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.
(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.