There are three risk-neutral bidders with valuations independently drawn from the uniform

distribution on [0, 100]. Consider a sealed-bid auction for a single object. Each bidder i simul-taneously and independently submits a bid bi

for an object.

First, consider a standard first-price auction. The bidder with the highest bid gets the object

and pays her bid. The losers pay nothing. Assume that all three bidders follow linear bidding

strategy of the form

bi(vi) = vi for all i = 1, 2, 3 and 0 < 1

Hint: Review slides 27-34 from the Auction Theory (3a) slide pack to answer parts (a) - (c).

You need to repeat all steps making adjustments to account for three bidders.

(a) Derive the probability of winning the object for bidder 1 assuming that b1 100 .

Hint: Bidder 1 needs to outbid bidder 2 and bidder 3 simultaneously. The probability

of simultaneously outbidding two bidders (with independent values) is just a product of

probability of outbidding bidder 2 and probability of outbidding bidder 3:

P r(B1 wins) = P r(b1 > b2, b1 > b3) = P r(b1 > b2) P r(b1 > b3)

(b) Determine the Bayesian Nash equilibrium bid functions for all bidders. Compare bid functions to the ones we derived in class for a standard first-price auction with two bidders

(N = 2). Provide Intuition.

(c) Calculate expected revenue from this auction (you will need the lecture slide "Order Statistics").

Now, consider a standard second-price auction. The bidder with the highest bid gets the object

and pays the second-highest bid. The losers pay nothing.

Hint: Review slides 18-22 from the Auction Theory (3a) slide pack to answer parts (d) - (f ).

Argue that the equilibrium strategies in a second-price auction do not depend on the number of

bidders.

(d) Determine the equilibrium bid functions for all bidders. Compare bid functions to the ones

we derived in class for a standard second-price auction. Provide Intuition.

(e) Calculate expected revenue from this auction (you will need the lecture slide "Order Statistics").

(f) Compare revenues generates by the first-price auction and the second-price auction.