In this problem, we consider a special case of the first-price, sealed-bid auction and show what the

In this problem, we consider a special case of the
first-price, sealed-bid auction and show what the equilibrium amount of bid
shading should be. Consider a first-price, sealed-bid auction with n
risk-neutral bidders. Each bidder has a private value independently drawn from
a uniform distribution on [0,1]. That is, for each bidder, all values between 0
and 1 are equally likely. The complete strategy of each bidder is a “bid
function” that will tell us, for any value v, what amount b(v) that bidder will
choose to bid. Deriving the equilibrium bid functions requires solving a
differential equation, but instead of asking you to derive the equilibrium
using a differential equation, this problem proposes a candidate equilibrium
and asks you to confirm that it is indeed a Nash equilibrium. It is proposed
that the equilibrium-bid function for n = 2 is b(v) 5 v2 for each of the two
bidders. That is, if we have two bidders, each should bid half her value, which
represents considerable shading.

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(a) Suppose you’re bidding against just one opponent
whose value is uniformly distributed on [0, 1], and who always bids half her
value. What is the probability that you will win if you bid b 5 0.1? If you bid
b 5 0.4? If you bid b 5 0.6?

(b) Put together the answers to part (a). What is the
correct mathematical expression for Pr(win), the probability that you win, as a
function of your bid b?

(c) Find an expression for the expected profit you make
when your value is v and your bid is b, given that your opponent is bidding
half her value. Remember that there are two cases: either you win the auction,
or you lose the auction. You need to average the profit between these two
cases.

(d) What is the value of b that maximizes your expected
profit? This should be a function of your value v. (e) Use your results to
argue that it is a Nash equilibrium for both bidders to follow the same bid
function b(v) 5 v2.

 

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