To determine the values of q that result in a dictator, we need to consider the given list [11, 5, 3]. In this context, a dictator is a player who has the power to veto any decision made by the other players.
1. The possible values of q that result in a dictator are:
- If q is less than or equal to 3, then player 1 with a power of 11 can veto any decision made by players 2 and 3.
- If q is less than or equal to 5, then player 2 with a power of 5 can veto any decision made by player 3.
2. To find the smallest value for q that results in exactly one player with veto power who is not a dictator, we need to consider the case where q is equal to the smallest power in the list, which is 3. In this case, player 1 has a veto power of 11, player 2 has a veto power of 5, and player 3 has a veto power of 3. Player 3 is the only player who does not have veto power, but there is still one player with veto power (player 1), so there is no value of q that satisfies this condition.
3. To find the smallest value for q that results in exactly two players with veto power, we need to consider the case where q is greater than the smallest power in the list, which is 3, and less than or equal to the second smallest power in the list, which is 5. In this case, player 1 has a veto power of 11, player 2 has a veto power of q, and player 3 has a veto power of 3. Only players 1 and 2 have veto power, so the smallest value for q that satisfies this condition is 4.
To summarize:
- Values of q that result in a dictator: q ≤ 3 or q ≤ 5.
- Smallest value for q that results in exactly one player with veto power who is not a dictator: There is no such value.
- Smallest value for q that results in exactly two players with veto power: q = 4.