Appendix 1
Proof of Proposition 1
To prove condition (7) of Proposition 1, we perform the maximization operator in (4) and invoke equation (6) to obtain the optimal strategies
$$v_{t}^{i(i)} = \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{1}{{(1 - \alpha_{i} )}}}} x_{t}^{i} \;{\text{and}}\;v_{t}^{i(j)} = \left( {\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)^{{\frac{1}{{(1 - \varepsilon_{i} )}}}} \varpi_{t}^{i(j)} x_{t}^{j} \;{\text{and}},\;{\text{for}}\,j \in \overline{K}(i);$$
$$u_{t}^{i} = \frac{{(\overline{A}_{t + 1}^{i(i)} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4(c_{t}^{i} )^{2} }}x_{t}^{i} ,\;{\text{for}}\;i \in N\;{\text{and}}\;t \in \{ 1,2, \cdots ,{T}\} .$$
(A1)
Substituting the optimal strategies in (A1) into (4) we obtain
$$\begin{aligned} & \left( {\overline{A}_{t}^{i(i)} x_{t}^{i} + \sum\limits_{\begin{subarray}{l} \ell \in N \\ \ell \ne i \end{subarray} } {\overline{A}_{t}^{i(\ell )} x_{t}^{\ell } + \overline{C}_{t}^{i} } } \right)\delta_{1}^{t} \\& = \left[ {p_{t}^{i(i)} \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} )x_{t}^{i} - \frac{{(\overline{A}_{t + 1}^{i(i)} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }}x_{t}^{i} } \right. \\ & + \left. {\sum\limits_{{j \in \overline{K}(i)}} {\left( {p_{t}^{i(j)} \left( {\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)^{{\frac{{\varepsilon_{i} }}{{(1 - \varepsilon_{i} )}}}} (1 - \varepsilon_{i} )\varpi_{t}^{i(j)} x_{t}^{j} + \gamma_{t}^{i(j)} x_{t}^{i} } \right)} } \right]\delta_{1}^{t} \\ & + \left( {\overline{A}_{t + 1}^{i(i)} x_{t + 1}^{i} + \sum\limits_{\begin{subarray}{l} \ell \in N \\ \ell \ne i \end{subarray} } {\overline{A}_{t + 1}^{i(\ell )} x_{t + 1}^{\ell } + \overline{C}_{t + 1}^{i} } } \right)\delta_{1}^{t + 1} ,{\text{for}}\;t \in \{ 1,2, \cdots ,{T}\} \;{\text{and}}\;i \in N, \\ \end{aligned}$$
(A2)
where \(x_{t + 1}^{i} = x^{i} + a_{t}^{i} \frac{{\overline{A}_{t + 1}^{i(i)} a_{t}^{i} \delta_{t}^{t + 1} }}{{2c_{t}^{i} }}x_{t}^{i} + \sum\limits_{{j \in \overline{K}(i)}} {\lambda_{t}^{i(j)} x_{{}}^{j} - \sigma_{t}^{i} x_{{}}^{i} ,}\) and
$$x_{t + 1}^{\ell } = x_{{}}^{\ell } + a_{t}^{\ell } \frac{{\overline{A}_{t + 1}^{\ell (\ell )} a_{t}^{\ell } \delta_{t}^{t + 1} }}{{2c_{t}^{\ell } }}x_{t}^{\ell } + \sum\limits_{{\varsigma \in \overline{K}(\ell )}} {\lambda_{t}^{\ell (\varsigma )} x_{{}}^{\varsigma } - \sigma_{t}^{\ell } x_{{}}^{\ell } } ,{\text{for}}\;\ell \in N\;{\text{and}}\;\ell \ne i.$$
(A3)
Note that both the LHS and RHS of equation (A2) are linear functions of \(x_{t}^{{}} = (x_{t}^{1} ,x_{t}^{2} , \cdots ,x_{t}^{n} )\).
We classify the state variables in (A2)-(A3) into three categories: (i) \(x_{{}}^{i}\), (ii) \(x_{{}}^{j}\), for \(j \in \overline{K}(i)\), and (iii) \(x_{{}}^{k}\), for \(k \notin \overline{K}(i)\). After grouping terms, we obtain the following results.
For (i) \(x_{{}}^{i}\),
$$\begin{aligned} \overline{A}_{t}^{i(i)} x_{t}^{i} = & p_{t}^{i(i)} \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} )x_{t}^{i} - \frac{{(\overline{A}_{t + 1}^{i(i)} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }}x_{t}^{i} + \sum\limits_{{j \in \overline{K}(i)}} {\gamma_{t}^{i(j)} x_{t}^{i} } \\ & + \delta_{t}^{t + 1} A_{t + 1}^{i(i)} \left( {a_{t}^{i} \frac{{\overline{A}_{t + 1}^{i(i)} a_{t}^{i} \delta_{t}^{t + 1} }}{{2c_{t}^{i} }}x_{t}^{i} + (1 - \sigma_{t}^{i} )x_{{}}^{i} + \sum\limits_{{j \in \overline{K}(i)}}^{{}} {} \overline{A}_{t + 1}^{i(j)} \lambda_{t}^{j(i)} x_{{}}^{i} } \right); \\ \end{aligned}$$
(A4)
(ii) \(x_{{}}^{j}\), for \(j \in \overline{K}(i)\),
$$\begin{aligned} \overline{A}_{t}^{i(j)} x_{t}^{j} = & p_{t}^{i(j)} \left( {\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)^{{\frac{{\varepsilon_{i} }}{{(1 - \varepsilon_{i} )}}}} (1 - \varepsilon_{i} )\varpi_{t}^{i(j)} x_{t}^{j} \\ & + \delta_{t}^{t + 1} \overline{A}_{t + 1}^{i(j)} \left( {a_{t}^{j} \frac{{\overline{A}_{t + 1}^{j(j)} a_{t}^{j} \delta_{t}^{t + 1} }}{{2c_{t}^{j} }}x_{t}^{j} + (1 - \sigma_{t}^{j} )x_{{}}^{j} } \right) + \delta_{t}^{t + 1} \sum\limits_{{\ell \in \vartheta_{{}}^{i(j)\ell } }}^{{}} {\overline{A}_{t + 1}^{i(\ell )} \lambda_{t}^{\ell (j)} x_{{}}^{j} ,} \\ \end{aligned}$$
(A5)
$${\text{where}}\;\vartheta_{{}}^{i(j)\ell } \;{\text{is }}\;{\text{the}}\;{\text{ set }}\;{\text{of}}\;\ell \;{\text{with}}\;\overline{K}(\ell )\;{\text{containing}}\;j,\;{\text{that}}\;{\text{ is}}\;\{ \ell :j \in \overline{K}(\ell )\};$$
(iii) \(x_{{}}^{k}\), for \(k \notin \overline{K}(i)\),
$$\begin{aligned} & \overline{A}_{t}^{i(k)} x_{t}^{k} = \delta_{t}^{t + 1} \overline{A}_{t + 1}^{i(k)} \left( {a_{t}^{k} \frac{{\overline{A}_{t + 1}^{k(k)} a_{t}^{k} \delta_{t}^{t + 1} }}{{2c_{t}^{k} }}x_{t}^{k} + (1 - \sigma_{t}^{k} )x_{{}}^{k} } \right) + \delta_{t}^{t + 1} \sum\limits_{{\ell \in \vartheta_{{}}^{i(k)\ell } }} {\overline{A}_{t + 1}^{i(\ell )} \lambda_{t}^{\ell (k)} x_{{}}^{k} ,} \hfill \\ & {\text{for}} \;k \in [N\backslash \overline{K}(i)], \hfill \\ \end{aligned}$$
(A6)
where \(\vartheta_{{}}^{i(k)\ell }\) is the set of \(\ell\) with \(\overline{K}(\ell )\) containing \(k\), that is \(\{ \ell :k \in \overline{K}(\ell )\}\).
Hence, for (A4)-(A6) to hold, it is required that:
$$\begin{aligned} \overline{A}_{t}^{i(i)} = & p_{t}^{i(i)} \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} ) - \frac{{(\overline{A}_{t + 1}^{i(i)} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }} + \sum\limits_{{j \in \overline{K}(i)}}^{{}} {\gamma_{t}^{i(j)} } \\ & + \delta_{t}^{t + 1} \overline{A}_{t + 1}^{i(i)} \left( {a_{t}^{i} \frac{{\overline{A}_{t + 1}^{i(i)} a_{t}^{i} \delta_{t}^{t + 1} }}{{2c_{t}^{i} }} + (1 - \sigma_{t}^{i} ) + \sum\limits_{{j \in \overline{K}(i)}}^{{}} {\overline{A}_{t + 1}^{i(j)} \lambda_{t}^{j(i)} } } \right),\overline{A}_{T + 1}^{i(i)} = q_{T + 1}^{i(i)} ; \\ \end{aligned}$$
(A7)
$$ {\begin{aligned} &\overline{A}_{t}^{i(j)} = p_{t}^{i(j)} \left( {\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)^{{\frac{{\varepsilon_{i} }}{{(1 - \varepsilon_{i} )}}}} (1 - \varepsilon_{i} )\varpi_{t}^{i(j)} \\ &\qquad\qquad + \delta_{t}^{t + 1} \overline{A}_{t + 1}^{i(j)} \left( {a_{t}^{j} \frac{{\overline{A}_{t + 1}^{j(j)} a_{t}^{j} \delta_{t}^{t + 1} }}{{2c_{t}^{j} }} + (1 - \sigma_{t}^{j} )} \right) \\ &\qquad\qquad + \delta_{t}^{t + 1} \sum\limits_{{\ell \in \vartheta_{{}}^{i(j)\ell } }}^{{}} {\overline{A}_{t + 1}^{i(\ell )} \lambda_{t}^{\ell (j)} ,\;\overline{A}_{T + 1}^{i(j)} = 0,} \, {\text{for}}\;j \in \overline{K}(i); \end{aligned}} $$
(A8)
$$\begin{aligned} & \overline{A}_{t}^{i(k)} = \delta_{t}^{t + 1} \overline{A}_{t + 1}^{i(k)} \left( {a_{t}^{k} \frac{{\overline{A}_{t + 1}^{k(k)} a_{t}^{k} \delta_{t}^{t + 1} }}{{2c_{t}^{k} }} + (1 - \sigma_{t}^{k} )} \right) + \delta_{t}^{t + 1} \sum\limits_{{\ell \in \vartheta_{{}}^{i(k)\ell } }}^{{}} {\overline{A}_{t + 1}^{i(\ell )} \lambda_{t}^{\ell (k)} ,\;\overline{A}_{T + 1}^{i(k)} = 0} , \hfill \\ & {\text{for}}\;k \in [N\backslash \overline{K}(i)]. \hfill \\ \end{aligned}$$
(A9)
Since \(\overline{A}_{T + 1}^{i(\ell )} = 0\), for \(\ell \ne i\), therefore according to (A9), \(\overline{A}_{t}^{i(k)} = 0\), for \(k \notin \overline{K}(i)\) and \(t \in \{ 1,2, \cdots ,{T}\}\), and with \(\overline{A}_{t}^{i(k)} = 0\), for \(k \notin \overline{K}(i)\) and \(t \in \{ 1,2, \cdots ,{T}\}\), we can express equation (A8) as
$$\begin{aligned} \overline{A}_{t}^{i(j)} = & \;p_{t}^{i(j)} \left( {\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)^{{\frac{{\varepsilon_{i} }}{{(1 - \varepsilon_{i} )}}}} (1 - \varepsilon_{i} )\varpi_{t}^{i(j)} \; + \delta_{t}^{t + 1} \overline{A}_{t + 1}^{i(j)} \left( {a_{t}^{j} \frac{{\overline{A}_{t + 1}^{j(j)} a_{t}^{j} \delta_{t}^{t + 1} }}{{2c_{t}^{j} }}\; + (1 - \sigma_{t}^{j} )} \right) \\ & + \delta_{t}^{t + 1} \sum\limits_{{\ell \in [\overline{K}(i) \cap \overline{K}(j)]}}^{{}} {\overline{A}_{t + 1}^{i(\ell )} \lambda_{t}^{\ell (j)} ,\,\overline{A}_{T + 1}^{i(j)} = 0} ,\;{\text{for}}\;j \in \overline{K}(i). \\ \end{aligned}$$
(A10)
Finally, we can obtain
$$\overline{C}_{T + 1}^{i} = M_{T + 1}^{i} ,{\text{ and}}\;\overline{C}_{t}^{i} = \delta_{t}^{t + 1} \overline{C}_{t + 1}^{i} {,}\;{\text{for}}\;t \in \{ 1,2, \cdots ,{T}\} \;{\text{and}}\;i \in N.$$
(A11)
Appendix 2
Proof of Proposition 2
To prove condition (18) of Proposition 2, we perform the maximization operator in (15) and invoke equation (17) to obtain the optimal cooperative strategies
$$\begin{aligned} & v_{t}^{i(i)} = \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{1}{{(1 - \alpha_{i} )}}}} x_{t}^{i} \;{\text{and}}\;v_{t}^{i(j)} = \left( {\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)^{{\frac{1}{{(1 - \varepsilon_{i} )}}}} \varpi_{t}^{i(j)} x_{t}^{j} ,{\text{ for}}\;j \in K(i); \hfill \\ & \quad u_{t}^{i} = \frac{{(\hat{A}_{t + 1}^{i} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4(c_{t}^{i} )^{2} }}x_{t}^{i} ,{\text{for}}\;i \in N\;{\text{and}}\;t \in \{ 1,2, \cdots ,{T}\} . \hfill \\ \end{aligned}$$
(B1)
Substituting the optimal strategies in (B1) into (15) we obtain
$$\begin{aligned} & \left( {\sum\limits_{i \in N}^{{}} {\hat{A}_{t}^{i} x_{t}^{i} + \hat{C}_{t}^{{}} } } \right)\delta_{1}^{t} = \sum\limits_{i \in N}^{{}} {\left[ {p_{t}^{i(i)} \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} )x_{t}^{i} - \frac{{(\hat{A}_{t + 1}^{i} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }}x_{t}^{i} } \right.} \hfill \\ & \quad + \left. {\left. {\sum\limits_{j \in K(i)}^{{}} {\left( {p_{t}^{i(j)} \left( {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right.\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)}^{{\frac{{\varepsilon_{i} }}{{(1 - \varepsilon_{i} )}}}} (1 - \varepsilon_{i} )\varpi_{t}^{i(j)} x_{t}^{j} + \sum\limits_{j \in K(i)}^{{}} {\gamma_{t}^{i(j)} x_{t}^{i} } } \right)} \right]\delta_{1}^{t} \hfill \\ & \quad + \left( {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right.\sum\limits_{i \in N}^{{}} {} \hat{A}_{t + 1}^{i} x_{t + 1}^{i} + \hat{C}_{t + 1}^{{}} \left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right)\delta_{1}^{t + 1} , \hfill \\ \end{aligned}$$
(B2)
where \(x_{t + 1}^{i} = x_{{}}^{i}\)\(+ a_{t}^{i} \frac{{\hat{A}_{t + 1}^{i} a_{t}^{i} \delta_{t}^{t + 1} }}{{2c_{t}^{i} }}x_{t}^{i}\)\(+ \sum\limits_{j \in K(i)}^{{}} {} \lambda_{t}^{i(j)} x_{{}}^{j} - \sigma_{t}^{i} x_{{}}^{i}\), for \(i \in N\).
Note that both the LHS and RHS of equation (B2) are linear functions of \(x_{t}^{{}} = (x_{t}^{1} ,x_{t}^{2} , \cdots ,x_{t}^{n} )\). Grouping terms involving \(x_{t}^{{}}\), we obtain the following results:
$$\begin{aligned} & \hat{A}_{t}^{i} x_{t}^{i} = p_{t}^{i(i)} \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} )x_{t}^{i} - \frac{{(A_{t + 1}^{i} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }}x_{t}^{i} + \sum\limits_{j \in K(i)}^{{}} {\gamma_{t}^{i(j)} x_{t}^{i} } \hfill \\ & \quad + \sum\limits_{j \in K(i)}^{{}} {p_{t}^{j(i)} \left( {\frac{{\varepsilon_{j}^{{}} p_{t}^{j(i)} }}{{c_{t}^{j(i)} }}} \right)^{{\frac{{\varepsilon_{j} }}{{(1 - \varepsilon_{j} )}}}} } (1 - \varepsilon_{j} )\varpi_{t}^{j(i)} x_{t}^{i} + \delta_{t}^{t + 1} A_{t + 1}^{i} \left( {x_{{}}^{i} + a_{t}^{i} \frac{{A_{t + 1}^{i} a_{t}^{i} \delta_{t}^{t + 1} }}{{2c_{t}^{i} }}x_{t}^{i} - \sigma_{t}^{i} x_{{}}^{i} } \right) \hfill \\ & \quad + \delta_{t}^{t + 1} \sum\limits_{j \in K(i)}^{{}} {\hat{A}_{t + 1}^{j} \lambda_{t}^{j(i)} x_{{}}^{i} }, {\text{for}} \, i \in N.\hfill \\ \end{aligned}$$
(B3)
.
For (B3) to hold, it is required that
\(\hat{A}_{T + 1}^{i} =\)\(q_{T + 1}^{i(i)}\), and
$$\begin{aligned} \hat{A}_{t}^{i} & = p_{t}^{i(i)} \left( {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right.\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}\left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right)^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} ) - \frac{{(\hat{A}_{t + 1}^{i} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }} + \sum\limits_{j \in K(i)}^{{}} {} \gamma_{t}^{i(j)} \hfill \\ & \quad + \sum\limits_{j \in K(i)}^{{}} {} p_{t}^{j(i)} \left( {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right.\frac{{\varepsilon_{j}^{{}} p_{t}^{j(i)} }}{{c_{t}^{j(i)} }}\left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right)^{{\frac{{\varepsilon_{j} }}{{(1 - \varepsilon_{j} )}}}} (1 - \varepsilon_{j} )\varpi_{t}^{j(i)} \hfill \\ & \quad + \delta_{t}^{t + 1} \hat{A}_{t + 1}^{i} \left( {(1 - \sigma_{t}^{i} )\; + a_{t}^{i} \frac{{\hat{A}_{t + 1}^{i} a_{t}^{i} \delta_{t}^{t + 1} }}{{2c_{t}^{i} }}} \right)\; + \delta_{t}^{t + 1} \sum\limits_{j \in K(i)}^{{}} {\hat{A}_{t + 1}^{j} \;\lambda_{t}^{j(i)} }, \hfill \\ \end{aligned}$$
(B4)
for \(t \in \{ 1,2, \cdots ,{T}\}\) and \(i \in N\).
Finally, we can obtain
\(\hat{C}_{T + 1}^{{}} = \sum\limits_{i \in N}^{{}} {M_{T + 1}^{i} }\), and
$$\hat{C}_{t}^{{}} \; = \delta_{t}^{t + 1} \hat{C}_{t + 1}^{{}} ,\;{\text{for}}\;t \in \{ 1,2, \cdots ,{T}\}.$$
(B5)
Hence Proposition 2 follows.
Appendix 3
Proof of Proposition 3
To prove condition (30) of Proposition 3, we perform the maximization operator in (26) and invoke equation (29) to obtain the optimal strategies of firm \(i \in S\) as
\(v_{t}^{i(i)} = \left( {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right.\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}\left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right)^{{\frac{1}{{(1 - \alpha_{i} )}}}} x_{t}^{i}\) and \(v_{t}^{i(j)} = \left( {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right.\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}\left. {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \right)^{{\frac{1}{{(1 - \varepsilon_{i} )}}}} \varpi_{t}^{i(j)} x_{t}^{j}\), for \(j \in K(i) \cap S\);
$$u_{t}^{i} = \frac{{(A_{t + 1}^{i} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4(c_{t}^{i} )^{2} }}x_{t}^{i} ,\;{\text{for}}\;t \in \{ 1,2, \cdots ,{T}\}.$$
(C1)
Substituting the optimal strategies in (C1) into (26)-(27) we obtain
$$\begin{aligned} & \left( {\sum\limits_{i \in S}^{{}} {A_{t}^{(S)i} x_{t}^{i} + C_{t}^{(S)} } } \right)\delta_{1}^{t} = \sum\limits_{i \in S}^{{}} {\left[\vphantom{\left( {p_{t}^{i(j)} \left( {\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)^{{\frac{{\varepsilon_{i} }}{{(1 - \varepsilon_{i} )}}}} (1 - \varepsilon_{i} )\varpi_{t}^{i(j)} x_{t}^{j} + \gamma_{t}^{i(j)} x_{t}^{i} } \right)} {p_{t}^{i(i)} \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)} \right.^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} )x_{t}^{i} - \frac{{(A_{t + 1}^{(S)i} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }}x_{t}^{i} } \hfill \\ &\quad + \left. {\sum\limits_{j \in K(i) \cap S}^{{}} {\left( {p_{t}^{i(j)} \left( {\frac{{\varepsilon_{i}^{{}} p_{t}^{i(j)} }}{{c_{t}^{i(j)} }}} \right)^{{\frac{{\varepsilon_{i} }}{{(1 - \varepsilon_{i} )}}}} (1 - \varepsilon_{i} )\varpi_{t}^{i(j)} x_{t}^{j} + \gamma_{t}^{i(j)} x_{t}^{i} } \right)} } \right]\delta_{1}^{t} \hfill \\ & \quad + \left( {\sum\limits_{i \in S}^{{}} {A_{t + 1}^{(S)i} x_{t + 1}^{i} + C_{t + 1}^{(S)} } } \right)\delta_{1}^{t + 1} , \hfill \\& \quad \, x_{t + 1}^{i} = x_{t}^{i} \; + \frac{{a_{t}^{i} (A_{t + 1}^{(S)i} a_{t}^{i} \delta_{t}^{t + 1} )}}{{2c_{t}^{i} }}x_{t}^{i} \; + \sum\limits_{j \in K(i) \cap S}^{{}} {} \lambda_{t}^{i(j)} x_{{}}^{j} - \sigma_{t}^{i} x_{{}}^{i} , \, \text{for}\, i \in S \end{aligned}$$
(C2)
and
$$\begin{aligned} & \left( {\sum\limits_{\ell \in N\backslash S}^{{}} {A_{t}^{(N\backslash S)\ell } x_{t}^{\ell } + C_{t}^{(N\backslash S)} } } \right)\delta_{1}^{t} \; \\& = \sum\limits_{\ell \in N\backslash S}^{{}} {\left[ {p_{t}^{\ell (\ell )} \left( {\frac{{\alpha_{\ell }^{{}} p_{t}^{\ell (\ell )} }}{{c_{t}^{\ell (\ell )} }}} \right)^{{\frac{{\alpha_{\ell } }}{{(1 - \alpha_{\ell } )}}}} (1 - \alpha_{\ell } )x_{t}^{\ell } \; - \frac{{(A_{t + 1}^{(N\backslash S)\ell } a_{t}^{\ell } \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{\ell } }}x_{t}^{\ell } } \right.} \\&\quad + \left. {\sum\limits_{j \in K(\ell ) \cap N\backslash S}^{{}} {\left( {p_{t}^{\ell (j)} \left( {\frac{{\varepsilon_{\ell }^{{}} p_{t}^{\ell (j)} }}{{c_{t}^{\ell (j)} }}} \right)^{{\frac{{\varepsilon_{\ell } }}{{(1 - \varepsilon_{\ell } )}}}} (1 - \varepsilon_{\ell } )\varpi_{t}^{\ell (j)} x_{t}^{j} + \gamma_{t}^{\ell (j)} } \right)} } \right]\delta_{1}^{t} \\& \quad + \left( {\sum\limits_{\ell \in N\backslash S}^{{}} {A_{t + 1}^{(N\backslash S)\ell } x_{t + 1}^{\ell } + C_{t + 1}^{(N\backslash S)} } } \right)\delta_{1}^{t + 1} , \\& \,\, x_{t + 1}^{\ell } = x_{t}^{\ell } + \frac{{a_{t}^{\ell } (A_{t + 1}^{(N\backslash S)\ell } a_{t}^{\ell } \delta_{t}^{t + 1} )}}{{2c_{t}^{\ell } }}x_{t}^{\ell } + \sum\limits_{j \in K(i) \cap N\backslash S}^{{}} {\lambda_{t}^{\ell (j)} x_{{}}^{j} - \sigma_{t}^{\ell } x_{{}}^{\ell } }, \, \text{for}\, \ell \in N\backslash S. \end{aligned}$$
(C3)
Note that (i) both the LHS and RHS of equation (C2) are linear functions of \(x_{t}^{S}\), and (ii) both the LHS and RHS of equation (C3) are linear functions of \(x_{t}^{N\backslash S}\). Grouping terms involving \(x_{t}^{i}\) for \(i \in S\) and \(x_{t}^{\ell }\) in \(\ell \in N\backslash S\), we obtain the following results:
$$ \begin{aligned} A_{t}^{(S)i} x_{t}^{i} & = p_{t}^{i(i)} \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} )x_{t}^{i} \; - \frac{{(A_{t + 1}^{(S)i} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }}x_{t}^{i} + \sum\limits_{j \in K(i) \cap S}^{{}} {\gamma_{t}^{i(j)} x_{t}^{i} } \\& \quad + \sum\limits_{j \in K(i) \cap S}^{{}} {p_{t}^{j(i)} \left( {\frac{{\varepsilon_{j}^{{}} p_{t}^{j(i)} }}{{c_{t}^{j(i)} }}} \right)^{{\frac{{\varepsilon_{j} }}{{(1 - \varepsilon_{j} )}}}} (1 - \varepsilon_{j} )\varpi_{t}^{j(i)} x_{t}^{i} } \\&\quad + A_{t + 1}^{(S)i} \delta_{t}^{t + 1} \left( {x_{t}^{i} + \frac{{a_{t}^{i} (A_{t + 1}^{(S)i} a_{t}^{i} \delta_{t}^{t + 1} )}}{{2c_{t}^{i} }}x_{t}^{i} - \sigma_{t}^{i} x_{{}}^{i} } \right) + \delta_{t}^{t + 1} \sum\limits_{j \in K(i) \cap S}^{{}} {A_{t + 1}^{(S)j} \lambda_{t}^{j(i)} x_{{}}^{i} }, \hfill \\ & {\text{for}}\;i \in S; \end{aligned}$$
(C4)
and
$$\begin{aligned} A_{t}^{(N\backslash S)\ell } x_{t}^{\ell } & = \,p_{t}^{\ell (\ell )} \sum\limits_{\ell \in N\backslash S}^{{}} {\left( {\frac{{\alpha_{\ell }^{{}} p_{t}^{\ell (\ell )} }}{{c_{t}^{\ell (\ell )} }}} \right)}^{{\frac{{\alpha_{\ell } }}{{(1 - \alpha_{\ell } )}}}} (1 - \alpha_{\ell } )x_{t}^{\ell } \; - \frac{{(A_{t + 1}^{(N\backslash S)\ell } a_{t}^{\ell } \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{\ell } }}x_{t}^{\ell }\hfill \\ & \quad + \sum\limits_{j \in K(\ell ) \cap N\backslash S}^{{}} {\gamma_{t}^{\ell (j)} x_{t}^{\ell } } \hfill \\& \quad + \sum\limits_{j \in K(\ell ) \cap N\backslash S}^{{}} {p_{t}^{j(\ell )} \left( {\frac{{\varepsilon_{j}^{{}} p_{t}^{j(\ell )} }}{{c_{t}^{j(\ell )} }}} \right)^{{\frac{{\varepsilon_{j} }}{{(1 - \varepsilon_{j} )}}}} (1 - \varepsilon_{j} )\varpi_{t}^{j(\ell )} x_{t}^{\ell } } \hfill \\ &\quad + A_{t + 1}^{(N\backslash S)\ell } \delta_{t}^{t + 1} \left( {x_{t}^{\ell } + \frac{{a_{t}^{\ell } (A_{t + 1}^{(N\backslash S)\ell } a_{t}^{\ell } \delta_{t}^{t + 1} )}}{{2c_{t}^{\ell } }}x_{t}^{\ell } - \sigma_{t}^{\ell } x_{{}}^{\ell } } \right) \hfill \\& \quad + \delta_{t}^{t + 1} \sum\limits_{j \in K(\ell ) \cap N\backslash S}^{{}} {A_{t + 1}^{(N\backslash S)j} \lambda_{t}^{j(\ell )} x_{{}}^{\ell } }{\text{for}}\;\;\ell \in N\backslash S . \end{aligned}$$
(C5)
For (C4)-(C5) and Proposition 3 to hold, it is required that
$$\begin{aligned} & A_{T + 1}^{(S)i} = \;q_{T + 1}^{i(i)} , \\ & \quad A_{t}^{(S)i} = p_{t}^{i(i)} \left( {\frac{{\alpha_{i}^{{}} p_{t}^{i(i)} }}{{c_{t}^{i(i)} }}} \right)^{{\frac{{\alpha_{i} }}{{(1 - \alpha_{i} )}}}} (1 - \alpha_{i} ) - \frac{{(A_{t + 1}^{(S)i} a_{t}^{i} \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{i} }} + \sum\limits_{j \in K(i) \cap S}^{{}} {\gamma_{t}^{i(j)} } \hfill \\ &\quad + \sum\limits_{j \in K(i) \cap S}^{{}} {p_{t}^{j(i)} \left( {\frac{{\varepsilon_{j}^{{}} p_{t}^{j(i)} }}{{c_{t}^{j(i)} }}} \right)^{{\frac{{\varepsilon_{j} }}{{(1 - \varepsilon_{j} )}}}} (1 - \varepsilon_{j} )\varpi_{t}^{j(i)} } \hfill \\ &\quad + A_{t + 1}^{(S)i} \delta_{t}^{t + 1} \left( {(1 - \sigma_{t}^{i} ) + \frac{{a_{t}^{i} (A_{t + 1}^{(S)i} a_{t}^{i} \delta_{t}^{t + 1} )}}{{2c_{t}^{i} }}} \right)\; + \,\delta_{t}^{t + 1} \sum\limits_{j \in K(i) \cap S}^{{}} {A_{t + 1}^{(S)j} \lambda_{t}^{j(i)} ,} \hfill \end{aligned}$$
(C6)
for \(i \in S; \) and
$$\begin{aligned} & A_{T + 1}^{(N\backslash S)\ell } = \;q_{T + 1}^{\ell (\ell )} ,\\ & \quad A_{t}^{(N\backslash S)\ell } = \;p_{t}^{\ell (\ell )} \sum\limits_{\ell \in N\backslash S}^{{}} {\left( {\frac{{\alpha_{\ell }^{{}} p_{t}^{\ell (\ell )} }}{{c_{t}^{\ell (\ell )} }}} \right)^{{\frac{{\alpha_{\ell } }}{{(1 - \alpha_{\ell } )}}}} (1 - \alpha_{\ell } )\; - \frac{{(A_{t + 1}^{(N\backslash S)\ell } a_{t}^{\ell } \delta_{t}^{t + 1} )^{2} }}{{4c_{t}^{\ell } }}\, + \sum\limits_{j \in K(\ell ) \cap N\backslash S}^{{}} {\gamma_{t}^{\ell (j)} } } \hfill \\ & \qquad + \sum\limits_{j \in K(\ell ) \cap N\backslash S}^{{}} {p_{t}^{j(\ell )} \left( {\frac{{\varepsilon_{j}^{{}} p_{t}^{j(\ell )} }}{{c_{t}^{j(\ell )} }}} \right)^{{\frac{{\varepsilon_{j} }}{{(1 - \varepsilon_{j} )}}}} (1 - \varepsilon_{j} )\varpi_{t}^{j(\ell )} x_{t}^{\ell } } \hfill \\ & \qquad + A_{t + 1}^{(N\backslash S)\ell } \delta_{t}^{t + 1} \left( {x_{t}^{\ell } \; + \frac{{a_{t}^{\ell } (A_{t + 1}^{(S)\ell } a_{t}^{\ell } \delta_{t}^{t + 1} )}}{{2c_{t}^{\ell } }}x_{t}^{\ell } \; - \sigma_{t}^{\ell } x_{{}}^{\ell } } \right)\; + \delta_{t}^{t + 1} \sum\limits_{j \in K(\ell ) \cap N\backslash S}^{{}} {A_{t + 1}^{(N\backslash S)j} \;\lambda_{t}^{j(\ell )} ,} \hfill \\ & \quad \end{aligned}$$
(C7)
for \(\ell \in N\backslash S.\)
Finally, we can obtain \(C_{T + 1}^{(S)} = \;\sum\limits_{i \in S}^{{}} {C_{T + 1}^{i} }\) and \(C_{T + 1}^{(N\backslash S)} = \;\sum\limits_{\ell \in N\backslash S}^{{}} {C_{T + 1}^{\ell } }\),
$$C_{t}^{(S)} \; = \delta_{t}^{t + 1} C_{t + 1}^{(S)} \;{\text{and}} \;\;C_{t}^{(N\backslash S)} \; = \delta_{t}^{t + 1} C_{t + 1}^{(N\backslash S)} \;{\text{for}}\;t \in \{ 1,2, \cdots ,{T}\} .$$
(C8)
Hence Proposition 3 follows.
Appendix 4
Proof of Proposition 4
Consider two distinct sets—\(S_{1}^{{}}\) and \(S_{2}^{{}}\). The characteristic function \(v(S_{1}^{{}} \cup S_{2}^{{}} ;t,x)\) is the maximized value of the payoff of coalition \(S_{1}^{{}} \cup S_{2}^{{}}\) which is the solution to the problem
$$\begin{aligned} &\mathop{\mathop{\mathop{\mathop{\max}\limits_{u_{\tau }^{i},v_{\tau}^{i(j)}}}\limits_{j \in K(i) \cap (S_{1} \cup S_{2})}}\limits_{i \in S_{1} \cup S_{2}}}\limits_{\tau \in \{ t,t + 1, \cdots ,{T}\}} \left\{ {\sum\limits_{{i \in S_{1} \cup S_{2} }}{\left[ {\sum\limits_{\tau = t}^{T} {\left. \left( {p_{\tau }^{i(i)} (v_{\tau }^{i(i)} )^{{\alpha_{i} }} (x_{\tau }^{i} )^{{1 - \alpha_{i} }} - c_{\tau }^{i(i)} (v_{\tau }^{i(i)} ) - c_{\tau }^{i} (u_{\tau }^{i} )} \right. \right.} } \right.} } \right. \\& + \left. {\sum\limits_{{j \in K(i) \cap (S_{1} \cup S_{2} )}}{\left( {p_{\tau }^{i(j)} (v_{\tau }^{i(j)} )^{{\varepsilon_{i} }} (\varpi_{\tau }^{i(j)} x_{\tau }^{j} )^{{(1 - \varepsilon_{i} )}} - c_{\tau }^{i(j)} (v_{\tau }^{i(j)} ) + \gamma_{\tau }^{i(j)} x_{t}^{i} } \right)} } \right)\delta_{1}^{\tau } \\& + \left. {\left. {\left( {q_{T + 1}^{i(i)} x_{T + 1}^{i} + M_{T + 1}^{i} } \right)\delta_{1}^{T + 1} } \right. \left. {\sum\limits_{{i \in S_{1} \cup S_{2} }}}\right]} \right\} \end{aligned}$$
(D1)
s.t. state dynamics
$$x_{t + 1}^{i} = x_{t}^{i} + a_{t}^{i} (u_{t}^{i} x_{t}^{i} )^{1/2} + \sum\limits_{{j \in K(i) \cap (S_{1} \cup S_{2} )}}^{{}} {\lambda_{t}^{i(j)} x_{t}^{j} - \sigma_{t}^{i} x_{t}^{i} } .$$
(D2)
Note that in Theorem 3, the maximization problem of coalition \(S_{1}^{{}} \cup S_{2}^{{}}\) is independent of the maximization problem of coalition \(N\backslash (S_{1}^{{}} \cup S_{2}^{{}} )\). Hence the characteristic function \(v(S_{1}^{{}} \cup S_{2}^{{}} ;t,x)\) can be obtained by solving the problem (D1)-(D2).
Similarly, the characteristic function \(v(S_{1}^{{}} ;t,x)\) is the maximized value of the payoff of coalition \(S_{1}^{{}}\) which is the solution to the problem
$$ \begin{aligned} & \mathop{\mathop{\mathop{\mathop{\max}\limits_{u_{\tau }^{i}
,v_{\tau }^{i(j)}}}\limits_{j \in K(i) \cap S_{1}}}\limits_{i \in
S_{1}}}\limits_{\tau \in \{ t,t + 1, \cdots ,{\text{T}}\}} \left\{ {\sum\limits_{{i \in S_{1} }} {\left[ {\sum\limits_{\tau = t}^{T} {\left({\sum_{11111}} {p_{\tau }^{i(i)} (v_{\tau }^{i(i)} )^{{\alpha_{i} }} (x_{\tau }^{i} )^{{1 - \alpha_{i} }} - c_{\tau }^{i(i)} (v_{\tau }^{i(i)} ) - c_{\tau }^{i} (u_{\tau }^{i} )} \right.} } \right.} } \right. \\& + \sum\limits_{{j \in K(i) \cap S_{1} }} {\left( {p_{\tau }^{i(j)} (v_{\tau }^{i(j)} )^{{\varepsilon_{i} }} (\varpi_{\tau }^{i(j)} x_{\tau }^{j} )^{{(1 - \varepsilon_{i} )}} - c_{\tau }^{i(j)} (v_{\tau }^{i(j)} ) + \gamma_{\tau }^{i(j)} x_{t}^{i} } \right)\left. \right)\delta_{1}^{\tau } } \\& + \left. {\left. {\left( {q_{T + 1}^{i(i)} x_{T + 1}^{i} + M_{T + 1}^{i} } \right)\delta_{1}^{T + 1} } {\sum_{11111}}\right]} {\sum_{11111}}\right\} \end{aligned} $$
(D3)
s.t. state dynamics
$$ x_{t + 1}^{i} = x_{t}^{i} + a_{t}^{i} (u_{t}^{i} x_{t}^{i} )^{1/2} + \sum\limits_{{j \in K(i) \cap S_{1} }}^{{}} {\lambda_{t}^{i(j)} x_{t}^{j} - \sigma_{t}^{i} x_{t}^{i} .} $$
(D4)
The characteristic function \(v(S_{2}^{{}} ;t,x)\) is the maximized value of the payoff of coalition \(S_{2}^{{}}\) which is the solution to the problem
$$ \begin{aligned} & \mathop{\mathop{\mathop{\mathop{\max}\limits_{u_{\tau }^{i}
,v_{\tau }^{i(j)}}}\limits_{j \in K(i) \cap S_{2}}}\limits_{i \in
S_{2}}}\limits_{\tau \in \{ t,t + 1, \cdots ,{\text{T}}\}} \left\{ {\sum\limits_{{i \in S_{2} }} {\left[ {\sum\limits_{\tau = t}^{T} {\left({\sum_{11111}} {p_{\tau }^{i(i)} (v_{\tau }^{i(i)} )^{{\alpha_{i} }} (x_{\tau }^{i} )^{{1 - \alpha_{i} }} - c_{\tau }^{i(i)} (v_{\tau }^{i(i)} ) - c_{\tau }^{i} (u_{\tau }^{i} )} \right.} } \right.} } \right. \\& + \left. {\sum\limits_{{j \in K(i) \cap S_{2} }}{\left( {p_{\tau }^{i(j)} (v_{\tau }^{i(j)} )^{{\varepsilon_{i} }} (\varpi_{\tau }^{i(j)} x_{\tau }^{j} )^{{(1 - \varepsilon_{i} )}} - c_{\tau }^{i(j)} (v_{\tau }^{i(j)} ) + \gamma_{\tau }^{i(j)} x_{t}^{i} } \right)} } \right)\delta_{1}^{\tau } \\& + \left. {\left. {\left( {q_{T + 1}^{i(i)} x_{T + 1}^{i} + M_{T + 1}^{i} } \right)\delta_{1}^{T + 1} } {\sum_{11111}}\right]} {\sum_{11111}}\right\} \end{aligned} $$
(D5)
s.t. state dynamics
$$ x_{t + 1}^{i} = x_{t}^{i} + a_{t}^{i} (u_{t}^{i} x_{t}^{i} )^{1/2} + \sum\limits_{{j \in K(i) \cap S_{2} }}^{{}} {\lambda_{t}^{i(j)} x_{t}^{j} - \sigma_{t}^{i} x_{t}^{i} .} $$
(D6)
Note that
(i) The payoff structure of coalition \(S_{1}^{{}} \cup S_{2}^{{}}\) exceeds the sum of the payoff structures of coalition \(S_{1}^{{}}\) and coalition \(S_{2}^{{}}\) by
$$\begin{gathered} \sum\limits_{\tau = 1}^{T} {\left( {\sum\limits_{{i \in S_{1} \cup S_{2} }} {\sum\limits_{{j \in K(i) \cap (S_{1} \cup S_{2} )}} {\left( {p_{\tau }^{i(j)} (v_{\tau }^{i(j)} )^{{\varepsilon_{i} }} (\varpi_{\tau }^{i(j)} x_{\tau }^{j} )^{{(1 - \varepsilon_{i} )}} - c_{\tau }^{i(j)} (v_{\tau }^{i(j)} ) + \gamma_{\tau }^{i(j)} x_{t}^{i} } \right)} } } \right)\delta_{1}^{\tau } } \hfill \\ - \left[ {\sum\limits_{\tau = 1}^{T} {\left( {\sum\limits_{{i \in S_{1} }} {\sum\limits_{{j \in K(i) \cap S_{1} }} {\left( {p_{\tau }^{i(j)} (v_{\tau }^{i(j)} )^{{\varepsilon_{i} }} (\varpi_{\tau }^{i(j)} x_{\tau }^{j} )^{{(1 - \varepsilon_{i} )}} - c_{\tau }^{i(j)} (v_{\tau }^{i(j)} ) + \gamma_{\tau }^{i(j)} x_{t}^{i} } \right)} } } \right)} \delta_{1}^{\tau } } \right. \hfill \\ + \left. {\sum\limits_{\tau = 1}^{T} {\left( {\sum\limits_{{i \in S_{2} }} {\sum\limits_{{j \in K(i) \cap S_{2} }} {\left( {p_{\tau }^{i(j)} (v_{\tau }^{i(j)} )^{{\varepsilon_{i} }} (\varpi_{\tau }^{i(j)} x_{\tau }^{j} )^{{(1 - \varepsilon_{i} )}} - c_{\tau }^{i(j)} (v_{\tau }^{i(j)} ) + \gamma_{\tau }^{i(j)} x_{t}^{i} } \right)} } } \right)} \;\delta_{1}^{\tau } } \right] \hfill \\ = \left[ {\sum\limits_{\tau = 1}^{T} {\left( {\sum\limits_{{i \in S_{1} }} {\sum\limits_{{j \in K(i) \cap S_{2} }} {\left( {p_{\tau }^{i(j)} (v_{\tau }^{i(j)} )^{{\varepsilon_{i} }} (\varpi_{\tau }^{i(j)} x_{\tau }^{j} )^{{(1 - \varepsilon_{i} )}} - c_{\tau }^{i(j)} (v_{\tau }^{i(j)} ) + \gamma_{\tau }^{i(j)} x_{t}^{i} } \right)} } } \right)} \;\delta_{1}^{\tau } } \right. \hfill \\ + \sum\limits_{\tau = 1}^{T} {\left. {\;\left( {\sum\limits_{{i \in S_{2} }} {\;\sum\limits_{{j \in K(i) \cap S_{1} }} {\left( {p_{\tau }^{i(j)} (v_{\tau }^{i(j)} )^{{\varepsilon_{i} }} (\varpi_{\tau }^{i(j)} x_{\tau }^{j} )^{{(1 - \varepsilon_{i} )}} - c_{\tau }^{i(j)} (v_{\tau }^{i(j)} ) + \gamma_{\tau }^{i(j)} x_{t}^{i} } \right)} } } \right)\delta_{1}^{\tau } } \right]} . \hfill \\ \end{gathered}$$
(D7)
(ii) the state dynamics of coalition \(S_{1}^{{}} \cup S_{2}^{{}}\) given in (D2) entail more technology spillover effects than the state dynamics of coalition \(S_{1}^{{}}\) and coalition \(S_{2}^{{}}\) in (D4) and (D6).
(iii) Hence the optimization scheme (D.1)-(D.2) would yield a higher payoff than the sum of payoffs from optimization (D.3)-(D.4) and optimization scheme (D5)-(D6).
Therefore, \(v(S_{1}^{{}} \cup S_{2}^{{}} ;t,x) \geqslant v(S_{1}^{{}} ;t,x) + v(S_{2}^{{}} ;t,x)\).
