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 show, for any value v, what amount b(v) that bidder will choose to bid.
It is proposed that the equilibrium bid function for n=2 is b(v)=v/2 for each of the two bidders. That is, if we have two bidders, each should bid half their value.
a) Suppose one is bidding against an opponent whose value is uniformly distributed on [0,1] and always bids half their value. What is the probability that they will win in they bid b=0.1? If they bid b=0.4? If they bid b=0.6?
b) Using the answers above: what is the correct mathematical expression for p(win), the probability that one will win, as a function of their bid b?
c) Find an expression for the expected profit one makes when their value is v and their bid is b, given that the opponent is bidding half of their value. Remember that there are two cases: either one wins the auction or they lose the auction, so it's needed to average the profits between these cases.
d) What is the value of b that maximizes one's expected profit? This should be a function of one's value v.
e) Using the results above, argue that it is a Nash Equilibrium for both bidders to follow the same bid function b(v)=v/2.