Find the Banzhaf power distribution of the weighted voting system
(33: 19, 15, 14, 3]
Give each player's power as a fraction or decimal value
P1 =


To find the Banzhaf power distribution of the weighted voting system \((33: 19, 15, 14, 3)\), we need to calculate the power index for each player. The Banzhaf Power Index measures the number of times a player can change an outcome from a loss to a win by joining a coalition. Here are the steps:

### Step 1: Define the System
- The quota is \(33\).
- The players' weights are: 
  - Player 1: \(19\)
  - Player 2: \(15\)
  - Player 3: \(14\)
  - Player 4: \(3\)

### Step 2: List All Possible Coalitions
All subsets of players need to be considered to see which coalitions are winning and identify each player’s critical status (i.e., when their vote turns a losing coalition into a winning one).

### Step 3: Identify Winning Coalitions and Critical Players
For each coalition, determine if it meets or exceeds the quota of \(33\), and identify which player(s) are critical.

### Step 4: Count the Banzhaf Power for Each Player
We will count how many times each player is critical across all winning coalitions.

### Step 5: Normalize the Banzhaf Index
Normalize the index by dividing each player’s count by the total number of critical counts.

Let's calculate the Banzhaf Power Index for each player.The Banzhaf power distribution for the weighted voting system \((33: 19, 15, 14, 3)\) is as follows:

- **P1 (19 votes):** 0.6
- **P2 (15 votes):** 0.2
- **P3 (14 votes):** 0.2
- **P4 (3 votes):** 0.0

This distribution shows that Player 1 has the most power, while Player 4 has no power since their vote never changes an outcome from losing to winning.

Would you like a detailed explanation of how each player's criticality was determined? Here are some related questions:

1. How is the Banzhaf Power Index calculated for each player?
2. What does it mean for a player to be critical in a coalition?
3. How would changing the quota affect the power distribution?
4. Can Player 4 gain power with a different set of weights?
5. How does the Banzhaf Index compare to other power indices, like Shapley-Shubik?

**Tip:** In weighted voting systems, players with weights near the quota are often critical more frequently, significantly impacting their power index.

Find the Banzhaf power distribution of the weighted voting system (33: 19, 15, 14, 3). Give each player's power as a fraction or decimal value. P1 =

Learn how to calculate the Banzhaf power distribution for the weighted voting system (33: 19, 15, 14, 3). This problem covers voting theory, combinatorics, and power indices, with detailed calculations for each player's power as a fraction or decimal. Suitable for students studying discrete mathematics or political science.

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