Optimization implementation of solution concepts for cooperative games with stochastic payoffs

Abstract

In this paper, we study solution concepts for cooperative games with stochastic payoffs. we define four kinds of solution concepts, namely the most coalitional (marginal) stable solution and the fairest coalitional (marginal) solution, by minimizing the total variance of excesses of coalitions (individual players). All these four concepts are optimal solutions of corresponding optimal problem under the least square criterion. It turns out that the fairest coalitional (marginal) solution belongs to the set of the most coalitional (marginal) stable solutions. Inspired by the definition of nucleolus, we propose various extended nucleolus based on lexicographic criterion. Furthermore, axiomatizations of the proposed solutions are exhibited through the linkage between the stochastic and deterministic models.

Appendices

Proof of Theorem 3.1

Proof

By calculating the Hessian matrix, it can be checked that the objective function of Problem $\widetilde{\text {1}}$ is convex, and so is the feasible set. Hence, there exists at most one optimal solution of Problem $\widetilde{\text {1}}$. It remains to verify the Lagrange conditions. The Lagrange function of Problem $\widetilde{\text {1}}$ is

$$\begin{aligned} L(r,\lambda )=\sum \limits _{S\subseteq N}(r^{2}(S)\mathbf{Var }[{\mathcal {V}}(N)]-2r(S)cov({\mathcal {V}}(S),{\mathcal {V}}(N)))+\lambda (r(N)-1). \end{aligned}$$

(37)

Taking partial derivative of $r_{i}$ in equation (37), then the Lagrangian conditions are

$$\begin{aligned} L_{i}(r,\lambda )=2\sum \limits _{S\ni i}(r(S)\mathbf{Var }[{\mathcal {V}}(N)]-cov({\mathcal {V}}(S),{\mathcal {V}}(N)))+\lambda =0, \forall i\in N. \end{aligned}$$

(38)

For any $i,j\in N$, we then have

$$\begin{aligned} \sum \limits _{S\ni i}r(S)\mathbf{Var }[{\mathcal {V}}(N)]-\sum \limits _{S\ni i}cov({\mathcal {V}}(S),{\mathcal {V}}(N))= & {} \sum \limits _{S\ni j}r(S)\mathbf{Var }[{\mathcal {V}}(N)]\nonumber \\&-\sum \limits _{S\ni j}cov({\mathcal {V}}(S),{\mathcal {V}}(N)). \end{aligned}$$

(39)

Denote $\sum \nolimits _{S\ni i}cov({\mathcal {V}}(S),{\mathcal {V}}(N))$ as $a_{i}({\mathcal {V}})$. Furthermore,

$$\begin{aligned} \sum \nolimits _{S\ni i}r(S)\mathbf{Var }[{\mathcal {V}}(N)]= & {} 2^{n-1}r_{i}+2^{n-2}(r(N)-r_{i})\mathbf{Var }[{\mathcal {V}}(N)]\nonumber \\= & {} 2^{n-2}(r_{i}+1)\mathbf{Var }[{\mathcal {V}}(N)]. \end{aligned}$$

(40)

Together with equation (40), the optimal solution of equations (39) is

$$\begin{aligned} r^{*}_{i}=\frac{1}{n}+\frac{na_{i}({\mathcal {V}})-\sum \nolimits _{j\in N}a_{j}({\mathcal {V}})}{n2^{n-2}\mathbf{Var }[{\mathcal {V}}(N)]},\quad \forall i\in N. \end{aligned}$$

(41)

In conclusion, the optimal solution of Problem 1 is given by

$$\begin{aligned} x^{*}_{i}=d^{*}_{i}+\left( \frac{1}{n}+\frac{na_{i}({\mathcal {V}})-\sum \nolimits _{j\in N}a_{j}({\mathcal {V}})}{n2^{n-2}\mathbf{Var }[{\mathcal {V}}(N)]}\right) ({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)]), \forall i\in N, \end{aligned}$$

(42)

where $d^{*}$ satisfies $\sum \nolimits _{i\in N}d^{*}_{i}=\mathbf{E }[{\mathcal {V}}(N)]$. $\square$

Proof of Lemma 3.1

Proof

Denote the objective function of Problem 2 as $P_{2}$, then we have

$$\begin{aligned} P_{2}= & {} \sum _{S\subseteq N}\mathbf{E }[(({\mathcal {V}}(S)-x(S))^{2}+{\bar{e}}({\mathcal {V}})^{2}-2({\mathcal {V}}(S)-x(S)){\bar{e}}({\mathcal {V}}))]\\= & {} \underbrace{\sum _{S\subseteq N}\mathbf{E }[(({\mathcal {V}}(S)-x(S))^{2}]}_{P_{21}}+\underbrace{\sum _{S\subseteq N}\mathbf{E }[{\bar{e}}({\mathcal {V}})^{2}]}_{P_{22}}-\underbrace{2\sum _{S\subseteq N}\mathbf{E }[({\mathcal {V}}(S)-x(S)){\bar{e}}({\mathcal {V}})]}_{P_{23}}\\= & {} P_{21}+P_{22}+P_{23}. \end{aligned}$$

Next we will make concretely analysis of the three parts respectively.

$$\begin{aligned} P_{21}= & {} \sum _{S\subseteq N}\mathbf{E }[(({\mathcal {V}}(S)-x(S))^{2}]\\= & {} \sum _{S\subseteq N}\mathbf{E }[({\mathcal {V}}(S)-d(S)-r(S){\mathcal {V}}(N)+r(S)\mathbf{E }[{\mathcal {V}}(N)])^{2}]\\= & {} \sum _{S\subseteq N}\mathbf{E }[({\mathcal {V}}(S)-d(S))^{2}+r^{2}(S)\mathbf{Var }[x(N)]\\&-2r(S)({\mathcal {V}}(S)-d(S))({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)])]\\= & {} \sum _{S\subseteq N}(\mathbf{E }[{\mathcal {V}}^{2}(S)]-2d(S)\mathbf{E }[{\mathcal {V}}(S)]+d^{2}(S)\\&+r^{2}(S)\mathbf{VarD} [{\mathcal {V}}(N)]-2r(S)cov({\mathcal {V}}(S),{\mathcal {V}}(N))). \end{aligned}$$

According to Proposition 3.1, the average excess ${\bar{e}}({\mathcal {V}})$ ia actually irrelevant to the allocation x, that is, it contains no unknown quantities d and r. Therefore eliminating the item $P_{22}$ from $P_{2}$ does not make any difference to the optimal solution of Problem 2. As to the last item $P_{23}$, we have

$$\begin{aligned} P_{23}= & {} 2\sum _{S\subseteq N}\mathbf{E }[({\mathcal {V}}(S)-x(S)){\bar{e}}({\mathcal {V}})]\\= & {} 2\sum _{S\subseteq N}\mathbf{E }[({\mathcal {V}}(S)-x(S)])(\frac{1}{2^{n}-1}\sum _{T\subseteq N}{\mathcal {V}}(T)-\frac{2^{n-1}}{2^{n}-1}{\mathcal {V}}(N))]\\= & {} \frac{2}{2^{n}-1}\sum _{S\subseteq N}\mathbf{E }[\sum _{T\subseteq N}{\mathcal {V}}(T)({\mathcal {V}}(S)-x(S)])]\\&-\frac{2^{n}}{2^{n}-1}\sum _{S\subseteq N}\mathbf{E }[({\mathcal {V}}(S)-x(S)]){\mathcal {V}}(N)]\\= & {} \frac{2}{2^{n}-1}\underbrace{\sum _{S\subseteq N}\mathbf{E }[\sum _{T\subseteq N}{\mathcal {V}}(T)({\mathcal {V}}(S)-d(S)-r(S){\mathcal {V}}(N)+r(S)\mathbf{E }[{\mathcal {V}}(N)])]}_{P^{1}_{23}}\\&-\frac{2^{n}}{2^{n}-1}\underbrace{\sum _{S\subseteq N}\mathbf{E }[({\mathcal {V}}(S)-d(S)-r(S){\mathcal {V}}(N)+r(S)\mathbf{E }[{\mathcal {V}}(N)]){\mathcal {V}}(N)]}_{P^{2}_{23}}\\= & {} \frac{2}{2^{n}-1}P^{1}_{23}-\frac{2^{n}}{2^{n}-1}P^{2}_{23}. \end{aligned}$$

Further, the first item $P^{1}_{23}$ can be rewrite as follows:

$$\begin{aligned} P^{1}_{23}= & {} \sum _{S\subseteq N}\mathbf{E }[\sum _{T\ne N}{\mathcal {V}}(T)({\mathcal {V}}(S)-d(S)-r(S){\mathcal {V}}(N)+r(S)\mathbf{E }[{\mathcal {V}}(N)])]\\&+\sum _{S\subseteq N}\mathbf{E }[{\mathcal {V}}(S){\mathcal {V}}(N)-d(S){\mathcal {V}}(N)-r(S){\mathcal {V}}^{2}(N)+r(S)\mathbf{E }[{\mathcal {V}}(N)]{\mathcal {V}}(N)]\\= & {} \sum _{S\subseteq N}\sum _{T\ne N}(\mathbf{E }[{\mathcal {V}}(T){\mathcal {V}}(S)]-d(S)\mathbf{E }[{\mathcal {V}}(T)]-r(S)cov({\mathcal {V}}(T),{\mathcal {V}}(N)))\\&+\sum _{S\subseteq N}(\mathbf{E }[{\mathcal {V}}(S){\mathcal {V}}(N)]-d(S)\mathbf{E }[{\mathcal {V}}(N)]-r(S)\mathbf{Var }[{\mathcal {V}}(N)]). \end{aligned}$$

Observe that the last item of the last equation is equivalent to $P^{2}_{23}$, thus we have

$$\begin{aligned} P_{23}= & {} \frac{2}{2^{n}-1}\sum _{S\subseteq N}\sum _{T\ne N}(\mathbf{E }[{\mathcal {V}}(T){\mathcal {V}}(S)]-d(S)\mathbf{E }[{\mathcal {V}}(T)]-r(S)cov({\mathcal {V}}(T),{\mathcal {V}}(N)))\\&-\frac{2^{n}-2}{2^{n}-1}\sum _{S\subseteq N}(\mathbf{E }[{\mathcal {V}}(S){\mathcal {V}}(N)]-d(S)\mathbf{E }[{\mathcal {V}}(N)]-r(S)\mathbf{Var }[{\mathcal {V}}(N)])\\= & {} \frac{2}{2^{n}-1}\sum _{S\subseteq N}(\sum _{T\ne N}(\mathbf{E }[{\mathcal {V}}(T){\mathcal {V}}(S)]-d(S)\mathbf{E }[{\mathcal {V}}(T)]-r(S)cov({\mathcal {V}}(T),{\mathcal {V}}(N)))\\&+(1-2^{n-1})(\mathbf{E }[{\mathcal {V}}(S){\mathcal {V}}(N)]-d(S)\mathbf{E }[{\mathcal {V}}(N)])+(2^{n-1}-1)r(S)\mathbf{Var }[{\mathcal {V}}(N)])\\= & {} \frac{2}{2^{n}-1}\sum _{S\subseteq N}(\sum _{T\subseteq N}(\mathbf{E }[{\mathcal {V}}(T){\mathcal {V}}(S)]-d(S)\mathbf{E }[{\mathcal {V}}(T)]-r(S)cov({\mathcal {V}}(T),{\mathcal {V}}(N)))\\&-2^{n-1}(\mathbf{E }[{\mathcal {V}}(S){\mathcal {V}}(N)]-d(S)\mathbf{E }[{\mathcal {V}}(N)]-r(S)\mathbf{Var }[{\mathcal {V}}(N)]). \end{aligned}$$

Note that the optimal solution of Problem 2 will not change by eliminating the constant items and corresponding positive coefficients in the objective function $P_{2}$. Denote the simplified objective function as $\widetilde{P_{2}}$, i.e.,

$$\begin{aligned} \widetilde{P_{2}}= & {} \sum _{S\subseteq N}(\mathbf{Var }[{\mathcal {V}}(N)]r^{2}(S)+d^{2}(S) +\left( \frac{2}{2^{n}-1}\sum _{T\subseteq N}\mathbf{E }[{\mathcal {V}}(T)]-2\mathbf{E }[{\mathcal {V}}(S)]\right. \\&\left. -\frac{2^{n}}{2^{n}-1}\mathbf{E }[{\mathcal {V}}(N)]\right) d(S)\\&+\left( \frac{2}{2^{n}-1}\sum _{T\subseteq N}cov({\mathcal {V}}(T),{\mathcal {V}}(N))-2cov({\mathcal {V}}(S),{\mathcal {V}}(N))-\frac{2^{n}}{2^{n}-1}\mathbf{Var }[{\mathcal {V}}(N)]\right) r(S)). \end{aligned}$$

In conclusion, Problem 2 is equivalent to the following modified problem:

Problem $\widetilde{\text {2}}$: Minimize $\widetilde{P_{2}}$ s.t. $x\in DR({\mathcal {V}})$. $\square$

Proof of Theorem 3.2

Proof

Similar to the proof of Theorem 3.1, one could verify that there exists at most one optimal solution of Problem ${\widetilde{2}}$. Imposing the Lagrangian multiplication factors $\lambda ,\omega$, we obtain the Lagrangian function of this problem

$$\begin{aligned} L(x,\lambda ,\omega )=\widetilde{P_{2}}+\lambda (d(N)-\mathbf{E }[{\mathcal {V}}(N)])+\omega (r(N)-1). \end{aligned}$$

(43)

Taking partial derivative of $d_{i}$ in equation (43), the Lagrangian conditions are

$$\begin{aligned} L_{d_{i}}(x,\lambda ,\omega )= & {} \sum _{S\ni i}\left( 2d(S)-2\mathbf{E }[{\mathcal {V}}(S))+\frac{2}{2^{n}-1}\sum _{T\subseteq N}\mathbf{E }[{\mathcal {V}}(T)]\right. \\&\left. -\frac{2^{n}}{2^{n}-1}\mathbf{E }[{\mathcal {V}}(N)]\right) +\lambda =0,\quad \forall i\in N. \end{aligned}$$

Let $e_{i}({\mathcal {V}})$ denote $\sum \nolimits _{S\ni i}\mathbf{E }[{\mathcal {V}}(S)]$. The Lagrangian conditions immediately give the following equations:

$$\begin{aligned} \sum _{S\ni i}d(S)-e_{i}({\mathcal {V}})=\sum _{S\ni j}d(S)-e_{j}({\mathcal {V}}), \quad \forall i,j\in N. \end{aligned}$$

(44)

Together with the constraint condition $d(N)=\mathbf{E }[{\mathcal {V}}(N)]$, we can obtain the solution $d_{i}^{*}$ of equations (44).

$$\begin{aligned} d_{i}^{*}=\frac{1}{n}\mathbf{E }[{\mathcal {V}}(N)]+\frac{1}{n2^{n-2}}[ne_{i}({\mathcal {V}})-\sum _{j\in N}e_{j}({\mathcal {V}})],\quad \forall i\in N. \end{aligned}$$

(45)

Taking the partial derivative of $r_{i}$ in equation (43), then the Lagrangian conditions of r are

$$\begin{aligned} L_{r_{i}}(x,\lambda ,\omega )= & {} \sum _{S\ni i}(2\mathbf{Var }[{\mathcal {V}}(N)]r(S)-\frac{2}{2^{n}-1}\sum _{T\subseteq N}cov({\mathcal {V}}(T),{\mathcal {V}}(N))\\&-2cov({\mathcal {V}}(S),{\mathcal {V}}(N))-\frac{2^{n}}{2^{n}-1}\mathbf{Var }[{\mathcal {V}}(N)])+\omega =0, \end{aligned}$$

which is equivalent to the following conditions: For all $i,j\in N$, it holds that

$$\begin{aligned}&\sum _{S\ni i}r(S)\mathbf{Var }[{\mathcal {V}}(N)]-\sum \limits _{S\ni i}cov({\mathcal {V}}(S),{\mathcal {V}}(N))\nonumber \\&\quad =\sum _{S\ni j}r(S)\mathbf{Var }[{\mathcal {V}}(N)]-\sum \limits _{S\ni j}cov({\mathcal {V}}(S),{\mathcal {V}}(N)). \end{aligned}$$

(46)

Similar to the analysis of equation (39), the optimal solution of equations (46) is

$$\begin{aligned} r^{*}_{i}=\frac{1}{n}+\frac{na_{i}({\mathcal {V}})-\sum \limits _{j\in N}a_{j}({\mathcal {V}})}{n2^{n-2}\mathbf{Var }[{\mathcal {V}}(N)]},\quad \forall i\in N. \end{aligned}$$

(47)

$\square$

Proof of Theorem 4.1

Proof

Denote the objective function of Problem 3 as $P_{3}$, then we have

$$\begin{aligned} P_{3}= & {} \sum _{i\in N}\sum _{S\ni i}\mathbf{E }[({\mathcal {V}}(S)-\mathbf{E }[{\mathcal {V}}(S)]-{\mathcal {V}}(S\backslash i)+\mathbf{E }[{\mathcal {V}}(S\backslash i)]-(x_{i}-\mathbf{E }[x_{i}]))^{2}]\\= & {} \sum _{i\in N}\sum _{S\ni i}\mathbf{E }[({\mathcal {V}}(S)-\mathbf{E }[{\mathcal {V}}(S)]-{\mathcal {V}}(S\backslash i)+\mathbf{E }[{\mathcal {V}}(S\backslash i)])^{2}+(x_{i}-\mathbf{E }[x_{i}])^{2}\\&-2\sum _{i\in N}\sum _{S\ni i}({\mathcal {V}}(S)-\mathbf{E }[{\mathcal {V}}(S)]-{\mathcal {V}}(S\backslash i)+\mathbf{E }[{\mathcal {V}}(S\backslash i)])(x_{i}-\mathbf{E }[x_{i}])]\\= & {} \sum _{i\in N}\sum _{S\ni i}(\mathbf{Var }[{\mathcal {V}}(S)]+\mathbf{Var }[{\mathcal {V}}(S\backslash i)])-2\sum _{i\in N}\sum _{S\ni i}cov({\mathcal {V}}(S),{\mathcal {V}}(S\backslash i))\\&+\sum _{i\in N}\sum _{S\ni i}r_{i}^{2}\mathbf{Var }[{\mathcal {V}}(N)]-2\sum _{i\in N}\sum _{S\ni i}r_{i}[cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N))]. \end{aligned}$$

Notice that the first two summations are constants for any given $(N,{\mathcal {V}})\in SG(N)$, hence the optimal solution of Problem 3 is equivalent to that of the following problem:

Problem $\widetilde{\text {3}}$: Minimize $\sum \nolimits _{i\in N}\sum \nolimits _{S\ni i}r_{i}^{2}\mathbf{Var }[{\mathcal {V}}(N)]-2\sum \nolimits _{i\in N}\sum \nolimits _{S\ni i}r_{i}[cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N))]$ s.t. $\sum \nolimits _{i\in N}r_{i}=1$.

We could verify that the objective function of Problem ${\tilde{3}}$ is convex and so is the feasible set, hence there exists at most one optimal point. Imposing the Lagrangian multiplication factors $\lambda$, then the Lagrangian function of Problem ${\tilde{3}}$ is

$$\begin{aligned} L(x,\lambda ,\omega )= & {} \sum \limits _{i\in N}\sum \limits _{S\ni i}r_{i}^{2}\mathbf{Var }[{\mathcal {V}}(N)]-2\sum \limits _{i\in N}\sum \limits _{S\ni i}r_{i}[cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N))]\nonumber \\&+\lambda (r(N)-1). \end{aligned}$$

(48)

Taking partial derivative of $r_{i}$, the Lagrangian conditions are

$$\begin{aligned} L_{r_{i}}(x,\lambda ,\omega )= & {} \sum _{S\ni i}2r_{i}\mathbf{Var }[{\mathcal {V}}(N)]\nonumber \\&-\sum _{S\ni i}2[cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N))]+\lambda =0, \forall i\in N. \end{aligned}$$

(49)

Denote $\sum \nolimits _{S\ni i}cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N))$ as $b_{i}({\mathcal {V}})$, then the above equations imply the following relations among players:

$$\begin{aligned} \sum _{S\ni i}r_{i}\mathbf{Var }[{\mathcal {V}}(N)]-b_{i}({\mathcal {V}})=\sum _{S\ni j}r_{j}\mathbf{Var }[{\mathcal {V}}(N)]-b_{j}({\mathcal {V}}), \forall i,j\in N. \end{aligned}$$

(50)

The solution of equations (50) is $r^{*}_{i}=\frac{1}{n}+\frac{nb_{i}({\mathcal {V}})-\sum \nolimits _{j\in N}b_{j}({\mathcal {V}})}{n2^{n-2}\mathbf{Var }[{\mathcal {V}}(N)]},\forall i\in N$, which complete the proof. $\square$

Proof of Lemma 4.1

Proof

Denote the objective function of Problem 4 as $P_{4}$, then we have

$$\begin{aligned} P_{4}= & {} \sum \limits _{i\in N}\sum \limits _{S\ni i}\mathbf{E }[(m_{i}(S,x)-{\overline{m}}({\mathcal {V}}))^{2}]\\= & {} \sum _{i\in N}\sum _{S\ni i}\mathbf{E }[m^{2}_{i}(S,x)+{\overline{m}}^{2}({\mathcal {V}})-2m_{i}(S,x){\overline{m}}({\mathcal {V}})]\\= & {} \sum _{i\in N}\sum _{S\ni i}\mathbf{E }[m^{2}_{i}(S,x)]+\underbrace{\sum _{i\in N}\sum _{S\ni i}\mathbf{E }[{\overline{m}}^{2}({\mathcal {V}})]}_{P_{4}^{1}}-\underbrace{2\sum _{i\in N}\sum _{S\ni i}\mathbf{E }[m_{i}(S,x){\overline{m}}({\mathcal {V}})]}_{P_{4}^{2}} \end{aligned}$$

Based on Proposition 4.1, ${\overline{m}}({\mathcal {V}})$ merely depends on the given game ${\mathcal {V}}$. So elimination of $P_{4}^{1}$ has no effect no the optimal solution of Problem 4. Now we focus on the item $P_{4}^{2}$, which can be rewritten as:

$$\begin{aligned} P_{4}^{2}= & {} 2\sum _{i\in N}\sum _{S\ni i}\mathbf{E }[({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)-x_{i}){\overline{m}}({\mathcal {V}})]\\= & {} 2\sum _{i\in N}\sum _{S\ni i}\mathbf{E }[({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)){\overline{m}}({\mathcal {V}})]\\&-2\sum _{i\in N}\sum _{S\ni i}d_{i}\mathbf{E }[{\overline{m}}({\mathcal {V}})]-2\sum _{i\in N}\sum _{S\ni i}r_{i}\mathbf{E }[({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)]){\overline{m}}({\mathcal {V}})]\\= & {} 2\sum _{i\in N}\sum _{S\ni i}\mathbf{E }[({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)){\overline{m}}({\mathcal {V}})]-2^{n}\mathbf{E }[{\mathcal {V}}(N)]\mathbf{E }[{\overline{m}}({\mathcal {V}})]\\&-2^{n}\mathbf{E }[({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)]){\overline{m}}({\mathcal {V}})] \end{aligned}$$

The last two equations are derived from the fact that $d(N)=\mathbf{E }[\{{\mathcal {V}}(N)]$ and $r(N)=1$. Once the game is given, $P_{4}^{2}$ is uniquely determined. therefore eliminating the constant items $P_{4}^{1}$ and $P_{4}^{2}$ from $P_{4}$ exerts no influence on the optimal solution with respect to Problem 4, which completes the proof. $\square$

Proof of Theorem 4.2

Proof of Theorem 4.2

Denote the objective function of Problem 5 as $P_{5}$, then we have

$$\begin{aligned} P_{5}= & {} \sum _{i\in N}\sum _{S\ni i}\mathbf{E }[({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)-d_{i}-r_{i}({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)])^{2}]\\= & {} \sum _{i\in N}\sum _{S\ni i}\mathbf{E }[({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)-d_{i})^{2}+r_{i}^{2}(({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)])^{2}\\&-2r_{i}({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)-d_{i})({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)])]\\= & {} \sum _{i\in N}\sum _{S\ni i}(\mathbf{E }[({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i))^{2}]+d_{i}^{2}\\&-2d_{i}\mathbf{E }[{\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)]+r_{i}^{2}\mathbf{Var }[{\mathcal {V}}(N)]\\&-2r_{i}cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N)]))\\= & {} \sum _{i\in N}\sum _{S\ni i}(\mathbf{E }[({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i))^{2}]+d_{i}^{2}+r_{i}^{2}\mathbf{Var }[{\mathcal {V}}(N)])\\&-2\sum _{i\in N}d_{i}c_{i}({\mathcal {V}})-2\sum _{i\in N}r_{i}b_{i}({\mathcal {V}}) \end{aligned}$$

where $b_{i}({\mathcal {V}})=\sum \nolimits _{S\ni i}cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N))$ and $c_{i}({\mathcal {V}})=\sum \nolimits _{S\ni i}\mathbf{E }[{\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)]$. Due to the convexity of objective function $P_{5}$ and the feasible set, there exists at most one optimal point. Imposing the Lagrangian multiplication factors $\lambda ,\omega$, the Lagrangian function of Problem 5 is

$$\begin{aligned} L(x,\lambda ,\omega )=P_{5}+\lambda (d(N)-\mathbf{E }[{\mathcal {V}}(N)])+\omega (r(N)-1). \end{aligned}$$

(51)

Taking partial derivative of $d_{i}$ and $r_{i}$ in equation (51), the Lagrangian conditions are

$$\begin{aligned}&L_{d_{i}}(x,\lambda ,\omega )=2\sum _{S\ni i}\ d_{i}-2c_{i}({\mathcal {V}})+\lambda =0, \forall i\in N. \end{aligned}$$

(52)

$$\begin{aligned}&L_{r_{i}}(x,\lambda ,\omega )=2\sum _{S\ni i}r_{i}\mathbf{Var }[{\mathcal {V}}(N)]-2b_{i}({\mathcal {V}})+\omega =0, \forall i\in N. \end{aligned}$$

(53)

Equations in (52) and (53) imply the following relations among players,

$$\begin{aligned}&\sum _{S\ni i}\ d_{i}-c_{i}({\mathcal {V}})=\sum _{S\ni j}\ d_{j}-c_{j}({\mathcal {V}}), \forall i,j\in N. \end{aligned}$$

(54)

$$\begin{aligned}&\sum _{S\ni i}r_{i}\mathbf{Var }[{\mathcal {V}}(N)]-b_{i}({\mathcal {V}})=\sum _{S\ni j}r_{j}\mathbf{Var }[{\mathcal {V}}(N)]-b_{j}({\mathcal {V}}), \forall i,j\in N. \end{aligned}$$

(55)

The optimal solution of equations (54) and (55) are

$$\begin{aligned} r^{*}_{i}=\frac{1}{n}+\frac{nb_{i}({\mathcal {V}})-\sum \limits _{j\in N}b_{j}({\mathcal {V}})}{n2^{n-2}\mathbf{Var }[{\mathcal {V}}(N)]}, \quad d^{*}_{i}=\frac{1}{n}\mathbf{E }[{\mathcal {V}}(N)]+\frac{nc_{i}({\mathcal {V}})-\sum \limits _{j\in N}c_{j}({\mathcal {V}})}{n2^{n-2}}, \forall i\in N. \end{aligned}$$

$\square$

Extension of the models

For any weight function m, let us consider the following problems, which are extensions of optimization models in Section 3 and 4.

Problem 6: Minimize $\sum \nolimits _{S\subseteq N}m(s)\mathbf{Var }[e(S,x)]$ s.t. $x\in DR({\mathcal {V}})$.

Problem 7: Minimize $\sum \nolimits _{S\subseteq N}m(s)\mathbf{E }[(e(S,x)-{\bar{e}}({\mathcal {V}},x))^{2}]$ s.t. $x\in DR({\mathcal {V}})$.

Problem 8: Minimize $\sum \nolimits _{i\in N}\sum \nolimits _{S\ni i}m(s)\mathbf{E }[(m_{i}(S,x)-\mathbf{E }[m_{i}(S,x)])^{2}]$ s.t. $x\in DR({\mathcal {V}})$.

Problem 9: Minimize $\sum \nolimits _{i\in N}\sum \nolimits _{S\ni i}m(s)\mathbf{E }[(m_{i}(S,x)-{\overline{m}}({\mathcal {V}},x))^{2}]$ s.t. $x\in DR({\mathcal {V}})$.

The method to obtain the optimal solutions of the above Problems is the same as that of Theorem 3.1, 3.2, 4.1 and 4.2. Here we omit the detailed and lengthy proofs of the following results.

Proposition G.1

For any $(N,{\mathcal {V}})\in SG(N)$ and weight function $m:{\mathcal {P}}(N)\backslash \emptyset \rightarrow {\mathbb {R}}$, the optimal solution of Problem 6 is

$$\begin{aligned} x^{*}_{i}=d^{*}_{i}+\left( \frac{1}{n}+\frac{na^{m}_{i}({\mathcal {V}})-\sum \limits _{j\in N}a^{m}_{j}({\mathcal {V}})}{n\alpha \mathbf{Var }[{\mathcal {V}}(N)]}\right) ({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)]), \end{aligned}$$

where $a^{m}_{i}({\mathcal {V}})=\sum \nolimits _{S\ni i}m(s)cov({\mathcal {V}}(S),{\mathcal {V}}(N))$, $\alpha =\sum \nolimits _{s=1}^{n-1}C_{n-2}^{s-1}m(s)$ and $d^{*}$ satisfies $\sum \nolimits _{i\in N}d^{*}_{i}=\mathbf{E }[{\mathcal {V}}(N)]$.

Proposition G.2

For any $(N,{\mathcal {V}})\in SG(N)$ and weight function $m:{\mathcal {P}}(N)\backslash \emptyset \rightarrow {\mathbb {R}}$, there exists a unique optimal solution $x^{*}$ of Problem 7, which is

$$\begin{aligned} x^{*}_{i}=d^{*}_{i}+r^{*}_{i}({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)]), \end{aligned}$$

where

$$\begin{aligned}&r^{*}_{i}=\frac{1}{n}+\frac{na^{m}_{i}({\mathcal {V}})-\sum \limits _{j\in N}a^{m}_{j}({\mathcal {V}})}{n\alpha \mathbf{Var }[{\mathcal {V}}(N)]},\forall i\in N, \\&d_{i}^{*}=\frac{1}{n}\mathbf{E }[{\mathcal {V}}(N)]+\frac{ne^{m}_{i}({\mathcal {V}})-\sum _{j\in N}e^{m}_{j}({\mathcal {V}})}{n\alpha },\forall i\in N, \end{aligned}$$

$a^{m}_{i}({\mathcal {V}})=\sum \nolimits _{S\ni i}m(s)cov({\mathcal {V}}(S),{\mathcal {V}}(N))$, $\alpha =\sum \nolimits _{s=1}^{n-1}C_{n-2}^{s-1}m(s)$ and $e^{m}_{i}({\mathcal {V}})=\sum \nolimits _{S\ni i}m(s)\mathbf{E }[{\mathcal {V}}(S)]$.

Proposition G.3

For any $(N,{\mathcal {V}})\in SG(N)$ and weight function $m:{\mathcal {P}}(N)\backslash \emptyset \rightarrow {\mathbb {R}}$, the optimal solution of Problem 8 is

$$\begin{aligned} x^{*}_{i}=d^{*}_{i}+\left( \frac{1}{n}+\frac{nb^{m}_{i}({\mathcal {V}})-\sum \limits _{j\in N}b^{m}_{j}({\mathcal {V}})}{n\alpha \mathbf{Var }[{\mathcal {V}}(N)]}\right) ({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)]), \end{aligned}$$

where $b^{m}_{i}({\mathcal {V}})=\sum \nolimits _{S\ni i}m(s)cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N))$, $\alpha =\sum \nolimits _{s=1}^{n-1}C_{n-2}^{s-1}m(s)$ and $d^{*}$ satisfies $\sum \nolimits _{i\in N}d^{*}_{i}=\mathbf{E }[{\mathcal {V}}(N)]$.

Proposition G.4

For any $(N,{\mathcal {V}})\in SG(N)$ and weight function $m:{\mathcal {P}}(N)\backslash \emptyset \rightarrow {\mathbb {R}}$, there exists a unique optimal solution $x^{*}$ of Problem 9, which is

$$\begin{aligned} x^{*}_{i}=d^{*}_{i}+r^{*}_{i}({\mathcal {V}}(N)-\mathbf{E }[{\mathcal {V}}(N)]), \end{aligned}$$

where

$$\begin{aligned} r^{*}_{i}&=\frac{1}{n}+\frac{nb^{m}_{i}({\mathcal {V}})-\sum \limits _{j\in N}b^{m}_{j}({\mathcal {V}})}{n\alpha \mathbf{Var }[{\mathcal {V}}(N)]},\forall i\in N,\\ d_{i}^{*}&=\frac{1}{n}\mathbf{E }[{\mathcal {V}}(N)]+\frac{nc^{m}_{i}({\mathcal {V}})-\sum \limits _{j\in N}c^{m}_{j}({\mathcal {V}})}{n2^{n-2}},\forall i\in N, \end{aligned}$$

$b^{m}_{i}({\mathcal {V}})=\sum \nolimits _{S\ni i}m(s)cov({\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i),{\mathcal {V}}(N))$, $\alpha =\sum \nolimits _{s=1}^{n-1}C_{n-2}^{s-1}m(s)$ and $c^{m}_{i}({\mathcal {V}})=\sum \nolimits _{S\ni i}m(s)\mathbf{E }[{\mathcal {V}}(S)-{\mathcal {V}}(S\backslash i)]$.

Once the weight function is taken into account, the optimal solutions are highly unified with the original solutions in form. Models in Section 3 and 4 can be seen as the special cases when all the coalitions have the same weight. In this way, we extend the most coalitional (marginal) stable and the fairest coalitional (marginal) solutions and obtain a family of solution concepts for stochastic cooperative games.