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I am facing a problem to understand the calculation of shapley value on the below example:

Question: The parliament of Micronesia is made up of four political parties, $A$, $B$, $C$, and $D$, which have $45$, $25$, $15$, and $15$ representatives, respectively. They are to vote on whether to pass a $\$100$ million spending bill and how much of this amount should be controlled by each of the parties. A majority vote, that is, a minimum of $51$ votes, is required in order to pass any legislation, and if the bill does not pass then every party gets zero to spend.

Solution: we get the payoff division $(50, 16.66, 16.66, 16.66)$, which adds up to the entire $\$100$ million.

I didn't understand the solution of it how those value comes? Need Help! Thanks in advance!

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The four parties $A, B, C$ and $D$ have $45,25,15$ and $15$ representatives in the parliament respectively. To pass a bill $51$ votes are necessary, called a quota. This is a weighted majority game. Let us denote the total number of votes as $w(N) = \sum_{k \in N}\, w_{k} \in \mathbb{N}$, hence $w(N)=100$. Here $N$ denotes the player set, that is, $\{A,B,C,D\}$. For passing a bill at least $0 < qt \le w(N)$ votes are needed. A simple game is referred to a weighted majority game, if there exists a quota $ qt > 0$ and weights $w_{k} \ge 0$ for all $k \in N$ such that for all $S \subseteq N$ it holds either $v(S) = 1$ if $w(S) \ge qt$, or $v(S) = 0$ otherwise. Such a game is formally represented as $[qt; w_{1}, \ldots, w_{n}]$. The weights vector in this example game is given by $\vec{w} =\{45,25,15,15\}$ and the quota by $qt=51$. Then the weighted majority game (TU game) with its coalitional values is given by

$$ sv =\{0,0,0,0,1,1,1,0,0,0,1,1,1,1,1\},$$

whereas the set of all permissible coalitions are $$\{\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},N\}.$$

From this game the Shapley value is computed, which is

$$sh_{sv}=\{1/2,1/6,1/6,1/6\}, $$

multiplying this result by $100$ million gives the "payoff division" $\{50,16.66,16.66,16.66\}$. This indicates the power of the parties in the parliament.

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As the question does not state that any of the parties defect, we will assume that all the teams voted in favor of the bill. therefore, the winning combination will be considered as ABCD.

First lets calculate the payoff value of A.

-if A is added first, it adds 0 to the payoff.

-if A is added second, it adds 100 mill to the payoff.(it does not matter what the first party was as no party can get majority)

-if A is added third, it adds 100 mill to the payoff.(it does not matter what the first 2 parties were as no 2 parties from B,C,D can get majority)

-if A is added forth, it adds 0 to the payoff.(because the other 3 parties can get a majority without A as 55>51)

Thus, payoff value for A is 1/4*(0+100+100+0)=50 mill.

lets calculate the payoff value for C

-if C is added first, it adds 0 to the payoff.

-if C is added second, it adds 100 mill to the payoff if first party was A and 0 otherwise. The probability of A being the first party to be added is 1/3(there are total 3 parties other than C).

-if C is added third, it adds 100 mill to the payoff if first 2 parties were B,D and 0 otherwise. The probability of B,D being in the first 2 parties to be added is 1/3(there are total 3 combination of parties that can be added in the first 2 places: A,B ; A,D ; B,D).

-if C is added forth, it adds 0 to the payoff.(because the other 3 parties can get a majority without A as 85>51)

Thus, payoff value for C is 1/4*[0+1/3(100)+1/3(100)+0]=16.66 mill.

The payoff value for D is same as the payoff value for C as they both have the same number of seats.

Thus, payoff value for D is 1/4*[0+1/3(100)+1/3(100)+0]=16.66 mill.

The payoff value for B can be calculated using the same method which is used to calculate the payoff value of C. Or the simpler way is to subtract the payoff value of the other 3 parties from the total money to be distibuted.

Thus, payoff value for B is 100-50-16.66-16.66 = 16.66 mill.

Thus, the distribution is (50,16.6,16.6,16.6)

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  • $\begingroup$ Can you please clarify how is this answer is lacking in the explaining the solution to the question? And can you please also point out if there is any mistake in my explanation. $\endgroup$ – Sneh Shah Jun 27 at 4:12

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