Q4 Sealed-Bid Auctions and Revenue Equivalence

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

tion on [0, 100]. Consider a sealed-bid auction for a single object. Each bidder i simultaneously

and independently submits a bid bi for an object.

Hint: Review slides 18{22 from the Auction Theory (3a) slide pack to answer this question.

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

bidders.

(a) Determine the equilibrium bid functions for all bidders in a standard second-price auction.

(b) Calculate expected revenue from this auction (you will need the lecture slide \Order Statis-

tics").

It is well-known that both rst-price auction and second-price auction in the canonical model

are ecient, and the revenue equivalence theorem applies.

(c) Find Bayesian Nash equilibrium bid functions for a standard-rst-price auction of the form

bi(vi) = vi using the fact that the revenue generated by the rst-price auction should

be the same as the one generated by the second-price auction.

Q4 Sealed-Bid Auctions and Revenue Equivalence There are / risk-neutral bidders with valuations independently drawn from the uniform distribu- tion on [0, 100]. Consider a sealed-bid auction for a single object. Each bidder i simultaneously and independently submits a bid by for an object. Hint: Review slides 18-22 from the Auction Theory (3a) slide pack to answer this question. Argue that the equilibrium strategies in a second-price auction do not depend on the number of bidders. (a) Determine the equilibrium bid functions for all bidders in a standard second-price auction. (b) Calculate expected revenue from this auction (you will need the lecture slide "Order Statis- tics"). It is well-known that both first-price auction and second-price auction in the canonical model are efficient, and the revenue equivalence theorem applies. (c) Find Bayesian Nash equilibrium bid functions for a standard-first-price auction of the form bi(vi) = B* v; using the fact that the revenue generated by the first-price auction should be the same as the one generated by the second-price auction