which would be used in this PPI or BPI: Consider the weighted voting system [51: 30, 30, 30, 10]. Write a few sentences explaining to a friend who hasn't taken this class why it might not make sense to say that Player 4 has 10% of the power in this system even though they have 10% of the votes. You can assume that your friend understands how a weighted voting system works.

In a weighted voting system, the power of a player isn’t just about the percentage of votes they control, but rather how often their vote is pivotal in deciding the outcome. This is where Player 4's situation comes in. While they hold 10% of the votes (10 out of 100), their actual power depends on how frequently their vote changes the result from losing to winning. In this system, the other three players each hold 30 votes, so most decisions could be made without Player 4. Therefore, Player 4's influence, or power, might be much less than 10%, even though their vote count is exactly 10% of the total. The player’s power is better captured by indices like the Banzhaf Power Index (BPI) or the Shapley-Shubik Power Index (PPI), which measure the player's real decision-making influence rather than just their share of the votes.

Would you like more details or have any questions?

Here are five related questions to consider:
1. How does the Banzhaf Power Index calculate the power of each player?
2. What is the difference between the BPI and the Shapley-Shubik Power Index?
3. Can Player 4 form any winning coalitions in the given system?
4. In which situations might Player 4 have a pivotal role despite their low vote count?
5. How would removing Player 4 affect the overall voting system?

**Tip:** In weighted voting, smaller players often gain power by being part of pivotal coalitions, not just by the number of votes they control.

Consider the weighted voting system [51: 30, 30, 30, 10]. Write a few sentences explaining to a friend who hasn't taken this class why it might not make sense to say that Player 4 has 10% of the power in this system even though they have 10% of the votes.

In this problem, we explore how Player 4, with 10% of the votes in the weighted voting system [51: 30, 30, 30, 10], might not have 10% of the power. Concepts such as the Banzhaf Power Index (BPI) and Shapley-Shubik Power Index (PPI) are introduced to explain voting power and coalition theory in detail.

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