Endogenous Minimum Participation in International Environmental Agreements: An Experimental Analysis

Abstract

Almost all international environmental treaties require a minimum number of countries to ratify the treaty before it enters into force. Despite the wide-spread use of this mechanism, little is known about its effectiveness at facilitating cooperation. We analyze an agreement formation game that includes an endogenously determined minimum participation constraint and then test the predictions using economic experiments. We demonstrate theoretically that players will vote to implement an efficient coalition size as the membership requirement and this coalition will form. Experimental tests of the theory demonstrate that the minimum participation mechanism is highly effective at facilitating cooperation when efficiency requires the participation of all players. However, when efficiency requires only a subset of players to participate, profitable coalitions are often deliberately blocked. In light of our results it is possible that equity concerns can impede the formation of international agreements when membership requirements allow free riders.

Appendix

Here we incorporate inequality aversion into our model to demonstrate that minimum profitable coalitions are weakly larger for inequality averse individuals than for individuals with standard preferences. We follow Fehr and Schmidt (1999) in modeling preferences over inequality. Suppose at first that $\bar{s}=n$ and $s_p =0$. Given the financial payoffs (2) and (3) with these restrictions, define the utility of a member of an effective coalition with $s$ members as:

$$\begin{aligned} u_{i}^{m} (s)=\pi _{i}^{m} (s)-\frac{\alpha _i}{n-1}\sum _{j\ne i} {\max \left( {\pi _j -\pi _{i}^{m} (s),0} \right) } -\frac{\beta _i}{n-1}\sum _{j\ne i} {\max \left( {\pi _{i}^{m} (s)-\pi _j ,0} \right) }, \end{aligned}$$

(7)

where $\alpha _i >0$ captures the player’s loss from disadvantageous inequality and $\beta _i >0$ captures her loss from advantageous inequality. Since $\pi _j^{nm} (s)-\pi _{i}^{m} (s)=c$ and $\pi _j^m (s)-\pi _{i}^{m} (s)=0$ from (2) and (3), (7) can be written as:

$$\begin{aligned} u_{i}^{m} (s)=A+bs-c-\frac{\alpha _i c(n-s)}{n-1}. \end{aligned}$$

(8)

Similarly, the utility of a nonmember of an effective coalition with $s$ members is:

$$\begin{aligned}&u_i^{nm} (s)=\pi _i^{nm} (s)-\frac{\alpha _i}{n-1}\sum _{j\ne i} {\max \left( {\pi _j -\pi _i^{nm} (s),0} \right) }\nonumber \\&\quad -\frac{\beta _i}{n-1}\sum _{j\ne i} {\max \left( {\pi _i^{nm} (s)-\pi _j ,0} \right) }, \end{aligned}$$

(9)

which can be written as

$$\begin{aligned} u_i^{nm} (s)=A+bs-\frac{\beta _i sc}{n-1}. \end{aligned}$$

(10)

It is straightforward to show that the free-riding incentive is preserved in this model if $(n-1)(\beta _i -1)<\alpha _i$; that is, as long as the aversion to advantageous inequality is not too strong relative to the aversion to disadvantageous inequality. Incorporating the efficient coalition size $\bar{{s}}\le n$ to determine individual payoffs yields:

$$\begin{aligned} u_{i}^{m} (s)&= \left\{ {{ \begin{array}{ll} {A+bs-c-\frac{\alpha _i (n-s)c}{n-1}} &{}\quad {\hbox {for}\; s\le \bar{{s}};}\\ {A+b\bar{{s}}-c-\frac{\alpha _i (n-s)c}{n-1}} &{} \quad {\hbox {for}\; s>\bar{{s}};} \end{array}}} \right. \end{aligned}$$

(11)

$$\begin{aligned} u_i^{nm} (s)&= \left\{ {{ \begin{array}{ll} {A+bs-\frac{\beta _i sc}{n-1}} &{} \quad {\hbox {for}\; s\le \bar{{s}};} \\ {A+b\bar{{s}}-\frac{\beta _i sc}{n-1}} &{} \quad {\hbox {for}\; s>\bar{{s}}.} \end{array}}}\right. \end{aligned}$$

(12)

Using (11), an individual’s minimum profitable coalition size can be characterized as:

$$\begin{aligned} \tilde{s}_i \left\{ {{ \begin{array}{l} {=\hat{{s}}_{i}\, \hbox {if}\, \hat{{s}}_i \le \bar{{s}}} \\ {>\hat{{s}}_{i}\,\hbox {if}\, \hat{{s}}_i >\bar{{s}},} \\ \end{array}}} \right. \quad \hbox {where} \quad \hat{{s}}_i =\frac{c(n-1)+\alpha _i cn}{b(n-1)+\alpha _i c}. \end{aligned}$$

(13)

To demonstrate $\tilde{s}_{i}$, we first derive $\hat{{s}}_i$ as the solution to:

$$\begin{aligned} A+bs-c-\frac{\alpha _i (n-s)c}{n-1}=A. \end{aligned}$$

(14)

Since $u_{i}^{m} (s)$ in (11) is increasing in $s$, if $\hat{{s}}_i \le \bar{{s}}$ then $\hat{{s}}_i$ is $i$’s minimum profitable coalition size. However, if $\hat{{s}}_i >\bar{{s}}$, then $\tilde{s}_i$ must be the solution to

$$\begin{aligned} A+b\bar{{s}}-c-\frac{\alpha _i (n-s)c}{n-1}=A. \end{aligned}$$

(15)

Plug $\hat{{s}}_i$ into (14) and $\tilde{s}_i$ into (15), set the resulting equations equal to each other and collect terms to obtain $b(\hat{{s}}_i -\bar{{s}})=\alpha _i c(\tilde{s}_i -\hat{{s}}_i )/(n-1)$, which implies that $\tilde{s}_i >\hat{{s}}_i$ if $\hat{{s}}_i >\bar{{s}}$.

Recall from (4) that the minimum profitable coalition size for an individual with standard preferences is $s_{\min } =\min \{s|s\ge c/b\}$. From (13), $\tilde{s}_i \ge \hat{{s}}_i$, $\hat{{s}}_{i}$ is increasing in $\alpha _i$, and $\hat{{s}}_i =c/b$ for $\alpha _i =0$. Together, these imply $\tilde{s}_i \ge s_{\min }$ for an individual with disadvantageous inequality aversion (i.e., $\alpha _i >0$). Therefore, such an individual has a weakly higher minimum profitable coalition size than an individual with standard preferences.

By substituting in our experimental parameters into equation (13) we can demonstrate how large $\alpha _i$ must be to increase the minimum profitable coalition size beyond the efficient size. In the treatment with an efficient coalition of three members, if $\alpha _i$ exceeds 0.584, then the minimum profitable coalition for an individual will exceed three members. As a frame of reference, at least 40 % of players in Fehr and Schmidt’s analysis were estimated to have $\alpha > 0.50$.