Find the Shapley-Shubik power index for the weighted voting system [36: 20, 17, 15]

To calculate the **Shapley-Shubik power index** for the weighted voting system \([36: 20, 17, 15]\), we follow these steps:

### Step 1: Define the voting system
- **Quota** = 36
- **Weights of voters**:
  - \( W_1 = 20 \)
  - \( W_2 = 17 \)
  - \( W_3 = 15 \)

This means we have 3 players with voting weights 20, 17, and 15 respectively, and the decision is passed if the total votes exceed or equal 36.

### Step 2: List all permutations of the players
There are \(3!\) (i.e., \(3 \times 2 \times 1 = 6\)) different possible orders in which the players can vote:

- \( P_1, P_2, P_3 \)
- \( P_1, P_3, P_2 \)
- \( P_2, P_1, P_3 \)
- \( P_2, P_3, P_1 \)
- \( P_3, P_1, P_2 \)
- \( P_3, P_2, P_1 \)

### Step 3: Identify pivotal players
A pivotal player is the one whose vote causes the total to reach or exceed the quota (36). We calculate the cumulative total for each order and see which player is pivotal:

1. \( P_1, P_2, P_3 \):
   - \( P_1 = 20 \) (total = 20, not enough)
   - \( P_2 = 17 \) (total = 20 + 17 = 37, **pivotal**)
   - \( P_3 = 15 \) (not pivotal)
   
2. \( P_1, P_3, P_2 \):
   - \( P_1 = 20 \) (total = 20, not enough)
   - \( P_3 = 15 \) (total = 20 + 15 = 35, not enough)
   - \( P_2 = 17 \) (total = 35 + 17 = 52, **pivotal**)
   
3. \( P_2, P_1, P_3 \):
   - \( P_2 = 17 \) (total = 17, not enough)
   - \( P_1 = 20 \) (total = 17 + 20 = 37, **pivotal**)
   - \( P_3 = 15 \) (not pivotal)
   
4. \( P_2, P_3, P_1 \):
   - \( P_2 = 17 \) (total = 17, not enough)
   - \( P_3 = 15 \) (total = 17 + 15 = 32, not enough)
   - \( P_1 = 20 \) (total = 32 + 20 = 52, **pivotal**)
   
5. \( P_3, P_1, P_2 \):
   - \( P_3 = 15 \) (total = 15, not enough)
   - \( P_1 = 20 \) (total = 15 + 20 = 35, not enough)
   - \( P_2 = 17 \) (total = 35 + 17 = 52, **pivotal**)
   
6. \( P_3, P_2, P_1 \):
   - \( P_3 = 15 \) (total = 15, not enough)
   - \( P_2 = 17 \) (total = 15 + 17 = 32, not enough)
   - \( P_1 = 20 \) (total = 32 + 20 = 52, **pivotal**)

### Step 4: Count pivotal occurrences
Now, count how many times each player is pivotal:

- \( P_1 \) (Weight 20) is pivotal in: 3 out of 6 permutations.
- \( P_2 \) (Weight 17) is pivotal in: 2 out of 6 permutations.
- \( P_3 \) (Weight 15) is pivotal in: 1 out of 6 permutations.

### Step 5: Calculate the Shapley-Shubik power index
The power index is the fraction of times each player is pivotal:

- \( \text{Power of } P_1 = \frac{3}{6} = 0.50 \)
- \( \text{Power of } P_2 = \frac{2}{6} = 0.33 \)
- \( \text{Power of } P_3 = \frac{1}{6} = 0.17 \)

### Conclusion:
The **Shapley-Shubik power index** for the system \([36: 20, 17, 15]\) is:
- \( P_1: 0.50 \)
- \( P_2: 0.33 \)
- \( P_3: 0.17 \)

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Would you like more details or clarifications on any part? Here are some follow-up questions to consider:

1. How is the quota (36) chosen in a weighted voting system?
2. How does the Shapley-Shubik index differ from the Banzhaf power index?
3. What are some real-life applications of the Shapley-Shubik index?
4. How would the power distribution change if the weights or quota were adjusted?
5. What are other methods for determining power in voting systems?

**Tip:** In weighted voting systems, the quota is typically set to over 50% of the total weight to ensure decisions are passed by a majority.

Discover how to calculate the Shapley-Shubik power index for the weighted voting system [36: 20, 17, 15]. This solution covers game theory concepts, including player pivotal roles and permutations, making it suitable for advanced high school and university students.

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