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The voting system described is \([19: 13, 6, 4, 2]\), where:

- 19 is the quota (the minimum number of votes needed to pass a motion),
- The players have the following weights: Player 1 has 13 votes, Player 2 has 6 votes, Player 3 has 4 votes, and Player 4 has 2 votes.

Let's analyze the system for the following aspects:

### a) Identifying Dictators
A **dictator** is a player whose votes alone are enough to meet or exceed the quota. If a player has more votes than or equal to the quota, they are a dictator.

In this system:
- The quota is 19.
- Player 1 has 13 votes, which is less than the quota, so Player 1 is not a dictator.
- None of the other players have enough votes to meet or exceed the quota on their own.

**Conclusion**: There are no dictators in this system.

### b) Identifying Players with Veto Power
A player has **veto power** if the motion cannot pass without their participation. In other words, if a player’s votes are necessary to meet the quota, they have veto power.

To see if any player has veto power, we need to check if removing any one player's votes makes it impossible to reach the quota of 19.

- Without Player 1 (13 votes), the remaining players have \(6 + 4 + 2 = 12\) votes, which is less than 19. Thus, Player 1 has veto power.
- Without Player 2 (6 votes), the remaining players have \(13 + 4 + 2 = 19\) votes, so Player 2 does not have veto power.
- Without Player 3 (4 votes), the remaining players have \(13 + 6 + 2 = 21\) votes, so Player 3 does not have veto power.
- Without Player 4 (2 votes), the remaining players have \(13 + 6 + 4 = 23\) votes, so Player 4 does not have veto power.

**Conclusion**: Player 1 has veto power.

### c) Identifying Dummies
A **dummy** is a player whose votes never affect the outcome. If a player’s votes are not needed to meet or exceed the quota in any coalition, they are a dummy.

- Player 1 has veto power, so they are not a dummy.
- Player 2's votes can be helpful in reaching the quota, for example, in a coalition of Player 1 and Player 2 (which totals \(13 + 6 = 19\)), so Player 2 is not a dummy.
- Player 3's votes can also help reach the quota, for example, in a coalition of Player 1, Player 3, and Player 4 (\(13 + 4 + 2 = 19\)), so Player 3 is not a dummy.
- Player 4's votes are needed in the coalition with Player 1 and Player 3 to reach 19 votes, so Player 4 is also not a dummy.

**Conclusion**: There are no dummies in this system.

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Would you like further details or explanations? Here are some related questions to explore:

1. How do you mathematically define the concept of veto power in weighted voting systems?
2. Can a system have more than one player with veto power?
3. How do you identify dummies in larger voting systems with more players?
4. What are the properties of a dictator in more complex voting systems?
5. How do changes in the quota affect the classification of players?

**Tip**: In weighted voting systems, identifying key players (dictators, veto players, and dummies) helps to understand the distribution of power and influence in decision-making processes.

Consider the weighted voting system [19: 13, 6, 4, 2].
a. Identify the dictators, if any.
b. Identify players with veto power, if any.
c. Identify dummies, if any.

Explore the weighted voting system [19: 13, 6, 4, 2] to identify dictators, players with veto power, and dummy players. Understand the power dynamics within the system, including when a player has the power to block or ensure the passage of a motion. Suitable for advanced high school and undergraduate students.

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