil, Canada, China, Germany, India, Japan, Russia, South Africa, and Spain.

For our cooperative game, let’s assign arbitrary values to the parameters for each country based on hypothetical scenarios related to their economic capacities, emissions levels, and cost-benefit analyses. Given these values, we can compute the net benefit for each country using the equation \eqref{eq:pi}. Then, the benefit of any coalition $S \subseteq N$ in this game is the sum of the benefit of ervery member $i \in $ in the coalition \eqref{eq:pi_s}, conditioned to the fact that all of the members are part of it when computing each country benefit through \eqref{eq:q}.

\[\begin{equation} \pi(S) = \sum_{i \in S}{\pi_{i}} \label{eq:pi_s} \end{equation}\]From the computed net benefit values of each posible coalition in the game, now we can deduce the Shapley value for each country, which represents the equitable distribution of benefits derived from the grand coalition emission reduction effort. To ensure a dynamic and interactive experience, every value in the sample table can be edited. Simply click on the desired cell you wish to change, modify its content and review the solution of the cooperative game at the column *$\phi_{i}$ (Shapley Profit Share)*. This flexibility allows for real-time adjustments and experimentation, enabling users to explore various scenarios and outcomes.

This experimental example serves as a proof of concept for the mathematical model and demonstrates how benefit redistribution among nations can be computed in practice. It’s essential to note that these values are hypothetical and serve only as an example to illustrate the mathematical solution. Real-world values would require comprehensive data collection and analysis.

## References

[1]M. McGinty, “Leadership and Free-Riding: Decomposing and Explaining the Paradox of Cooperation in International Environmental Agreements,” *Environmental and Resource Economics*, vol. 77, no. 2, pp. 449–474, Oct. 2020, doi: 10.1007/s10640-020-00505-1.

[2]L. S. Shapley, “17. A Value for n-Person Games,” in *Contributions to the Theory of Games (AM-28), Volume II*, H. W. Kuhn and A. W. Tucker, Eds. Princeton: Princeton University Press, 1953, pp. 307–318.