A seller has one item that he would like to sell. He can at any time sell it to a dealer who will pay $50 for the item. He is also considering two other ways to try to sell the item: posting a price of $100 or using a second price auction. He knows that there are two people interested in buying or bidding on the item. He also knows that these two people have independent, private values for the item and that the value for any person is either $100 or $80 with equal probability. [These values are independent, so the probability of each of the four possible pairs of values (100, 100), (100, 80), (80, 100), (80, 80) is 1/4.] If the seller posts a price of $100 then the item will be sold $100 if at least one of the people interested in the item has a value for it of $100; otherwise it will not be purchased at the posted price and our seller will have to sell it to the dealer for $50. If the seller runs a second price auction the two people interested in the item will bid optimally in the auction. (a) Suppose the seller attempts to sell the item at a posted price of $100. (i) What is the probability that someone will buy the item at the price of $100? (ii) What will be the seller's expected revenue from trying to sell the item at a posted price of $100? (b) Suppose that instead of using the posted price the seller runs a second price auction. (i) For each of the possible pairs of values for the individuals how much will they each bid in the second price auction?

(ii) What will be the seller's expected revenue from running a second price auction?