find the shapley-shubik power distribution the the weighted voting system [6:5,4,2]

To find the Shapley-Shubik power distribution for the weighted voting system \([6:5, 4, 2]\), we'll go through the following steps:

1. **Identify the Players and Weights**:
   - Player 1: Weight = 5
   - Player 2: Weight = 4
   - Player 3: Weight = 2
   - Quota: 6 (The minimum number of votes required to pass a motion)

2. **List All Permutations of Players**:
   The total number of permutations of the three players is \(3! = 6\).
   The permutations are:
   - \(P_1, P_2, P_3\)
   - \(P_1, P_3, P_2\)
   - \(P_2, P_1, P_3\)
   - \(P_2, P_3, P_1\)
   - \(P_3, P_1, P_2\)
   - \(P_3, P_2, P_1\)

3. **Determine the Pivotal Player in Each Permutation**:
   The pivotal player is the one who, when added to the preceding players, meets or exceeds the quota.

   - **Permutation 1: \(P_1, P_2, P_3\)**
     - \(P_1\) (5) alone does not reach the quota.
     - \(P_1 + P_2\) (5 + 4 = 9) meets the quota. \(P_2\) is pivotal.
   - **Permutation 2: \(P_1, P_3, P_2\)**
     - \(P_1\) (5) alone does not reach the quota.
     - \(P_1 + P_3\) (5 + 2 = 7) meets the quota. \(P_3\) is pivotal.
   - **Permutation 3: \(P_2, P_1, P_3\)**
     - \(P_2\) (4) alone does not reach the quota.
     - \(P_2 + P_1\) (4 + 5 = 9) meets the quota. \(P_1\) is pivotal.
   - **Permutation 4: \(P_2, P_3, P_1\)**
     - \(P_2\) (4) alone does not reach the quota.
     - \(P_2 + P_3\) (4 + 2 = 6) meets the quota. \(P_3\) is pivotal.
   - **Permutation 5: \(P_3, P_1, P_2\)**
     - \(P_3\) (2) alone does not reach the quota.
     - \(P_3 + P_1\) (2 + 5 = 7) meets the quota. \(P_1\) is pivotal.
   - **Permutation 6: \(P_3, P_2, P_1\)**
     - \(P_3\) (2) alone does not reach the quota.
     - \(P_3 + P_2\) (2 + 4 = 6) meets the quota. \(P_2\) is pivotal.

4. **Count the Number of Times Each Player is Pivotal**:
   - Player 1: Pivotal in 2 out of 6 permutations.
   - Player 2: Pivotal in 2 out of 6 permutations.
   - Player 3: Pivotal in 2 out of 6 permutations.

5. **Calculate the Shapley-Shubik Power Index**:
   The Shapley-Shubik power index is calculated as the fraction of permutations in which each player is pivotal.
   - Player 1: \(\frac{2}{6} = \frac{1}{3}\)
   - Player 2: \(\frac{2}{6} = \frac{1}{3}\)
   - Player 3: \(\frac{2}{6} = \frac{1}{3}\)

So, the Shapley-Shubik power distribution for the weighted voting system \([6: 5, 4, 2]\) is:
- Player 1: \( \frac{1}{3} \)
- Player 2: \( \frac{1}{3} \)
- Player 3: \( \frac{1}{3} \)

Would you like further details or have any questions?

### Related Questions
1. What is the significance of the Shapley-Shubik power index?
2. How is the Shapley-Shubik power index different from the Banzhaf power index?
3. How can the Shapley-Shubik index be applied in real-world voting systems?
4. What are the computational challenges in calculating the Shapley-Shubik index for large voting systems?
5. How would the power distribution change if the quota were different in the given system?
6. Can the Shapley-Shubik power index ever be zero for a player?
7. How does the concept of pivotal players extend to cooperative game theory?
8. What are the limitations of the Shapley-Shubik power index?

### Tip
When calculating power indices in weighted voting systems, understanding the concept of pivotal players is crucial as it directly influences the power distribution.

Learn how to calculate the Shapley-Shubik power distribution for a weighted voting system with players having different weights. This problem involves applying the Shapley-Shubik Power Index Theorem to determine the power distribution among players in a voting system with a quota. Suitable for advanced students and those interested in cooperative game theory.

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