# (Optional) This exercise is a continuation of Exercise S4; it looks at the general case where n is..

(Optional) This exercise is a continuation of Exercise

S4; it looks at the general case where n is any positive integer. It is

proposed that the equilibrium-bid function with n bidders is b(v) 5 v(n 2

1)n. For n 5 2, we have the case explored in Exercise S4: each of the

bidders bids half of her value. If there are nine bidders (n 5 9), then each

should bid 910 of her value, and so on.

(a) Now there are n 2 1 other bidders bidding against

you, each using the bid function b(v) 5 v(n 2 1)n. 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 b.

(c) Recall that there are n – 1 other bidders, all using

the same bid function. What is the probability that your bid b is larger than

all of the other bids? That is, find an expression for Pr(win), the probability

that you win, as a function of your bid b.

(d) Use this result to find an expression for your

expected profit when your value is v and your bid is b. (e) What is the value

of b that maximizes your expected profit? Use your results to argue that it is

a Nash equilibrium for all n bidders to follow the same bid function b(v) 5 v(n

– 1)n.