The Joint Belief Function and Shapley Value for the Joint Cooperative Game

Abstract

In this paper, the characterization of the joint distribution of random set vectors by the belief function is investigated and the joint game in terms of the characteristic function is given. The bivariate Shapley value of a joint cooperative game is obtained through both cores and games. Formulas for the Shapley value derived from two different methods are shown to be identical. For illustration of our main results, several examples are given.

Appendix

Proof of Lemma 4.4

Note that the left hand side of (18) is equal to

$$\begin{aligned} \phi _{ij}(\nu )&= \sum \limits _{A\subseteq E_1}\sum \limits _{B\subseteq E_2}c_{_{A,B}}\phi _{ij}(\omega _{A,B}) =\sum \limits _{A\ni i}\sum \limits _{B \ni j}\frac{c_{_{A,B}}}{|A||B|}\nonumber \\&= \sum \limits _{A\ni i}\sum \limits _{B \ni j}\frac{1}{|A||B|}\sum \limits _{C\subseteq A}\sum \limits _{D\subseteq B}(-1)^{|A{\setminus } C|+|B{\setminus } D|}\nu (C,D)\\&= \alpha _1-\alpha _2-\alpha _3+\alpha _4,\nonumber \end{aligned}$$

(19)

where

$$\begin{aligned} \alpha _1&= \sum \limits _{C\ni i}\sum \limits _{D\ni j}\left[ \sum \limits _{C\subseteq A}\sum \limits _{D\subseteq B}(-1)^{|A{\setminus } C|+|B{\setminus } D|}\frac{1}{|A||B|}\right] \nu (C,D)\\&= \sum \limits _{C\ni i}\sum \limits _{D\ni j}\left[ \sum \limits _{s=|C|}^{n_1}\sum \limits _{t=|D|}^{n_2}(-1)^{(s-|C|)+(t-|D|)} \frac{1}{st}\left( \begin{array}{c} n_1-|C|\\ s-|C|\\ \end{array} \right) \left( \begin{array}{c} n_2-|D|\\ t-|D|\\ \end{array} \right) \right] \nu (C,D), \end{aligned}$$

$$\begin{aligned} \alpha _2&= \sum \limits _{C\ni i}\sum \limits _{D\not \ni j}\left[ \sum \limits _{C\subseteq A}\sum \limits _{D\cup \{j\}\subseteq B}(-1)^{|A{\setminus } C|+|B{\setminus } D|}\frac{1}{|A||B|}\right] \nu (C,D)\\&= \sum \limits _{C\ni i}\sum \limits _{D\not \ni j}\left[ \sum \limits _{s=|C|}^{n_1}\sum \limits _{t=|D|+1}^{n_2}(-1)^{(s-|C|)+(t-|D|)} \frac{1}{st}\left( \begin{array}{c} n_1-|C|\\ s-|C|\\ \end{array} \right) \left( \begin{array}{c} n_2-|D|-1\\ t-|D|-1\\ \end{array} \right) \right] \\&\nu (C,D), \end{aligned}$$

$$\begin{aligned} \alpha _3&= \sum \limits _{C\not \ni i}\sum \limits _{D\ni j}\left[ \sum \limits _{C\cup \{i\}\subseteq A}\sum \limits _{D\subseteq B}(-1)^{|A{\setminus } C|+|B{\setminus } D|}\frac{1}{|A||B|}\right] \nu (C,D)\\&= \sum \limits _{C\not \ni i}\sum \limits _{D\ni j}\left[ \sum \limits _{s=|C|+1}^{n_1}\sum \limits _{t=|D|}^{n_2}(-1)^{(s-|C|)+(t-|D|)} \frac{1}{st}\left( \begin{array}{c} n_1-|C|-1\\ s-|C|-1\\ \end{array} \right) \left( \begin{array}{c} n_2-|D|\\ t-|D|\\ \end{array} \right) \right] \\&\nu (C,D), \end{aligned}$$

and

$$\begin{aligned} \alpha _4&= \sum \limits _{C\not \ni i}\sum \limits _{D\not \ni j}\left[ \sum \limits _{C\cup \{i\}\subseteq A}\sum \limits _{D\cup \{j\}\subseteq B}(-1)^{|A{\setminus } C|+|B{\setminus } D|}\frac{1}{|A||B|}\right] \nu (C,D)\\&= \sum \limits _{C\not \ni i}\sum \limits _{D\not \ni j}\left[ \sum \limits _{s=|C|+1}^{n_1}\sum \limits _{t=|D|+1}^{n_2}(-1)^{(s-|C|)+(t-|D|)}\right. \\&\left. \qquad \qquad \quad \cdot \frac{1}{st}\left( \begin{array}{c} n_1-|C|-1\\ s-|C|-1\\ \end{array} \right) \left( \begin{array}{c} n_2-|D|-1\\ t-|D|-1\\ \end{array} \right) \right] \nu (C,D). \end{aligned}$$

Now by using the following equality,

$$\begin{aligned} \sum \limits _{s=c}^{n}\frac{1}{s}(-1)^{s-c}\left( \begin{array}{c} n-c\\ s-c\\ \end{array} \right) =\frac{(n-c)!(c-1)!}{n!}, \end{aligned}$$

$\alpha _i$’s can be reduced so that the $\phi _{ij}(\nu )$ is

$$\begin{aligned} \phi _{ij}(\nu )&= \sum \limits _{C\ni i}\sum \limits _{D\ni j}\left[ \frac{(n_1-|C|)!(|C|-1)!}{n_1 !}\frac{(n_2-|D|)!(|D|-1)!}{n_2 !}\right] \nu (C,D)\\&\quad -\,\sum \limits _{C\ni i}\sum \limits _{D\not \ni j}\left[ \frac{(n_1-|C|)!(|C|-1)!}{n_1 !}\frac{(n_2-|D|-1)!|D|!}{n_2 !}\right] \nu (C,D)\\&\quad -\,\sum \limits _{C\not \ni i}\sum \limits _{D\ni j}\left[ \frac{(n_1-|C|-1)!|C|!}{n_1 !}\frac{(n_2-|D|)!(|D|-1)!}{n_2 !}\right] \nu (C,D)\\&\quad +\,\sum \limits _{C\not \ni i}\sum \limits _{D\not \ni j}\left[ \frac{(n_1-|C|-1)!|C|!}{n_1 !}\frac{(n_2-|D|-1)!|D|!}{n_2 !}\right] \nu (C,D). \end{aligned}$$

Therefore, the bivariate Shapley value formula is given, after simplification, by

$$\begin{aligned} \phi _{ij}(\nu )&= \sum \limits _{C\ni i}\sum \limits _{D\ni j}\left[ \frac{(n_1-|C|)!(|C|-1)!}{n_1 !}\frac{(n_2-|D|)!(|D|-1)!}{n_2 !}\right] \nonumber \\&\quad \times \,\left[ \nu (C,D)-\nu (C,D{\setminus }\{j\})-\nu (C{\setminus }\{i\},D)+\nu (C{\setminus }\{i\},D{\setminus }\{j\})\right] . \end{aligned}$$

(20)

Note that Shavley value given in (20) is equivalent to the one given by the cores of the joint belief function $H$, in (17).    $\square$