In a weighted voting system, a dummy player is one whose vote does not affect the outcome of any possible coalition. In this system, the weights are represented as $[15: 11, 7, 5, 2]$, where 15 is the quota, and the players have weights of 11, 7, 5, and 2, respectively.

To determine which player is a dummy, let's check the impact of each player's vote on coalitions that could reach or exceed the quota of 15 votes.

Possible coalitions:

• Player 1 (11) + Player 2 (7): 11 + 7 = 18 (This exceeds 15, so no other player is needed)
• Player 1 (11) + Player 3 (5): 11 + 5 = 16 (This exceeds 15, so no other player is needed)
• Player 1 (11) + Player 4 (2): 11 + 2 = 13 (This is less than 15, so Player 1 and Player 4 can't form a coalition alone)
• Player 2 (7) + Player 3 (5): 7 + 5 = 12 (This is less than 15, so they can't form a coalition alone)
• Player 2 (7) + Player 4 (2): 7 + 2 = 9 (This is less than 15)
• Player 3 (5) + Player 4 (2): 5 + 2 = 7 (This is less than 15)

Analysis:

• Player 1 (11) can form a winning coalition with any of Player 2 or Player 3 alone.
• Player 2 (7) can form a winning coalition only when combined with Player 1.
• Player 3 (5) can form a winning coalition only when combined with Player 1.
• Player 4 (2) cannot form a winning coalition with anyone. Even if Player 4 is added to any coalition, their vote does not help reach or exceed the quota (e.g., 11 + 2 = 13, which is less than 15).

Conclusion:

Player 4 is a dummy player because their vote does not contribute to any winning coalition.

Would you like further clarification or more details?

Here are 5 related questions:

1. What is the definition of a weighted voting system?
2. How are dummy players identified in different voting systems?
3. How does changing the quota affect the identification of dummy players?
4. What role do coalitions play in weighted voting systems?
5. Can there be more than one dummy player in a voting system?

Tip: In weighted voting systems, a player's power is often different from their weight. Power indices like the Shapley-Shubik or Banzhaf index can provide a more accurate representation of a player's influence.