Two or three players are about to play a game: with probability 1/2 the game involves Players 1 and 2 and with probability 1/2 the game involves Players 1, 2, and 3. Players 2 and 3 know which game is being played. In contrast, Player 1, who participates in the game under all conditions, does not know whether he is playing against Player 2 alone, or against both Players 2 and 3. If the game involves Players 1 and 2 the game is given by the following matrix, where Player 1 chooses the row, and Player 2 chooses the column:
with Player 3 receiving no payoff. If the game involves all three players, the game is given by the following two matrices, where Player 1 chooses the row, Player 2 chooses the column, and Player 3 chooses the matrix:
(a) What are the states of nature in this game?
(b) How many pure strategies does each player have in this game?
(c) Depict this game as a game with incomplete information.
(d) Describe the game in extensive form.
(e) Find two Bayesian equilibria in pure strategies.
(f) Find an additional Bayesian equilibrium by identifying a strategy vector in which all the players of all types are indifferent between their two possible actions.