We have the game of incomplete information: The payoffs are given by either game

Question

We have the game of incomplete information: The payoffs are given by either game 1 (the matrix on the top) or game 2 (the matrix on the bottom). The probability is 1/2 that it is either game. Player 1 (the row player) does not know which game is being played while player 2 (the column player) knows. The players do not move simultaneously; player 2 chooses L or R, and player 1 chooses U or M after having seen player 2's move.

(a) Is there a Bayes Nash equilibrium (not necessarily weak sequential equilibrium) for this game in which player 2 pools on R (that is, player 2 chooses R for both types he may be)? Either exhibit such an equilibrium or explain why there is not one.

(b) Is there a Bayes Nash equilibrium (not necessarily weak sequential equilibrium) for this game in which player 2 pools on L? Either exhibit such an equilibrium or explain why there is not one.

(c) Is there a weak sequential equilibrium for this game in which player 2 pools and which satisfies the equilibrium domination test?

(d) Is there a separating weak sequential equilibrium for this game? Either exhibit such an equilibrium or explain why there is not one.

Explanation / Answer

a) Here in this game player 1 will always make his move after player 2 makes his move. That is whatever is the choice of player 2, player 1 will always make his move after player 2 makes his choice. So even if player 2 chooses 'R' for both the types even then player 1 will wait for player 2 to make his choice and will take his decision to make the move based on the knowledge of the move of player 2. So this is a Bayes Nash Equilibrium.

b) Similar to the above explanation, if player 2 chooses 'L' in either case then also it will be Bayes Nash Equilibrium.

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