Consider the independent private value auction model where there are 2 bidders and each bidder's valuation, Vi, is uniformly distributed on [0, 1], for i = 1, 2. That is, the distribution function of both V1 and V2 is F(v) = v for v [0,1].

(a) In the first price auction, show that it is a Nash equilibrium for each bidder to bid Bi(v) = B(1)(v) = 21v, for i = 1,2.

(b) Suppose the seller sells the product by posted price, instead. For any price p [0, 1] the seller sets, what is the expected revenue of the seller? What is the p that maximizes his expected revenue?

(c) Compare his maximized revenue from posted-price selling with his ex- pected revenue from the first price auction, 1/3. Can you think of a reason why the seller wants to sell the good through first price auction, even if it may mean lower expected revenue than posted-price selling?