Sealed-Bid Auctions and Revenue Equivalence

There are N 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 simultaneously and independently submits a bid bi for an object.

(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 Statistics").

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) = vi 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.