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.