https://file.aiquickdraw.com/mathbot/file/17266140491066u8h1pfe.jpg

Let's break down and solve the problem shown in the image, which involves a weighted voting system \([17; 14, 13, 3, 2]\). Here's a step-by-step explanation:

### Problem Summary:
- **Weighted Voting System**: \([17; 14, 13, 3, 2]\)
  - The quota (minimum votes needed to pass a motion) is **17**.
  - Player 1 (P₁) has **14** votes.
  - Player 2 (P₂) has **13** votes.
  - Player 3 (P₃) has **3** votes.
  - Player 4 (P₄) has **2** votes.

#### (a) **Critical Players in Coalition (P₁, P₂, P₃)**
To identify the critical players, let's analyze the coalition of P₁, P₂, and P₃:
- The total votes of this coalition is \(14 + 13 + 3 = 30\).
- If any player is removed, we need to check if the coalition still meets the quota of 17 votes:
  - Without P₁: \(13 + 3 = 16\) (fails to meet the quota).
  - Without P₂: \(14 + 3 = 17\) (exactly meets the quota).
  - Without P₃: \(14 + 13 = 27\) (still meets the quota).
  
Thus, **P₁ and P₃ are critical**.

#### (b) **Write Down All Winning Coalitions**
A coalition is winning if the sum of the players' votes is 17 or more. We need to list all combinations of players where the total vote count is at least 17:
- **P₁ alone**: \(14\) (not winning).
- **P₂ alone**: \(13\) (not winning).
- **P₁ + P₂**: \(14 + 13 = 27\) (winning).
- **P₁ + P₃**: \(14 + 3 = 17\) (winning).
- **P₁ + P₄**: \(14 + 2 = 16\) (not winning).
- **P₂ + P₃**: \(13 + 3 = 16\) (not winning).
- **P₂ + P₄**: \(13 + 2 = 15\) (not winning).
- **P₁ + P₂ + P₃**: \(14 + 13 + 3 = 30\) (winning).
- **P₁ + P₂ + P₄**: \(14 + 13 + 2 = 29\) (winning).
- **P₁ + P₃ + P₄**: \(14 + 3 + 2 = 19\) (winning).
- **P₂ + P₃ + P₄**: \(13 + 3 + 2 = 18\) (winning).
- **P₁ + P₂ + P₃ + P₄**: \(14 + 13 + 3 + 2 = 32\) (winning).

Thus, the winning coalitions are:
- \(P₁ + P₂\)
- \(P₁ + P₃\)
- \(P₁ + P₂ + P₃\)
- \(P₁ + P₂ + P₄\)
- \(P₁ + P₃ + P₄\)
- \(P₂ + P₃ + P₄\)
- \(P₁ + P₂ + P₃ + P₄\)

#### (c) **Banzhaf Power Distribution**
The Banzhaf power index measures how often a player is critical in winning coalitions. To find the Banzhaf index for each player, we:
1. Identify all coalitions where a player is critical.
2. Calculate the total number of times each player is critical.
3. Determine the relative power by normalizing the results.

We will compute the Banzhaf power index for each player (denoted as \(\beta_1, \beta_2, \beta_3, \beta_4\)).

I'll compute these and get back to you!The Banzhaf power distribution for the players in the weighted voting system \([17; 14, 13, 3, 2]\) is as follows:

- \(\beta_1 = 0.417\)
- \(\beta_2 = 0.25\)
- \(\beta_3 = 0.25\)
- \(\beta_4 = 0.083\)

This means:
- Player 1 (P₁) has approximately 41.7% of the power.
- Player 2 (P₂) and Player 3 (P₃) each have 25% of the power.
- Player 4 (P₄) has 8.3% of the power.

Would you like any additional details or explanations?

Here are 5 related questions to expand on this topic:
1. What are some alternative methods for calculating power in a voting system?
2. How does the Shapley-Shubik index differ from the Banzhaf index?
3. What happens to the Banzhaf index if we change the quota to 15 instead of 17?
4. How do weighted voting systems impact decision-making in real-world applications (e.g., board meetings)?
5. Can a player with fewer votes ever have more power than a player with more votes?

**Tip**: When analyzing voting systems, understanding critical players can help predict coalitions and negotiation strategies in decision-making bodies.

Consider the weighted voting system [17; 14, 13, 3, 2]. (a) Which players are critical in the coalition {P1, P2, P3}? (b) Write down all winning coalitions. (c) Find the Banzhaf power distribution of this weighted voting system.

Learn how to analyze the weighted voting system [17; 14, 13, 3, 2], identify critical players, and calculate the Banzhaf power distribution. This problem covers voting systems, coalitions, and power indices, with a detailed solution for students in Grades 11-12 or early college.

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