Consider the following first-price, sealed bid, private value auction of an object to two bidders.
In the auction, the highest bidder wins the bid and pays her bid.
Suppose that for i = 1, 2, bidder i's valuation ofthe object, θi, is independently and uniformly distributed on [0, 1]. In this case, the common cumulativedistribution function of θiis F(θi) = θi on [0, 1]. The bidders are risk neutral. The above set-up is commonknowledge.
a. Explain briefly how the auction can be treated as a Bayesian game. What is a (pure) strategy of a bidder?
b. Define a pure strategy Bayesian Nash equilibrium of the above bidding game.
c. Derive a symmetric Bayesian Nash equilibrium of the game.
d. Now, consider instead a second-price, sealed bid auction. In this auction, the highest bidder wins theobject and pays the second highest bid. The rest of the set-up is the same as before. What is the dominantstrategy Bayesian Nash equilibrium of this bidding game?
e. Give an expression for the expected revenue of the auctioneer for each of the above auction schemes.
f. Calculate the expected revenue of the auctioneer in each of the above auction schemes. Show that theyare the same.