Consider the weighted voting system [11: 7, 4, 1]

Find the Shapley-Shubik power distribution of this weighted voting system.

List the power for each player as a fraction:
p1: 
  
p2: 
  
p3: 
  

The Shapley-Shubik power index is a way to measure the power of each player in a weighted voting system. To find the Shapley-Shubik power distribution, we need to consider all possible permutations of the players and determine the pivotal player in each permutation. The pivotal player is the first one in the permutation whose addition makes the total votes exceed the quota (11 in this case).

The voting system is \[11: 7, 4, 1\], meaning:

- Player 1 has 7 votes
- Player 2 has 4 votes
- Player 3 has 1 vote

### Steps to calculate the Shapley-Shubik power index:

1. **List all possible permutations** of the players.
2. **Identify the pivotal player** in each permutation (the player who causes the sum to meet or exceed 11).
3. **Count the pivotal occurrences** for each player.
4. **Calculate the Shapley-Shubik power index** by dividing the number of times each player is pivotal by the total number of permutations.

#### Step 1: Permutations of the players
The players can be ordered in \( 3! = 6 \) different ways:

- (P1, P2, P3)
- (P1, P3, P2)
- (P2, P1, P3)
- (P2, P3, P1)
- (P3, P1, P2)
- (P3, P2, P1)

#### Step 2: Identify the pivotal player
For each permutation, we check when the total number of votes reaches or exceeds the quota (11 votes).

- **(P1, P2, P3)**: P1 has 7 votes, P2 adds 4 (7 + 4 = 11), so **P2 is pivotal**.
- **(P1, P3, P2)**: P1 has 7 votes, P3 adds 1 (7 + 1 = 8), but P2 adds 4 (8 + 4 = 12), so **P2 is pivotal**.
- **(P2, P1, P3)**: P2 has 4 votes, P1 adds 7 (4 + 7 = 11), so **P1 is pivotal**.
- **(P2, P3, P1)**: P2 has 4 votes, P3 adds 1 (4 + 1 = 5), but P1 adds 7 (5 + 7 = 12), so **P1 is pivotal**.
- **(P3, P1, P2)**: P3 has 1 vote, P1 adds 7 (1 + 7 = 8), but P2 adds 4 (8 + 4 = 12), so **P2 is pivotal**.
- **(P3, P2, P1)**: P3 has 1 vote, P2 adds 4 (1 + 4 = 5), but P1 adds 7 (5 + 7 = 12), so **P1 is pivotal**.

#### Step 3: Count the pivotal occurrences
- Player 1 (P1) is pivotal in 3 permutations.
- Player 2 (P2) is pivotal in 3 permutations.
- Player 3 (P3) is pivotal in 0 permutations.

#### Step 4: Calculate the Shapley-Shubik power index
Since there are 6 total permutations, the power index for each player is:

- \( p_1 = \frac{3}{6} = \frac{1}{2} \)
- \( p_2 = \frac{3}{6} = \frac{1}{2} \)
- \( p_3 = \frac{0}{6} = 0 \)

### Final Shapley-Shubik power distribution:
- Player 1: \( \frac{1}{2} \)
- Player 2: \( \frac{1}{2} \)
- Player 3: \( 0 \)

Would you like more details or have any further questions?

Here are 5 related questions:
1. How does the Shapley-Shubik index differ from the Banzhaf power index?
2. What happens if we change the quota in the voting system?
3. How can the Shapley-Shubik power index be used in real-world decision-making scenarios?
4. What does it mean for a player to have zero power in a voting system?
5. Can the Shapley-Shubik index be generalized for larger voting systems?

**Tip**: The Shapley-Shubik index considers the order of players in a coalition, which makes it useful for analyzing decision-making where sequence matters.

Consider the weighted voting system [11: 7, 4, 1]. Find the Shapley-Shubik power distribution of this weighted voting system. List the power for each player as a fraction.

Learn how to calculate the Shapley-Shubik power distribution for the weighted voting system [11: 7, 4, 1]. This tutorial walks through identifying pivotal players in different permutations and provides the Shapley-Shubik power index for each player. Suitable for students studying game theory and voting systems at the undergraduate level.

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