Suppose there are two buyers with independent private values uniformly distributed over [0, 1]….

Suppose there are two buyers with independent private values uniformly distributed over [0, 1]. Buyer 2 faces budget limitations, and the maximal sum that he can bid is  . Buyer 1 is free of budget limitations. Answer the following questions:

(a) Find an equilibrium if the buyers are participating in a sealed-bid second-price auction.

(b) Compute the seller’s expected revenue, given the equilibrium you found.

Consider what happens if the buyers participate instead in a sealed-bid first-price auction. To avoid a situation in which buyer 1 bids a price  + ε, where ε > 0 is very small, define the function p according to which if both buyers bid 1 4 buyer 1 is declared the winner (and if both buyers submit an identical bid that is lower than  , each of them is chosen the winner with equal probability  ). Answer the following questions, and justify your answers:

(c) Is the following strategy vector (β1, β2) an equilibrium?

(d) If your answer to item (c) is negative, find a nondecreasing equilibrium.

(f) Does Corollary 12.50 (page 500) enable you to deduce that the seller’s expected revenue under the nondecreasing equilibrium in this case equals the expected revenue that you found in item (b)?

(g) Does Theorem 12.59 (page 504) enable you to deduce that the individually rational, incentive-compatible direct selling mechanism that maximizes the seller’s expected revenue is a sealed-bid second-price auction with a reserve price?