This exercise is a continuation of Exercise S6; it looks at the general case where n is any positive integer. It is proposed that the equilibrium bid function with bidders is () = ( 1)/. For = 2, we have the case explored in Exercise S6: Each of the bidders bids half of her value. If there are nine bidders ( = 9), then each should bid 9/10 of her value, and so on.

(a) Now there are 1 other bidders bidding against you, each using the bid function () = ( 1)/. For the moment, let's focus on just one of your rival bidders. What is the probability that she will submit a bid less than 0.1? Less than 0.4? Less than 0.6? (b) Using the above results, find an expression for the probability that the other bidder has a bid less than your bid amount . (c) Recall that there are 1 other bidders, all using the same bid function. What is the probability that your bid is larger than all of the other bids? That is, find an expression for (), the probability that you win, as a function of your bid . (d) Use this result to find an expression for your expected profit when your value is and your bid is . (e) What is the value of that maximizes your expected profit? (f) Use your results to argue that it is a Nash equilibrium for all bidders to follow the same bid function () = ( 1)/.