A.1 Proof of Theorem 2.3
For every cooperator
\(i\in C\), we define her best response correspondence as a mapping
\(\gamma _{i} :X_{-i} \times Y \times Z \rightarrow 2^{X_{i}}\) given by
$$\begin{aligned} \gamma _{i}(x_{-i},y,z)=\underset{x_{i}\in X_{i}}{\arg \max } \, u_{i}(x_{i},x_{-i},y,z) \end{aligned}$$
(21)
for any
\((x_{-i},y,z)\in X_{-i}\times Y\times Z\).
Similarly, for the coalition of cooperators
C itself, we define the collective best response correspondence
\(\delta :X\times Z \rightarrow 2^Y\) by
$$\begin{aligned} \delta (x,z)=\underset{y\in Y}{\arg \max } \, U(x,y,z) \end{aligned}$$
(22)
for any
\((x,z)\in X\times Z\), where
U is the aggregated payoff function for
C defined in (
2).
Furthermore, for every non-cooperator
\(j\in N\), we define her best response correspondence
\(\varepsilon _{j} :X \times Y \times Z_{-j}\rightarrow 2^{Z_{j}}\) by
$$\begin{aligned} \varepsilon _{j}(x,y,z_{-j}) =\underset{z_{j}\in Z_{j}}{\arg \max } \, v_{j} (x,y,z_{j},z_{-j}) \end{aligned}$$
(23)
for any
\((x,y,z_{-j})\in X \times Y \times Z_{-j}\).
This allows us to introduce the joint best response correspondence
\({\mathbf {B}} :A \rightarrow 2^A\), such that for any
\({\hat{a}}=({\hat{x}},{\hat{y}},{\hat{z}})\in A\):
$$\begin{aligned} {\mathbf {B}} \left( {\hat{a}} \right) = \left\{ (x,y,z)\in A \, \left| \, \begin{array}{l} x_{i}\in \gamma _{i}({\hat{x}}_{-i},{\hat{y}},{\hat{z}}) \quad \text { for all } i\in C, \\ y \in \delta ({\hat{x}},{\hat{z}}), \quad \text {and} \\ z_{j}\in \varepsilon _{j}({\hat{x}},{\hat{y}},{\hat{z}}_{-j}) \quad \text {for all } j\in N \end{array} \right. \right\} . \end{aligned}$$
(24)
It is clear that a fixed point of this best response correspondence—defined as some
\(a^\star \in A\), such that
\(a^\star \in {\mathbf {B}}(a^\star )\)—corresponds to a partial cooperative equilibrium of the generalised partial cooperative game
\(\Gamma \).
We proceed by showing that \({\mathbf {B}}\), indeed, possesses such a fixed point.
First, we show that the best response correspondences \(\gamma _{i}\) (\(i \in C \)), \(\delta \), and \(\varepsilon _{j}\) (\(j \in N\)) are all non-empty valued.
Given that \(X_{i}\) is compact and \(u_i\) continuous on \(X_i\), for every \(i\in C\), applying the Weierstrass Theorem implies that \(u_i\) indeed admits a maximum and, thus, \(\gamma _{i} (x_{-i} , y,z) \ne \varnothing \) for all \(a = (x,y,z) \in A\), where \(i \in C\).
Next, since the aggregator \(\Lambda \) is continuous and all \(u_i\), \(i \in C\), are continuous on Y, it follows that U is continuous on Y as well. From compactness of Y, it follows that \(\delta \) is, therefore, non-empty valued.
Finally, the compactness of \(Z_{j}\) and continuity of \(v_j\) on \(Z_j\), for all \(j\in N\), implies that \(\varepsilon _{j}\) is non-empty valued as well for all \(j\in N \).
Therefore, combining these facts, it follows that \({\mathbf {B}}\) is a non-empty valued correspondence.
Next, we show that \({\mathbf {B}}\) is convex-valued.
First, we claim that each of the correspondences \(\gamma _{i}\) (\(i\in C\)), \(\delta \), and \(\varepsilon _{j}\) (\(j\in N\)) is convex-valued. To see this, consider \(\gamma _{i}\) for some \(i \in C\). For any \((x_{-i},y,z)\in X_{-i} \times Y \times Z\), \(\gamma _{i}(x_{-i},y,z)\) is the set of maxima of the quasi-concave function \(u_{i}(\cdot ,x_{-i},y,z)\) mapped onto the convex set \(X_{i}\). Hence, \(\gamma _{i}(x_{-i},y,z)\) is, indeed, a convex set.
Similar arguments can be used to show that, for any \((x,y,z_{-j}) \in X \times Y \times Z_{-j}\), the set \(\varepsilon _{j}(x,y,z_{-j})\) is convex.
Finally, consider
\(\delta \). Since
\(\Lambda \) is Paretian and quasi-concave, it aggregates quasi-concave utility functions in a quasi-concave function.
13 Therefore,
U is quasi-concave on
Y.
Furthermore, for every \((x,z)\in X \times Z\), \(\delta (x,z)\) is the set of maxima of the quasi-concave function \(U (x, \cdot ,z)\) mapped on a convex set Y, implying that \(\delta (x,z)\) is, indeed, convex.
This implies that \({\mathbf {B}}\) is, indeed, convex-valued.
Finally, we prove \({\mathbf {B}}\) is upper hemi-continuous.
Consider a sequence \({\hat{a}}_{p}=({\hat{x}}_{p},{\hat{y}}_{p},{\hat{z}}_{p})\) converging to some \({\hat{a}}=({\hat{x}},{\hat{y}},{\hat{z}})\), as well as a sequence \({\tilde{a}}_{p}=({\tilde{x}}_{p},{\tilde{y}}_{p}, {\tilde{z}}_{p})\) converging to some \({\tilde{a}}=({\tilde{x}},{\tilde{y}} ,{\tilde{z}})\), such that \(({\tilde{x}}_{p},{\tilde{y}}_{p},{\tilde{z}}_{p})\in {\mathbf {B}} ({\hat{x}}_{p},{\hat{y}}_{p},{\hat{z}}_{p})\) for all \(p \in {\mathbb {N}}\). We now prove that \(({\tilde{x}},{\tilde{y}}, {\tilde{z}}) \in {\mathbf {B}} ({\hat{x}},{\hat{y}},{\hat{z}})\), implying that the correspondence \({\mathbf {B}}\) is closed and, thus, since A is compact, it follows that \({\mathbf {B}}\) is, indeed, upper hemi-continuous.
By definition, it follows for all
\(i \in C\) that for every
\(x_{i}\in X_{i}\):
$$\begin{aligned} u_{i}({\tilde{x}}_{p,i}, {\hat{x}}_{p,-i}, {\hat{y}}_{p}, {\hat{z}}_{p})-u_{i} (x_{i}, {\hat{x}}_{p,-i}, {\hat{y}}_{p},{\hat{z}}_{p}) \geqslant 0. \end{aligned}$$
(25)
For every
\(y\in Y\):
$$\begin{aligned} U({\hat{x}}_{p},{\tilde{y}}_{p},{\hat{z}}_{p})- U({\hat{x}}_{p},y,{\hat{z}}_{p}) \geqslant 0. \end{aligned}$$
(26)
Finally, for every
\(j \in N\) and all
\(z_{j}\in Z_{j}\):
$$\begin{aligned} v_{j}({\hat{x}}_{p},{\hat{y}}_{p},{\tilde{z}}_{p,j},{\hat{z}}_{p,-j})-v_{j}({\hat{x}}_{p},{\hat{y}}_{p},z_{j},{\hat{z}}_{p,-j})\geqslant 0. \end{aligned}$$
(27)
From these conclusions, it follows immediately that
$$\begin{aligned} u_{i}({\tilde{x}}_{i},{\hat{x}}_{-i},{\hat{y}},{\hat{z}})-u_{i}(x_{i},{\hat{x}}_{-i},{\hat{y}},{\hat{z}})\geqslant 0&\text { for every } i \in C \text { and }x_{i}\in X_{i}; \end{aligned}$$
(28)
$$\begin{aligned} U({\hat{x}},{\tilde{y}},{\hat{z}})-U({\hat{x}},y,{\hat{z}})\geqslant 0 \quad&\text { for every }y\in Y; \end{aligned}$$
(29)
$$\begin{aligned} v_{j}({\hat{x}},{\hat{y}},{\tilde{z}}_{j},{\hat{z}}_{-j})-v_{j}({\hat{x}},{\hat{y}},z_{j},{\hat{z}}_{-j})\geqslant 0&\text { for every } j \in N \text { and } z_{j}\in Z_{j}. \end{aligned}$$
(30)
This proves that
\(({\tilde{x}},{\tilde{y}},{\tilde{z}})\in {\mathbf {B}} ({\hat{x}},{\hat{y}},{\hat{z}})\).
Hence, we conclude that the correspondence
\({\mathbf {B}} :A\rightarrow 2^A\) is a convex-valued and upper hemi-continuous correspondence. From Kakutani’s fixed point theorem (Border
1985, page 72), it can be concluded that
\({\mathbf {B}}\) admits a fixed point and, hence, a partial cooperative equilibrium exists for the generalised partial cooperative game
\(\Gamma \).
A.2 Proof of Theorem 2.7
The proof of Theorem
2.7 is based on Berge’s Theorem. For completeness, we state this fundamental result here.
Next, consider a generalised partial cooperative game
\(\Gamma =\left\langle C,N,X,Y,Z,u,v, \Lambda \right\rangle \). First, we define the Nash correspondence which is equal to the feasibility correspondence
E defined in Sect.
2 of the paper. Formally, we introduce the mapping
\(E :Y\rightarrow 2^{X\times Z}\) by
\(E(y)=E_{y}\) for every collective action
\(y \in Y\). We show that, under standard assumptions, this correspondence is non-empty and compact valued as well as upper hemi-continuous.
Proof
For an arbitrary collective action \(y\in Y\), consider the reduced normal-form game \(\Gamma _{y}= \langle C \cup N,X \times Z,w \rangle \) as given before. Assume that \(X_{i}\) is a compact and convex subset of a Euclidean space for each \(i\in C\), and \(Z_{j}\) is a compact and convex subset of a Euclidean space for all \(j\in N\).
For any \(i \in C\), the function \(w^y_{i}\) is equal to a section of \(u_{i}\) and, hence, by assumption its section \(w^y_{i}(\cdot ,x_{-i},z)\) is quasi-concave on \(X_{i}\).
For any \(j\in N\), the function \(w^y_{j}\) is equal to a section of \(v_{j}\) and, consequentially, by assumption its section \(w^y_{j}(x, \cdot ,z_{-j})\) is quasi-concave on \(Z_{j}\).
Moreover, both \(w^y_{i}\) (\(i \in C\)) and \(w^y_{j}\) (\(j\in N\)) are continuous on \(X\times Z\).
Hence, we conclude that the game
\(\Gamma _{y}= \langle C\cup N,X\times Z,w \rangle \) satisfies all conditions of a standard Nash equilibrium existence theorem (Fudenberg and Tirole
1991, page 34) and, therefore, admits a Nash equilibrium. Thus,
\(E_{y}\ne \varnothing \) for all
\(y\in Y\) implying that
\(E :Y\rightarrow 2^{X\times Z}\) is non-empty valued.
Next, we show that the correspondence E is closed, and since \(X \times Z\) is compact, E is upper hemi-continuous, as well. Consider a convergent sequence \({\hat{y}}_{p} \rightarrow {\hat{y}}\) and a convergent sequence \(({\hat{x}}_{p},{\hat{z}}_{p}) \rightarrow ({\hat{x}},{\hat{z}})\), such that \(({\hat{x}}_{p},{\hat{z}}_{p})\in E( {\hat{y}}_{p})\). We prove that \(({\hat{x}},{\hat{z}})\in E({\hat{y}} )\).
Indeed, by definition, for all \(i\in C\), we have \(w^y_{i}({\hat{x}}_{p,i},{\hat{x}}_{p,-i},{\hat{z}}_{p})-w^y_{i}(x_{i},{\hat{x}}_{p,-i}, {\hat{z}}_{p}) \geqslant 0\) for all \(x_{i}\in X_{i}\). Similarly, for all \(j\in N\), \(w^y_{j}( {\hat{x}}_{p}, {\hat{z}}_{p,j}, {\hat{z}}_{p,-j})-w^y_{j}({\hat{x}}_{p},z_{j},{\hat{z}}_{p,-j})\geqslant 0\) for all \(z_{j}\in Z_{j} \). It follows immediately that both \(w^y_{i}(\cdot ,\cdot ,\cdot )-w^y_{i}(x_{i},\cdot ,\cdot ) :X_{i} \times X_{-i} \times Z \rightarrow {\mathbb {R}}\) and \(w^y_{j}(\cdot ,\cdot ,\cdot )-w^y_{j}(\cdot ,z_{j},\cdot ) :X \times Z_{j} \times Z_{-j} \rightarrow {\mathbb {R}}\) are continuous functions.
Hence, for every \(i\in C\), we have that \(w^y_{i}( {\hat{x}}_{i}, {\hat{x}}_{-i}, {\hat{z}} )- w^y_{i}( x_{i},{\hat{x}}_{-i}, {\hat{z}}) \geqslant 0\) and, for all \(j\in N\), we have \(w^y_{j}({\hat{x}}, {\hat{z}}_{j}, {\hat{z}}_{-j}) -w^y_{j} ({\hat{x}}, z_{j}, {\hat{z}}_{-j}) \geqslant 0\). This implies \(({\hat{x}}, {\hat{z}}) \in E({\hat{y}})\). Thus, E is, indeed, a closed correspondence.
Finally, take any \(y\in Y\). We show that E(y) is compact. Since E is a closed correspondence, it is, therefore, closed-valued. Hence, E(y) is a closed subset of a compact set \(X \times Z\) and, thus, E(y) is compact. \(\square \)
Next, consider the function
\(U^{s} :Y \rightarrow {\mathbb {R}}\) as defined in (
8). From the continuity of
\(\Lambda \) and
\(u_i\) for each
\(i \in C\), we conclude that
U is continuous on
A. Since, from Lemma
A.2, the set
\(E_{y}\) is compact for all
\(y\in Y\), it then follows that
\({\arg \max }_{(x,y)\in E_{y}} \, U(x,y,z)\) exists for every
\(y\in Y\). Hence,
\(U^{s}\) is a well-defined function. It remains to show that
\(U^{s}\) is an upper semi-continuous function.
We are now in the position to complete the proof of Theorem
2.7.
From Lemma
A.3,
\(U^{s} :Y \rightarrow {\mathbb {R}}\) is an upper semi-continuous function defined on a compact set
Y. Hence, it follows from standard results that
\(\Phi ^{s}\) is non-empty.
Take any
\(y^{*}\in \Phi ^{s}\). From Lemma
A.2,
\(E_{y^{*}}\) is non-empty as well as compact. Moreover,
\(U \left( \cdot ,y^{*},\cdot \right) :X \times Z \rightarrow {\mathbb {R}}\) is a continuous function. Hence, from the Weierstrass Theorem:
$$\begin{aligned} \Psi = \underset{(x,z)\in E_{y^{*}}}{\arg \max } \, U(x,y^{*},z) \ne \varnothing . \end{aligned}$$
Take any
\((x^{*},z^{*}) \in \Psi \). Then, it is easy to establish that
\((x^{*},y^{*},z^{*})\) constitutes a leadership equilibrium in the generalised partial cooperative game
\(\Gamma \).
This completes the proof of Theorem
2.7.
A.3 Proof of Proposition 3.2
We proceed by straightforwardly compute the first-order conditions from the optimisation problems for the treaty countries
\(i \in C\), the independent countries
\(j \in N\) and for
C as a collective. First, notice that without loss of generality, we can replace the inequality in (
20) by an equality and re-write the constrained maximisation problem given by Eqs. (
16)–(
20) as a straightforward maximisation of a single function with two variables given by
$$\begin{aligned} \max _{L_{2h},d_{h}}\,u_{h} = \sqrt{{\overline{L}}-L_{2h}} + \sqrt{d_{h} L_{2h}}-\tfrac{1}{2} \left( \sum ^k_{i=1} d_{i} + \sum ^n_{j=1} d_{j} \right) . \end{aligned}$$
(31)
The first-order conditions of (
31) are given by:
$$\begin{aligned} \frac{\partial u_{h}}{\partial L_{2h}}= & {} -\frac{1}{2 \sqrt{{\overline{L}}-L_{2h}}} +\frac{d_{h}}{2 \sqrt{d_{h} L_{2h}}} \equiv 0 \end{aligned}$$
(32)
$$\begin{aligned} \frac{\partial u_{h}}{\partial d_{h}}= & {} \frac{L_{2h}}{2 \sqrt{d_{h} L_{2h}}}- \tfrac{1}{2} \equiv 0. \end{aligned}$$
(33)
We re-call that we already established that, for the treaty countries
\(i \in C\), the equilibrium level of individual country emissions is given by
$$\begin{aligned} d_i = \frac{L_{2i}}{k^2} , \end{aligned}$$
which, through substitution in (
32), implies that
$$\begin{aligned} \frac{1}{2 \sqrt{{\overline{L}} - L_{2i}}} = \tfrac{1}{2k}. \end{aligned}$$
We first show assertion (b) for
\({\overline{L}} > k^2\). Hence,
\(L^{PE}_{2i} = {\overline{L}}-k^2 >0\) and
\(d^{PE}_i = \frac{{\overline{L}}-k^2}{k^2}\). In turn, we then establish that
\(L^{PE}_{1i} = {\overline{L}} - L_{2i} = k^2\).
The first-order conditions for the independent countries \(j \in N\) now imply the computed outcomes for the Nash equilibrium, i.e., \(L^{PE}_{1j} = 1\) and \(L^{PE}_{2j} = d^{PE}_j = {\overline{L}}-1\).
Therefore,
$$\begin{aligned} \Delta ^{PE} = k \left( \frac{{\overline{L}} - k^2}{k^2} \right) + n \left( {\overline{L}}-1 \right) = ( n+\tfrac{1}{k} ) {\overline{L}} - (k+n) . \end{aligned}$$
This implies now that
$$\begin{aligned} u^{PE}_i= & {} k + \frac{{\overline{L}}-k^2}{k} - \tfrac{1}{2} ( n+\tfrac{1}{k} ) {\overline{L}} + \tfrac{1}{2} (k+n) \\= & {} \tfrac{1}{2} (k+n) - \tfrac{1}{2} \left( n - \tfrac{1}{k} \right) {\overline{L}} \\ u^{PE}_j= & {} {\overline{L}} - \tfrac{1}{2} \left( n+\tfrac{1}{k} \right) {\overline{L}} + \tfrac{1}{2} (k+n) . \end{aligned}$$
Now, since
\(\Delta ^{PE} < \Delta ^{NE}\), we derive immediately that for every
\(j \in N :u^{PE}_j > u^{NE}_j\). Furthermore, for
\(i \in C\), it holds that
\(u^{PE}_i > u^{NE}_i\) if and only if
$$\begin{aligned} \frac{{\overline{L}}-k^2}{k} + \tfrac{1}{2}(3k+n) - \tfrac{1}{2} \left( n+\tfrac{1}{k} \right) {\overline{L}} > {\overline{L}} - \tfrac{1}{2} (k+n) {\overline{L}} + \tfrac{1}{2} (k+n) \end{aligned}$$
if and only if
$$\begin{aligned} {\overline{L}} > \left( 1 - \tfrac{k}{2} + \tfrac{1}{2k} \right) \, k \, {\overline{L}} . \end{aligned}$$
Obviously,
\({\overline{L}} > 0\), so the inequality simplifies to
$$\begin{aligned} \left( 1 - \tfrac{k}{2} + \tfrac{1}{2k} \right) \, k <1 \quad \text{ or } \quad (k-1)^2 >0 . \end{aligned}$$
This is obviously the case for any value of
k in the assumed range
\(k \geqslant 2\), showing assertion (b).
Next, we show assertion (a) for \(1 < {\overline{L}} \leqslant k^2\).
In that case, there is a corner solution for the equilibrium conditions described by \(d^{PE}_i = L^{PE}_{2i} =0\) and \(L^{PE}_{1i} = {\overline{L}}\) for the treaty countries \(i \in C\).
For the independent countries \(j \in N\), we derive again \(L^{PE}_{1j} = 1\) and \(L^{PE}_{2j} = d^{PE}_j = {\overline{L}}-1\).
Thus,
\(\Delta ^{PE} = n ( {\overline{L}}-1)\) and
$$\begin{aligned} u^{PE}_i = \sqrt{{\overline{L}}} - \tfrac{n}{2} ( {\overline{L}}-1) \quad \text{ and } \quad u^{PE}_j = {\overline{L}} - \tfrac{n}{2} ( {\overline{L}}-1) = \left( 1- \tfrac{n}{2} \right) {\overline{L}} + \tfrac{n}{2}. \end{aligned}$$
To investigate the Pareto domination between the PE and the NE outcomes, we note again that
\(u^{PE}_j > u^{NE}_j\) as
\({\overline{L}} >1\) and now
\(u^{PE}_i > u^{NE}_i\) if and only if
$$\begin{aligned} \sqrt{{\overline{L}}} - \tfrac{n}{2} ( {\overline{L}}-1) > {\overline{L}} - \tfrac{1}{2} (k+n) ( {\overline{L}} -1), \end{aligned}$$
if and only if
$$\begin{aligned} \sqrt{{\overline{L}}} > {\overline{L}} - \tfrac{k}{2} \left( {\overline{L}} -1 \right) , \end{aligned}$$
if and only if
$$\begin{aligned} 2 \sqrt{{\overline{L}}} > k - (k-2) {\overline{L}} . \end{aligned}$$
Now,
\({\overline{L}} >1\) and
\(k \geqslant 2\) imply that
$$\begin{aligned} 2 \sqrt{{\overline{L}}}> 2 > k - (k-2) {\overline{L}}. \end{aligned}$$
Hence,
\(u^{PE}_i > u^{NE}_i\), thus confirming the assertion of (a) in Proposition
3.2.