8.12 Refinements of perfect Bayesian equilibrium Recall the job-market signaling game in Example 8.9.

a. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the firm offers an uneducated worker a job. Be sure to specify beliefs as well as strategies.

b. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the firm does not offer an uneducated worker a job. What is the lowest posterior belief that the worker is low-skilled conditional on obtaining an education consistent with this pooling equilibrium? Why is it more natural to think that a low-skilled worker would never deviate to E and thus an educated worker must be high-skilled? Cho and Kreps’s intuitive criterion is one of a series of complicated refinements of perfect Bayesian equilibrium that rule out equilibria based on unreasonable posterior beliefs as identified in this part; see I. K. Cho and D. M. Kreps, ‘‘Signalling Games and Stable Equilibria,’’ Quarterly Journal of Economics 102 (1987): 179–221.