(Solved): 4. Consider a game between two players 1 and 2. Player 1 can be one of three types: A, B and C. Pla ...

4. Consider a game between two players 1 and 2. Player 1 can be one of three types: A, B and C. Player 1 knows her type but player 2 does not. First player 1 chooses between action \( X \) and \( Y \). If player 1 chooses \( X \), both players get 0 regardless of player 1 's type. If player 1 chooses \( Y \), the players play a simultaneous game \( G \). The payoff matrices of \( G \), conditional on player 1 's type, are as follows: 2 \( \begin{array}{ll}1 & \text { U } \\ & \text { D (with probability 0.2) } \\ & \text { U } \\ & \text { D } \\ & \text { B (with probability } 0.4)\end{array} \) (1) Find a PBE in which player 1 chooses \( U \) regardless of her type and player 2 chooses \( L \) in game G. Remember to include a belief of player \( 2,(p, q, 1-p-q) \) on the type of player 1 being \( A, B \) and \( C \) respectively. (2) Is there a PBE in which player 2 chooses \( R \) in game G? Explain.