The Banzhaf power index measures the power of each voter (or party) in a weighted voting system. The Banzhaf power of a voter is calculated by determining how many times that voter is critical to the outcome of a vote.

We have the weighted voting system $[10: 6, 5, 1]$, which means:

• The quota (minimum number of votes needed to win) is 10.
• Player 1 has 6 votes.
• Player 2 has 5 votes.
• Player 3 has 1 vote.

Step-by-Step Process

Step 1: List all possible coalitions

A coalition is any combination of players. For 3 players, the possible coalitions are:

1. $\{1\}$
2. $\{2\}$
3. $\{3\}$
4. $\{1, 2\}$
5. $\{1, 3\}$
6. $\{2, 3\}$
7. $\{1, 2, 3\}$

Step 2: Determine which coalitions are winning

A winning coalition has a total of votes equal to or greater than the quota (10 votes).

• $\{1\} = 6$ (Not winning)
• $\{2\} = 5$ (Not winning)
• $\{3\} = 1$ (Not winning)
• $\{1, 2\} = 6 + 5 = 11$ (Winning)
• $\{1, 3\} = 6 + 1 = 7$ (Not winning)
• $\{2, 3\} = 5 + 1 = 6$ (Not winning)
• $\{1, 2, 3\} = 6 + 5 + 1 = 12$ (Winning)

So, the winning coalitions are $\{1, 2\}$ and $\{1, 2, 3\}$.

Step 3: Identify critical players

A critical player is one who, if removed from a winning coalition, causes the coalition to lose.

• In $\{1, 2\}$, both Player 1 and Player 2 are critical. If either is removed, the coalition no longer has 10 votes.
• In $\{1, 2, 3\}$, both Player 1 and Player 2 are critical. Removing Player 1 results in a total of 6 votes, and removing Player 2 results in a total of 7 votes, both of which are less than 10. Player 3 is not critical because removing them still leaves 11 votes, which is more than 10.

Step 4: Count critical occurrences

• Player 1 is critical in both $\{1, 2\}$ and $\{1, 2, 3\}$, so Player 1 is critical 2 times.
• Player 2 is critical in both $\{1, 2\}$ and $\{1, 2, 3\}$, so Player 2 is critical 2 times.
• Player 3 is not critical in any coalition.

Step 5: Compute Banzhaf power index

The Banzhaf power index for each player is calculated as the ratio of times the player is critical to the total number of critical player occurrences.

• Player 1: $\frac{2}{4} = 0.5$
• Player 2: $\frac{2}{4} = 0.5$
• Player 3: $\frac{0}{4} = 0$

Thus, the Banzhaf power distribution is:

$\text{Player 1: } 0.5, \text{ Player 2: } 0.5, \text{ Player 3: } 0$

Would you like further explanation, or to proceed with part (b)? Also, here are 5 questions to expand on this topic:

1. How does the Banzhaf power index compare to the Shapley-Shubik index in voting systems?
2. Can the Banzhaf power distribution ever be equal for all players?
3. What real-world systems rely on weighted voting mechanisms?
4. How would changing the quota in this system affect the Banzhaf power distribution?
5. How is the Banzhaf power index used in political science and economics?

Tip: The Banzhaf index is most useful when analyzing situations where coalitions form unpredictably, as it assumes each coalition is equally likely to form.