1. (a) Find the equilibrium bidding function for the standard
first-price auction where the iid values are exponentially distributed
over [0,∞] and there are two bidders. That is, the cdf of the
value distribution is F(v) = 1 - e -λ v,
where v can be any nonnegative real number and λ > 0 .
Assume in this part that there is no reserve. Sketch in the
support range [0, ∞].
(b) Assuming the item has no value to the seller, what choice of
reserve maximizes revenue in the auction of part (a)?
(c) When this optimal reserve is put into effect, is
the new equilibrium bidding function for allowable bids
(i) uniformly greater than the equilibrium bidding function
with no reserve, (ii) uniformly less, or (iii) neither?
2. Show by exhibiting a counterexample that in the usual IPV auction in Riley and Samuelson's class a Nash Equilibrium is not necessarily dominant, even if there are only two bidders.
3. Now 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 as bidder 1 use Δ12 as the criterion (instead of just 1's surplus),
the resultant equilibrium bidding function is a dominant strategy (can't be beat!).
4. Find the equilibrium (and hence dominant strategy) in the pairwise second-price tournament using the criterion described in Problem 3 when valuations are distributed uniformly on [0,1]. Is it a linear function of value?