Exercise 5 -- Three Problems

You may discuss this assignment with classmates, but all your code and written work must be your own. Reference all sources. Please hand in hard copy.

1. Consider the pairwise second-price tournament, the first two tournaments we ran in class. Define the criterion

E[Δ₁₂] = E[surplus of 1 if 1 wins - surplus of 2 if 2 wins]

That is, the expected difference in surplus between bidder 1 and bidder 2. Show that if we use Δ₁₂ as the criterion, the resultant equilibrium bidding function is a dominant strategy (can't be beat!).

2. [At least if you do this the way I did it, it's a little tricky. Hint: the answer is linear in v but may not look it.]
Find the equilibrium (and hence dominant strategy) in the pairwise second-price tournament using the criterion described above when valuations are distributed uniformly on [0,1].

3. Repeat question 2 for general value distribution F. Simplify the answer as much as you can. It will, however, involve at least one integral that must be left in general form.

4. [Extra Credit and hard]
Consider the Average-of-Other-Bids (AVO) auction, as described in assignment 3, questions 1 and 2. Is there a value distribution F for which there is no equilibrium? If there is, give an example; if there is not, prove it.