Appendix: S is the Proposer
The Appendix discusses the case in which S makes an ultimatum offer to N. This offer consists of a transfer t from S to N as a compensation for N to implement the efficient amount of abatement. For space considerations we concentrate on the case of interior solutions.
If the offer is rejected then players N and S choose the amounts of mitigation that maximize their own rents and no transfers are paid. These amounts of mitigation are \(\left( m_{{ NN}}^{1},m_{{ SS}}^{*}\right) \) or \(\left( m_{{ NN}}^{2},m_{{ NS}}^{2},m_{{ SS}}^{*}\right) \), depending on the regime. If the offer is accepted then \(\left( m_{{ NN}}^{3},m_{{ NS}}^{3},m_{{ SS}}^{*}\right) \) is implemented and country S pays t to country N. The net benefits for proposer S change from \(\theta _{S}^{i}\) to \(\theta _{S}^{3}+\sigma -t\) and N’s payoff changes from \(\theta _{N}^{i}\) to \(\theta _{N}^{3}+t+\nu \). Player N accepts the offer if \(\theta _{N}^{3}+t+\nu -\theta _{N}^{i}\ge 0\), or
$$\begin{aligned} \nu \ge \theta _{N}^{i}-\left( \theta _{N}^{3}+t\right) \end{aligned}$$
(13)
for \(i\in \{1,2\}\). This inequality describes a one-to-one relationship between t and the critical \(\nu \) such that N accepts the transfer demanded if the transfer is at least equal to the t that solves (13) with equality. We denote this critical \(\nu ^{i}\) as a function of t as
$$\begin{aligned} \nu ^{i}(t)=\theta _{N}^{i}-\theta _{N}^{3}-t\quad \text {for}\quad i\in \{1,2\}. \end{aligned}$$
(14)
This function is linear in t and has a slope of \(-1\).
The bargaining-success probability as a function of t is \(1-G\left( \theta _{N} ^{i}-\theta _{N}^{3}-t\right) \). Proposer S chooses t to maximize
$$\begin{aligned} E\pi _{S,i}=\left( \theta _{S}^{3}+\sigma -t-\theta _{S}^{i}\right) \cdot \left[ 1-G\left( \theta _{N}^{i}-\theta _{N}^{3}-t\right) \right] . \end{aligned}$$
This expected payoff is the product of S’s actual gain in case of successful bargaining, times the probability of acceptance of the proposal. Differentiating with respect to t yields a first-order condition which can be rewritten as
$$\begin{aligned} \frac{1-G\left( \theta _{N}^{i}-\theta _{N}^{3}-t\right) }{G^{\prime }\left( \theta _{N}^{i} -\theta _{N}^{3}-t\right) }=\theta _{S}^{3}+\sigma -t-\theta _{S}^{i}. \end{aligned}$$
(15)
If this condition has an interior solution, we denote it by \({\hat{t}} ^{i}(\sigma )\) to distinguish it from the function \(t^{i}(\nu )\) in the previous section. Note that the left-hand side is weakly increasing in the transfer offered (due to Assumption (HR) of a non-increasing hazard rate), and the right-hand side is strictly monotonically decreasing in the transfer. Accordingly, there is at most one transfer level \({\hat{t}}^{i}(\sigma )\) that fulfills this first-order condition for each regime i and a given \(\sigma \). We further note that
$$\begin{aligned} {\hat{t}}^{1}(\sigma )>{\hat{t}}^{2}(\sigma ) \end{aligned}$$
(16)
Recall the one-to-one correspondence between t and a value of \(\nu \) such that responder N accepts transfer offer t if \(\nu >\nu ^{i}(t)\) and rejects the transfer offer if N’s true value of \(\nu \) is smaller than \(\nu ^{i}(t)\) in the interior range of \(({\underline{\nu }},{\bar{\nu }})\). With a mild abuse of notation, let \(\nu ^{i}(t)\) denote the political benefit necessary to make country N indifferent between accepting and rejecting offer t in regime i. As shown by (14), the respective functions \(\nu ^{i}(t)\) are linear and have a slope of −1. Further, \(\nu ^{2}(t)-\nu ^{1}(t)=\theta _{N} ^{2}-\theta _{N}^{1}>0\). Together with (16) and (14) this implies
$$\begin{aligned} \nu ^{1}({\hat{t}}^{1}(\sigma ))<\nu ^{2}({\hat{t}}^{2}(\sigma ))\text {. } \end{aligned}$$
The probability of bargaining failure is higher if cross-border abatement opportunities are available in the non-cooperative fallback.