Final answer:
In the weighted voting system [17:12, 11, 5, 2], P1 is the only critical player in the coalition (P1, P2, P3), as their participation is necessary to meet the decision quota of 17.
Step-by-step explanation:
The question relates to a weighted voting system which is a mathematical concept used in decision-making processes where different participants have different levels of influence or voting power. In the system [17:12, 11, 5, 2], the numbers represent the weights of each player's vote, with 17 being the quota for a decision to be made. Players P1, P2, and P3 have weights of 12, 11, and 5, respectively. A critical player in a coalition is one whose support is necessary for the coalition to reach the quota and make decisions.
To determine which players are critical in the coalition (P1, P2, P3), we need to see if the coalition reaches the quota with and without each player. In this case:
- Without P1 (11 + 5 = 16), the quota is not met.
- Without P2 (12 + 5 = 17), the quota is just met, so P2 is not critical as the decision can be made without their individual vote.
- Without P3 (12 + 11 = 23), the quota is exceeded, so P3 is not critical either.
Therefore, P1 is the only critical player in the coalition of (P1, P2, P3), as their participation is necessary for the coalition to achieve the required quota.