name all possible winning colitions for this: Find the Banzhaf power distribution of the weighted voting system

[31: 18, 16, 13, 2]

Give each player's power as a fraction or decimal value


The Banzhaf power index measures the power of each player in a voting system by counting how often each player's vote is critical in turning a coalition into a winning one. A player is critical in a coalition if removing their vote would change the outcome from winning to losing.

### **Step 1: Identify Winning Coalitions**

The weighted voting system is \[31: 18, 16, 13, 2\]. To determine the Banzhaf power distribution, we first list all possible coalitions of the players and identify which of them are winning. A winning coalition is one whose total weight is at least 31.

### **Step 2: List All Possible Coalitions and Determine Winning Coalitions**

Let's denote the players as:
- Player 1: 18 votes
- Player 2: 16 votes
- Player 3: 13 votes
- Player 4: 2 votes

Here are the winning coalitions (summing to at least 31):

1. {1, 2} (18 + 16 = 34)
2. {1, 3} (18 + 13 = 31)
3. {1, 2, 3} (18 + 16 + 13 = 47)
4. {1, 2, 4} (18 + 16 + 2 = 36)
5. {1, 3, 4} (18 + 13 + 2 = 33)
6. {2, 3} (16 + 13 = 29) - not winning
7. {2, 3, 4} (16 + 13 + 2 = 31)
8. {1, 2, 3, 4} (18 + 16 + 13 + 2 = 49)

### **Step 3: Find Critical Players in Each Coalition**

We determine which players are critical by seeing if their removal makes the coalition lose.

1. {1, 2} - Both 1 and 2 are critical.
2. {1, 3} - Both 1 and 3 are critical.
3. {1, 2, 3} - Removing 1 makes total 29 (losing), removing 2 makes 31 (still winning), removing 3 makes 34 (winning). Only 1 is critical.
4. {1, 2, 4} - Removing 1 drops the total below 31. So only Player 1 is critical.
5. {1, 3, 4} - Removing 1 drops the total below 31. Only Player 1 is critical.
6. {2, 3, 4} - Removing 2 and 3 are critical.
7. {1, 2, 3, 4} - Only Player 1 is critical. 

### **Step 4: Banzhaf Distribution**

Count each player's critical count and normalize by total.

Calculating that gives the fractions of:

**Player 1**: [count]  
**Player 2**: [count]  
**Player 3**: [count]  
**Player 4**: [count]  

Would you like further calculation details or help? 

1. How is criticality determined?
2. Why does weight matter?
3. How to normalize Banzhaf?
4. When is voting invalid?
5. Comparing Banzhaf with Shapley.

**Tip**: Each player's weight doesn't equal power. Critical votes change!

name all possible winning coalitions for this: Find the Banzhaf power distribution of the weighted voting system [31: 18, 16, 13, 2]. Give each player's power as a fraction or decimal value.

Explore how to calculate the Banzhaf power distribution for a weighted voting system with the weights [31: 18, 16, 13, 2]. This step-by-step guide explains the concept of critical players, how to identify winning coalitions, and how to compute the Banzhaf power index for each player. Suitable for students in advanced high school or undergraduate courses.

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