Consider the weighted voting system: [12: w, 5, 3, 2, 1,]. Find a value for w for which P5 is not a dummy.

In the weighted voting system \([12: w, 5, 3, 2, 1]\), we are tasked with finding a value for \(w\) (the weight of the second player) such that \(P_5\) (the player with weight 1) is not a dummy. 

### Step-by-Step Approach:

1. **Quota and Voting Power**: 
   The first number, 12, is the quota, which is the number of votes required to pass a motion. For \(P_5\) not to be a dummy, there must be scenarios where the vote of \(P_5\) can change the outcome (i.e., their vote is critical).

2. **Definition of a Dummy**: 
   A player is a dummy if they never have the power to change the outcome of a vote. This means they never belong to a winning coalition where their participation is critical to winning.

3. **Key Concept**:
   A player's vote is critical if, without their vote, the coalition fails to meet the quota.

4. **Initial Conditions**: 
   The players' weights are \([w, 5, 3, 2, 1]\) with a quota of 12. So the sum of all players' votes is \(w + 5 + 3 + 2 + 1 = w + 11\).

### Critical Coalition Analysis:

We want \(P_5\)'s vote to matter, so we examine coalitions where the total votes are near the quota, but \(P_5\)'s vote is needed to push the total over 12.

- If \(w + 5 + 3 = w + 8\), this coalition lacks \(P_5\). For \(P_5\)'s vote to be critical, the total of \(w + 8\) should be less than 12, but adding \(P_5\)'s vote should make the total exceed 12.

Thus, we want:
\[
w + 8 < 12 \quad \text{(without } P_5\text{)}
\]
\[
w + 8 + 1 \geq 12 \quad \text{(with } P_5\text{)}
\]

From \(w + 8 < 12\), we get:
\[
w < 4
\]

From \(w + 9 \geq 12\), we get:
\[
w \geq 3
\]

Thus, \(w\) must be at least 3 but less than 4. So, \(w = 3\) is the solution where \(P_5\) is not a dummy.

### Conclusion:
The value of \(w\) for which \(P_5\) is not a dummy is \(w = 3\).

Would you like further explanation on any of these steps?

Here are five questions to expand on this:

1. What are coalitions and how are they formed in weighted voting systems?
2. Can you explain what it means for a player to be "critical" in more detail?
3. How do different values of \(w\) affect the power of other players?
4. How would the system change if the quota were higher than 12?
5. How can we calculate the Banzhaf power index for this system?

**Tip**: In weighted voting, a player's influence doesn't always correspond to their weight, especially when coalitions form and quotas change.

Consider the weighted voting system: [12: w, 5, 3, 2, 1]. Find a value for w for which P5 is not a dummy.

Explore how to determine the value of w in the weighted voting system [12: w, 5, 3, 2, 1] such that P5 is not a dummy player. This involves analyzing coalitions and determining critical players. Suitable for advanced mathematics learners, this problem applies key concepts in voting theory.

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