'1 Sealed-Bid Auctions with Three Bidders -.ere are three risk-neutral bidders with valuations independently drawn from the uniform .tribution on [0, 100]. Consider a sealed-bid auction for a single object. Each bidder i simul- 1eously and independently submits a bid 5.- for an object. rst, consider a standard rstprice auction. The bidder with the highest value gets the object d pays her bid. The losers pay nothing. Assume that all three bidders follow linear bidding ategy of the form b(u,;)=,6 *ui forall i =1 , 2,3 and 0 58 g 1 a) Derive the probability of winning the object for bidder 1 assuming that b1 3 100 a . 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: Pr(Bl wins): Pr(b1>b2, b1>b3)= Pr(b1>b2)*Pr(bl >b2) b) Determine the Bayesian Nash equilibrium bid functions for both bidders. Compare bid functionstotheoneswederivedinclassforastandardrst-priceauction.ProvideIntuition. 0) Calculate expected revenue from this auction (you might want to use the slide with order statistics) :w, consider astande secondprice auction. The bidder with the highest value gets the object d pays the second-highest value. The losers pay nothing. :1) Determine the Bayesian Nash equilibrium bid functions for both bidders. Compare bid functions to the ones we derived in class for a stande secondprice auction. Provide Intuition. e) Calculate expected revenue from this auction (you might want to use the slide with order statistics) 1) Compare revenues generates by the rstprice auction and the secondprice auction