# Find the Perfect Bayesian Equilibrium

Max has inherited a stamp collection from his grandfather which has quality q (from Max’s point of view, uniformly distributed on [10, 20]) and a “sentimental value” of 0.5; so Max’s utility from keeping the collection is

$$u_M$$ = 0.5 + q

Joe, a stamp collector, places a value of 1.2q on the collection.

a) Suppose first that Max knows the quality of the stamp collection. For which values of q will Max sell the stamp collection to Joe?

b) Suppose now that only Joe recognizes the value of the collection when he sees it. He then either declines to make an offer (after that move, the game ends), or makes an offer to pay a price p, which Max can accept of reject. Construct the Perfect Bayesian equilibrium with the most trade, and explain which quality types will be traded in equilibrium, and at which price. To do this, explain first how Max should update his beliefs about q, given p (including out of equilibrium beliefs). Then get a condition for those prices for which Max and Joe are both willing to trade, and then find the corresponding qualities.

For part a, Max is willing to sell the stamp collection for all values of q since both groups have perfect information on the value of the stamp collection.

For part b, I find that since Joe now knows q but Max does not, Max will have an expected value q = 15 since this is the average in the uniform distribution. Next, since Max has a sentimental value of 0.5 for his stamp collection, he will charge a price of 15.50 dollars. Since Joe values the stamp collection at 1.2q, this means that Joe is willing to purchase the stamps for as low of a value of q as 15.50/1.2, or roughly 12.91 dollars. Otherwise, Joe refuses the offer and no sale occurs. This implies that the lowest quality stamp collections are not sold in the market of imperfect information, which would otherwise be sold in a market when both Max and Joe have perfect information. Is this correct?