Consider the extensive-form game of incomplete information in the diagram that follows. There is a firm and a worker. In this game, nature first chooses the "type" of the firm (player 1).
With probability p, the firm is of high quality (H) and, with probability 1 - p, the firm is of low quality (L). The firm chooses either to offer a job to the worker (O) or not to offer a job (N).
If no job is offered, the game ends and both parties receive 0. If the firm offers a job, then the worker either accepts (A) or rejects (R) the offer. The worker's effort on the job brings the firm a profit of 2. If the worker rejects an offer of employment, then the firm gets a payoff of -1 (associated with being jilted). Rejecting an offer yields a payoff of 0 to the worker. Accepting an offer yields the worker a payoff of 2 if the firm is of high quality and -1 if the firm is of low quality. The worker does not observe the quality of the firm directly
(a) Is there a separating PBE in this game? If so, specify the equilibrium and explain under what conditions it exists. If not, briefly demonstrate why.
(b) Is there a pooling PBE in which both types of firms offer a job? If so, specify the equilibrium and explain under what conditions it exists. If not, briefly demonstrate why.
(c) Is there a pooling PBE in which neither type of firm offers a job? If so, specify the equilibrium and explain under what conditions it exists. If not, briefly demonstrate why.