Consider the weighted voting system [q: 9, 5, 1].
What is the smallest value for q that results in exactly two players with veto power?
Here is an example of a similiar problem: In the weighted voting system [q: 5, 4, 3, 1], for which values of q does P1 have veto power?
Answer: 9, 10, 11, 12, 13.    Explanation : Together, P2, P3, and P4 have 8 votes. If they all vote Yea, they can pass a resolution without P1 as long as q is no more than 8. Otherwise, P1 has veto power.

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**.

---

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.

Consider the weighted voting system [q: 9, 5, 1]. What is the smallest value for q that results in exactly two players with veto power?

Find the smallest quota (q) for the weighted voting system [q: 9, 5, 1] that gives exactly two players veto power. This problem covers key concepts like coalition formation and veto power in voting systems. Suitable for students in Grades 10-12.

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