Consider the auction model with a continuum of possible valuations. Bidder i's valuation, Vi, is drawn from the uniform distri- bution on [0,1], for i = 1,2,...,n. In other words, the cdf of Vi, can be defined as F (v) = v for v [0, 1] (and, of course, F (v) = 0 for v < 0 and F (v) = 1 for v > 1). Each bidder's valuation is independent of any other bidder's valuation. Consider the first-price auction. As I have ar- gued in class, the strategy profile in which Bi(v) = B(v) (n1)/nv for all v [0,1] and i = 1,2,...,n is a Nash equilibrium. For this ex- ercise, consider the case n = 3.

(a) Argue that the strategy profile in which each buyer bids according to the function B defined by B(v) = v/2 is not a Nash equilib- rium. Hint : Find a profitable deviation for some v [0, 1].

(b) Suppose the seller uses a posted price p. What is her expected revenue? Which price maximizes her expected revenue? Hint: What is the probability of at least one buyer is willing to pay p?

(c) Recall that in the first price auction, the seller's expected revenue is (n1)/(n+1). Compare the seller's revenue from the first-price auction and that from posted-price selling.