3. Gaming the Vickrey Auction: In the previous problem, you showed that submitting a bid equal to
one's valuation is a weakly dominant strategy in the Vickrey auction. In Lecture 7, I noted that if
all the bidders in a Vickrey auction play according to this strategy, then the bidders are playing a
symmetric Bayesian Nash equilibrium. In this problem, you will explore a different kind of BNE which
is severely detrimental to the seller. This problem should give you an appreciation of the risk of using
the Vickrey auction format.
a. Suppose, for simplicity, that there are just two bidders, Victor and Wendy, in a Vickrey auction.
Suppose further that the valuations of the two bidders are independently drawn from the uniform
distribution on $[0, 1]$. That is, every value between 0 and 1, inclusive, is equally likely. Come up
with pooling equilibrium strategies for Victor and Wendy that guarantee that Victor wins the
auction and pays zero. Make sure to confirm that the strategies you select for Victor and Wendy
are, in fact, equilibrium strategies (i.e. confirm that Victor and Wendy are both playing Bayesian
best responses). Once you have identified appropriate bidding strategies for both players, you
should notice that the distributional assumptions are actually irrelevant. You should also notice
that these strategies could easily be adapted for any number of bidders.
b. Suppose a second Vickrey auction is held for an identical item. Briefly argue how Victor and
Wendy can coordinate their bids so that Victor always wins the first auction at a price of zero
and Wendy always wins the second auction at a price of zero.
c. Briefly explain why the bidding strategies you came up with in part (a) for a Vickrey auction
would not work in an English auction.
