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 like \([51: 30, 30, 30, 10]\), it's tempting to think that the power each player has is proportional to their number of votes. However, power in such a system isn't just about the number of votes; it's about the ability to influence outcomes. Player 4 has 10% of the votes, but they might not have 10% of the power because the power depends on whether they are critical to forming a winning coalition.

In this system, the winning threshold is 51 votes. Player 4's 10 votes won't help form a winning coalition unless they join others with enough votes to reach 51. The other three players each have 30 votes, and two of them together already have 60 votes, which is more than the required 51. So, Player 4 is often not needed to form a winning coalition, meaning they have less power than their vote count suggests.

Would you like further details on how power is actually calculated, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How do we calculate the power of each player using the Banzhaf power index?
2. Can Player 4 have more power in different weighted voting systems?
3. What happens to Player 4's power if the quota (51) is changed?
4. How can a player with fewer votes sometimes have more power than others with more votes?
5. What is a dictator in a weighted voting system, and is there one here?

**Tip:** Power in weighted voting systems is often measured using tools like the Banzhaf power index, which counts how often a player is critical in winning coalitions.

Understand why Player 4 in the weighted voting system [51: 30, 30, 30, 10] doesn't have 10% of the power despite having 10% of the votes. This explanation dives into the concepts of coalition formation and the Banzhaf Power Index. Suitable for advanced high school or undergraduate students.

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