Question 1: Auctions

a. Using the second-price, sealed-bid auction model described in Lecture 1, but assuming buyers'

valuations are privately drawn from a uniform distribution over [0, 0], show that:

i. The optimal bidding strategy for any buyer is to bid their true valuation

and so:

ii. Explain how the expected revenue to the seller will increase with u.

b. Suppose instead that the seller implements a first-price, sealed-bid auction.

i. Show that the bidding strategy b' :

=

- (1)u, is

u, is an equilibrium bidding strategy when there

are two bidders in the auction (N = 2).

ii. What is the expected revenue in this auction?

c. Would a risk-averse seller prefer one auction-type over the other?

d. Consider a Position Auction discussed in lectures, with each buyer's valuation known to all other

buyers:

i. Show that the strategy b³ (as defined in lectures) is a best response to b¹ and, similarly,

that b² is a best response to b³ and b.

ii. Will a seller necessarily increase revenue by increasing the number of positions available

for advertising on a webpage?