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1. Consider the following auction: First a referee independently chooses two numbers s and t randomly from a uniform distribution on [0; 1]. He tells you the value of s and tells your opponent the value of t; then he puts $(s+t)/2 in a paper bag. You and your opponent engage in a frst-bid sealed auction for the contents of the bag. What is the Nash equilibrium bidding strategy (as a function of s)?
(Hint: You'll need to compute the expected value of t. But the value of t matters ONLY IF YOU WIN, so you'll want to compute the expected value of t ON THE ASSUMPTION THAT YOU WIN, which is to say ON THE ASSUMPTION that t < s. This is easy to compute: It's just the average value of all numbers between 0 and s; in other words, it's s=2.)
2. You're in an auction with two bidders, each of whom has a reservation price drawn uniformly from the interval [0; 1]. The rules of the auction are the same as those of a regular frst-price auction, except that you must pay a \bidding fee" of 1=4 if you submit a bid.
You are also entitled to walk away without paying the bidding fee.
Thus a bidding strategy consists of a) a minimal reservation price T below which you walk away, and b) a bidding function B(x) that tells you your bid for any reservation price x > T.
The goal of this problem is to compute that bidding strategy.
a) Write down an expression (x; b) for your expected gain as a function of your reservation price x and your bid b, on the assumption that you choose to pay the bidding fee and enter the auction. Only positive bids are allowed.
b) Evaluate 0(x) = (x;B(x)) in two di erent ways keeping in mind that a bidder with reservation price T must have an expected gain of zero!
c) Use parts a) and b) to write down the bidding function B in terms of x and T.
d) Note that a bidder with reservation price T would always bid exactly zero, because he knows that everyone with a lower reservation price is going to drop out of the auction, so he might as well drop his bid to zero. Use this observation to calculate T.
e) Now write down the optimal bidding strategy B as a function of x only.
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