Addressing cross-border environmental displacement: when can international treaties help?

Abstract

Environmental degradation is increasingly causing cross-border displacement of people, but countries have formed no treaties to facilitate collaboration on the issue. When is such collaboration feasible, and how should environmental displacement treaties be designed? We present a game-theoretic analysis. In our model, countries first decide on ratifying a treaty, and doing so commits them to helping other countries that face cases of environmental degradation in the future. The equilibrium analysis suggests that treaty formation is easier under conditions of mutual vulnerability than if some countries are at a greater risk of environmental degradation than others. Our most important finding is that contrary to the received wisdom, treaties imposing stringent demands on countries are easier to form than treaties that are easy to comply with. We also examine the benefits of using displacement treaties to build capacity for cooperation. We illustrate the utility of the analysis with a discussion of the Kampala Convention on environmental displacement and consider the potential for future treaty formation in Sub-Saharan Africa and Oceania.

Appendices

Mathematical Appendix

This appendix contains the mathematical analysis and extensions.

1.1 Equilibrium

The unique pair of equilibrium strategies is the following:

• 1. In the subgame without a treaty, the victim plays s _k  = θ and the non-victim plays s _l  = 0

• 2. In the subgame with a treaty, the victim plays s _k  = θ and the non-victim plays s _l  = αθ

• 3. At the treaty ratification stage, country i ratifies the treaty if and only if $\pi_{i} \int\limits_{0}^{1}f(\theta)\frac{1}{2}\theta^{2}{\rm d}\theta < \pi_{i} \int\limits_{0}^{1}f(\theta) \left[\frac{1}{2}\theta^{2} + \alpha\theta^{2} \right] {\rm d}\theta + \left(1 - \pi_{i}\right) \int\limits_{0}^{1} f(\theta) \left[\frac{1}{2}\alpha^{2}\theta^{2}- \alpha\theta^{2} \right] {\rm d}\theta$.

Consider first the subgame without a treaty. The first-order conditions for k, l are

$$ \begin{aligned} \theta -s_{k} \leq& \theta\\ -s_{l} \leq& 0, \\ \end{aligned} $$

each with equality whenever s _k , s _l  > 0. This completes the characterization of the first class of subgames.

Consider now the subgame with treaty. First, s _l  > θ can be rejected as strictly dominated strategy. Thus, the first-order conditions are

$$ \begin{aligned} \theta -s_{k} \leq& \theta \\ \alpha\theta -s_{l} \leq& \alpha\theta, \\ \end{aligned} $$

each with equality whenever s _k , s _l  > 0. This completes the characterization of the second class of subgames.

For the treaty ratification stage, note that the inequality given above has the expected payoff from non-ratification on the left and the expected payoff from ratification on the right side.

1.2 Proofs

To prove the first proposition, let $\pi_{A} \rightarrow \frac{1}{2}$. Each country’s expected payoff from the treaty is now arbitrarily close to the other country’s payoff. This payoff is given by

$$ \frac{1}{2} \int\limits_{0}^{1}f(\theta) \left[\frac{1}{2}\theta^{2} + \alpha\theta^{2} \right] {\rm d}\theta + \frac{1}{2} \int\limits_{0}^{1} f(\theta) \left[\frac{1}{2}\alpha^{2}\theta^{2}- \alpha\theta^{2} \right] {\rm d}\theta. $$

Without a treaty, each country’s payoff is

$$ \frac{1}{2} \int\limits_{0}^{1}f(\theta)\frac{1}{2}\theta^{2}{\rm d}\theta. $$

The marginal payoff from treaty formation is thus

$$ \frac{1}{2} \int\limits_{0}^{1}f(\theta) \alpha\theta^{2} {\rm d}\theta + \frac{1}{2} \int\limits_{0}^{1} f(\theta) \left[\frac{1}{2}\alpha^{2}\theta^{2}- \alpha\theta^{2} \right] {\rm d}\theta. $$

Combining terms and simplifying, we are left with

$$ \frac{1}{2} \int\limits_{0}^{1} f(\theta) \frac{1}{2}\alpha^{2}\theta^{2} {\rm d}\theta. $$

This expression is strictly positive, so the claim follows.

Now, we prove the second proposition. The less vulnerable country B receives a payoff of

$$ \pi_{B} \int\limits_{0}^{1}f(\theta) \left[\frac{1}{2}\theta^{2} + \alpha\theta^{2} \right] {\rm d}\theta + \left(1 - \pi_{B}\right) \int\limits_{0}^{1} f(\theta) \left[\frac{1}{2}\alpha^{2}\theta^{2}- \alpha\theta^{2} \right] {\rm d}\theta, $$

where $\pi_{B} \leq \frac{1}{2}$. Differentiating with respect to α, we obtain the effect of changing treaty bindingness α on country B’s payoff:

$$ \pi_{B} \int\limits_{0}^{1}f(\theta) \theta^{2} {\rm d}\theta + \left(1 - \pi_{B}\right) \int\limits_{0}^{1} f(\theta) \left[\alpha\theta^{2}- \theta^{2} \right] {\rm d}\theta. $$

The first term within an integral is strictly positive; the second term within an integral is strictly negative. Given the continuity of the expected value, it is clear that the value of the derivate is strictly increasing in π _B . To show that a unique threshold exists, it suffices to consider the extreme cases π _B  = 0 ⇔ π _A  = 1 and $\pi_{B} = \frac{1}{2} \Leftrightarrow pi_{A} = \frac{1}{2}$. With π _B  = 0, the expression is strictly negative because the first term disappears. With $\pi_{B} = \frac{1}{2}$, the expression simplifies to

$$ \frac{1}{2} \int\limits_{0}^{1} f(\theta) \alpha\theta^{2} {\rm d}\theta. $$

This is a strictly positive expression, so the claim follows.

1.3 Parametric example: uniform distribution

When θ is uniformly distributed on the [0,1] interval, country is expected payoff without treaty formation is given by:

$$ \pi_{i} \int\limits_{0}^{1}\frac{1}{2}\theta^{2}{\rm d}\theta = \frac{1}{6} \cdot \pi_{i}. $$

The expected payoff with treaty formation is

$$ \pi_{i} \int\limits_{0}^{1}f\left[\frac{1}{2}\theta^{2} + \alpha\theta^{2} \right] {\rm d}\theta + \left(1 - \pi_{i}\right) \int\limits_{0}^{1} \left[\frac{1}{2}\alpha^{2}\theta^{2}- \alpha\theta^{2} \right] {\rm d}\theta, $$

simplifying to

$$ \pi_{i}\cdot\left(\frac{1}{3}\alpha+\frac{1}{6}\right) +(1-\pi_{i})\cdot\left(\frac{1}{6}\alpha-\frac{1}{3}\right)\alpha. $$

The marginal payoff from treaty formation is obtained by subtracting the latter from the former. Simple algebraic manipulation shows that country i is exactly indifferent regarding the treaty whenever

$$ \pi_{i} = \frac{\alpha-2}{\alpha-4}. $$

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Extension: capacity building under mutual interest

To see that each country i’s payoff without a treaty does not depend on γ, simply note that it does not appear in the payoff equation. Consider now the case with a treaty. To see that country i’s payoff must increase, note that γ only appears in the term $-\gamma\frac{1}{2}s_{i}^{2}$, where i = k, l. As γ decreases, the value of this term increases for any given s _i . Thus, country i’s equilibrium payoff must increase whenever γ is decreased.

Extension: victim’s obligations

If the victim k plays s ^* _k  = θ, as in the original model, the reputational cost from $-\alpha\theta\cdot \left(\theta-s_{k}\right)$ is zero. With this term disappearing, the victim’s optimal strategy and behavior must remain unchanged.