To find the Shapley-Shubik power distribution for the weighted voting system [tex]\([21: 12, 7, 3, 1]\)[/tex], we need to determine the fraction of pivotal roles each player has in forming a winning coalition. Here is a step-by-step solution to arrive at the answer:
1. Define the System:
- Quota (total weight needed to win): [tex]\(21\)[/tex]
- Weights of players:
- Player 1: [tex]\(12\)[/tex]
- Player 2: [tex]\(7\)[/tex]
- Player 3: [tex]\(3\)[/tex]
- Player 4: [tex]\(1\)[/tex]
2. Identify All Permutations:
- There are [tex]\(4\)[/tex] players, so there are [tex]\(4! = 24\)[/tex] possible permutations of the players.
3. Determine Pivotal Player for Each Permutation:
- The pivotal player in a permutation is the one who causes the total weight to meet or exceed the quota when added.
- We list out all [tex]\(24\)[/tex] permutations and analyze which player is pivotal in each.
4. Count Pivotal Instances:
- After analyzing each permutation, we find the number of times each player is pivotal.
- In this system:
- Player 1 is pivotal in 8 permutations.
- Player 2 is pivotal in 8 permutations.
- Player 3 is pivotal in 8 permutations.
- Player 4 is pivotal in 0 permutations.
5. Calculate Shapley-Shubik Index:
- The index for each player can be calculated as the fraction of total pivotal instances divided by total permutations.
- Total permutations: [tex]\(24\)[/tex]
- Each player's Shapley-Shubik power index is calculated as follows:
[tex]\[
\sigma_i = \frac{\text{Number of times player } i \text{ is pivotal}}{\text{Total permutations}}
\][/tex]
6. Substitute Values:
[tex]\[
\sigma_1 = \frac{8}{24} = \frac{1}{3}
\][/tex]
[tex]\[
\sigma_2 = \frac{8}{24} = \frac{1}{3}
\][/tex]
[tex]\[
\sigma_3 = \frac{8}{24} = \frac{1}{3}
\][/tex]
[tex]\[
\sigma_4 = \frac{0}{24} = 0
\][/tex]
7. Conclusion:
- The Shapley-Shubik power indices for the players are as follows:
[tex]\[
\sigma_1 = \frac{1}{3}, \sigma_2 = \frac{1}{3}, \sigma_3 = \frac{1}{3}, \sigma_4 = 0
\][/tex]
Thus, the Shapley-Shubik power distribution for [tex]\([21: 12, 7, 3, 1]\)[/tex] is:
[tex]\[
\sigma_1 = \frac{1}{3}, \sigma_2 = \frac{1}{3}, \sigma_3 = \frac{1}{3}, \sigma_4 = 0
\][/tex]
