The winning coalitions for a weighted voting system with four players are given below. \( \left\{P_{1}, P_{2}, P_{3}\right\},\left\{P_{1}, P_{2}, P_{4}\right\},\left\{P_{1}, P_{3}, P_{4}\right\},\left\{P_{2}, P_{3}, P_{4}\right\},\left\{P_{1}, P_{2}, P_{3}, P_{4}\right\} \) Complete parts (a) through (b) below. (a) Find the critical players in each winning coalition. Choose the correct answer below.

In a weighted voting system, a critical player in a winning coalition is a player who, if removed, would cause the coalition to lose. Let's analyze the given winning coalitions: 1. **Coalition \(\{P_{1}, P_{2}, P_{3}\}\)**: Here, removing \(P_{1}\) results in \(\{P_{2}, P_{3}\}\), which may not be a winning coalition (depends on the voting system), so \(P_{1}\) is critical. Likewise, removing either \(P_{2}\) or \(P_{3}\) makes this coalition lose strength, so both are also critical. Thus, all players \(P_{1}\), \(P_{2}\), and \(P_{3}\) are critical. 2. **Coalition \(\{P_{1}, P_{2}, P_{4}\}\)**: Similar reasoning applies. Removing \(P_{1}\) loses the coalition, making \(P_{1}\) critical. The same goes for \(P_{2}\) and \(P_{4}\), so all three players are critical. 3. **Coalition \(\{P_{1}, P_{3}, P_{4}\}\)**: Again, removing \(P_{1}\) results in a losing coalition, marking \(P_{1}\) as critical. Removing \(P_{3}\) or \(P_{4}\) also leads to a loss, making them critical as well. Thus, \(P_{1}\), \(P_{3}\), and \(P_{4}\) are critical. 4. **Coalition \(\{P_{2}, P_{3}, P_{4}\}\)**: Here, if we remove \(P_{2}\), the coalition would be unable to win, making \(P_{2}\) critical. The same reasoning applies for \(P_{3}\) and \(P_{4}\), meaning all three are critical. 5. **Coalition \(\{P_{1}, P_{2}, P_{3}, P_{4}\}\)**: In this coalition, every player is critical since removing any one player causes a shift that could lead to losing, depending on the system's rules. In summary, all players in each coalition are critical players. Please select the correct player options if provided.