Answer Happy

In this problem we will ask how irrational behavior on the part
of one bidder affects optimal behavior for the other bidders in an
auction. In this auction the seller has one unit of the good which
will be sold using a second-price, sealed-bid auction. Assume that
there are three bidders who have independent, private values for
the good, v1, v2 v3, which are uniformly distributed on the
interval [0, 1].
(a) Suppose first that all bidders behave rationally; that is
they submit optimal bids.
Which bidder (in terms of values) wins the auction and how much
does this bidder pay
(again in terms of the bidder’s values)?
(b) Suppose now that bidder 3 irrationally bids more than his true
value for the
object; in particular, bidder 3’s bid is (v3 + 1)/2. All other
bidders know that bidder
3 is irrational in this way, although they do not know bidder 3’s
actual value for the
object. How does this affect the behavior of the other bidders?
(c) What effect does bidder 3’s irrational behavior have on the
expected payoffs of
bidder 1? Here the expectation is over the values of v2 and v3
which bidder 1 does
not know. You do not need to provide an explicit solution or write
a proof for your
answer; an intuitive explanation of the effect is fine. [Remember a
bidder’s payoff is
the bidder’s value for the object minus the price, if the bidder
wins the auction; or 0,
if the bidder does not win the auction.]