(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