Nick owns a company and would like to sell it to Sarah. The company's true value v is either 10, 11, or 12 (in millions of pounds). Nick offers to sell the company for a price p. If Sarah accepts the offer, then the company is sold, so Nick's payoff is p and Sarah's payoff is v p. If Sarah rejects the offer, then the company is not sold, so Nick's payoff is v and Sarah's payoff is 0.

(a) Suppose that neither Nick nor Sarah know the company's true value; both ascribe probability 1/3 to each possible value. All the above is common knowledge. Find a perfect Bayesian equilibrium.

(b) Suppose that Nick knows the company's true value, but Sarah does not. She ascribes probability 1/3 to each possible value. All the above is common knowledge. Find a perfect Bayesian equilibrium.

(c) Finally, suppose that Nick does not know the company's true value and ascribes probability 1/3 to each possible value. With probability p Sarah knows the true value, and with probability 1 p Sarah does not know the true value and ascribes probability 1/3 to each possible value, where 0 < p < 1. All the above is common knowledge. Describe a perfect Bayesian equilibrium in which Nick asks price p = 12.