Suppose that nature selects A with probability 1/2 and B with probability 1/2. If nature selects A, then players 1 and 2 interact according to matrix "A." If nature selects B, then the players interact according to matrix "B." These matrices are pictured here. Suppose that, before the players select their actions, player 1 observes nature's choice. That is, player 1 knows from which matrix the payoffs are drawn, and player 1 can condition his or her decision on this knowledge. Player 2 does not know which matrix is being played when he or she selects between L and R.
(a) Draw the extensive-form representation of this game. Also represent this game in Bayesian normal form. Compute the set of rationalizable strategies and find the Nash equilibria.
(b) Consider a three-player interpretation of this strategic setting in which each of player l's types is modeled as a separate player. That is, the game is played by players 1A, 1B, and 2. Assume that player 1A's payoff is zero whenever nature chooses B; likewise, player 1B's payoff is zero whenever nature selects A. Depict this version of the game in the extensive form (remember that payoff vectors consist of three numbers) and in the normal form. Compute the set of rationalizable strategies and find the Nash equilibria.
(c) Explain why the predictions of parts (a) and (b) are the same in regard to equilibrium but different in regard to rationalizability.