In a weighted voting system like $[q: 9, 5, 1]$, $q$ is the quota (the number of votes required to pass a resolution), and the numbers represent the voting weights of the players, i.e., Player 1 (P1) has 9 votes, Player 2 (P2) has 5 votes, and Player 3 (P3) has 1 vote.

We need to determine the smallest value of $q$ that results in exactly two players having veto power.

Definition of Veto Power:

A player has veto power if their vote is necessary to reach or exceed the quota $q$. In other words, a player has veto power if the other players combined cannot reach the quota without that player's votes.

Step-by-Step Process:

Step 1: Determine total votes.

The total number of votes in the system is: $9 + 5 + 1 = 15 \text{ votes}.$

Step 2: Consider cases for veto power.

• Player 1 (P1) has 9 votes.
• Player 2 (P2) has 5 votes.
• Player 3 (P3) has 1 vote.

We need to identify the smallest value of $q$ such that exactly two players (P1 and P2) have veto power.

Step 3: Find the conditions for veto power.

• P1 has veto power if the other two players (P2 and P3) together cannot reach $q$. This means that P2 and P3 together have 5 + 1 = 6 votes, so if $q > 6$, P1 will have veto power.
• P2 has veto power if P1 and P3 together cannot reach $q$. Since P1 and P3 together have 9 + 1 = 10 votes, P2 will have veto power if $q > 10$.

For exactly two players to have veto power, both P1 and P2 must have veto power, while P3 does not.

Step 4: Smallest $q$ value.

For P3 not to have veto power, P1 and P2 must be able to pass a resolution without P3. That is, $q \leq 9 + 5 = 14$.

Thus, the smallest $q$ must satisfy:

• $q > 6$ (so that P1 has veto power),
• $q > 10$ (so that P2 has veto power),
• $q \leq 14$ (so that P3 does not have veto power).

The smallest such $q$ is 11.

Conclusion:

The smallest value of $q$ that results in exactly two players (P1 and P2) having veto power is 11.

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Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. For which values of $q$ does P1 alone have veto power in the system $[q: 9, 5, 1]$?
2. What happens if $q = 7$? Who has veto power?
3. How does changing P3's vote from 1 to 2 affect the veto power in this system?
4. What is the total number of winning coalitions if $q = 11$?
5. How do weighted voting systems model real-world decision-making?

Tip: In weighted voting, veto power is crucial in determining which players are essential to form a coalition that can pass decisions. Adjusting the quota affects the power distribution significantly.