Consider the weighted voting system [q: 9, 5, 1].

Which values of q result in a dictator (list all possible values)

In a weighted voting system \([q: 9, 5, 1]\), the players have weights 9, 5, and 1, respectively, and \(q\) represents the quota — the number of votes needed to pass a motion. To determine the values of \(q\) that result in a dictator, we need to find the values of \(q\) such that a single player has enough votes to pass a motion on their own, without needing any other players' votes.

A **dictator** is a player who can determine the outcome of a vote by themselves. For a player to be a dictator, their weight must be at least equal to the quota \(q\). The key idea is that the dictator's weight must be greater than or equal to \(q\), and no other combination of players should be able to meet or exceed \(q\) without the dictator.

Here are the possible cases for a dictator:

1. **Player 1 (Weight = 9):**
   - Player 1 will be a dictator if \(q \leq 9\), because they can pass any motion on their own.
   - For Player 1 to be the only dictator, \(q\) must be greater than the combined weight of Player 2 and Player 3 (which is \(5 + 1 = 6\)). This ensures that Players 2 and 3 cannot pass any motion without Player 1.

   Therefore, for Player 1 to be the dictator, the quota must satisfy:
   \[
   6 < q \leq 9
   \]

2. **Player 2 (Weight = 5):**
   - Player 2 would be a dictator if \(q \leq 5\), but this is not possible, because Player 1 alone can always pass the motion with their weight of 9. Hence, Player 2 cannot be a dictator.

3. **Player 3 (Weight = 1):**
   - Similar to Player 2, Player 3 cannot be a dictator because Player 1 has a weight of 9, which far exceeds Player 3's weight, so Player 3 cannot meet the quota independently.

### Conclusion:
The only possible dictator is Player 1, and this occurs when the quota \(q\) satisfies:
\[
6 < q \leq 9
\]

Would you like more details, or do you have any further questions?

### Follow-up Questions:
1. What happens if \(q\) is less than or equal to 6?
2. How do we identify a veto player in this system?
3. Can there be more than one dictator in a weighted voting system?
4. What happens when \(q > 9\)?
5. How would this system change with a different set of weights?

**Tip:** In weighted voting, the sum of the weights and the value of \(q\) play a crucial role in determining the power dynamics among players.

Consider the weighted voting system [q: 9, 5, 1]. Which values of q result in a dictator (list all possible values)?

Explore the concept of a dictator in the weighted voting system [q: 9, 5, 1]. Discover which values of q make Player 1 the dictator and understand the conditions for having a dictator in a voting system. Learn key concepts in voting theory, including quotas and weighted voting.

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