Answer Happy

4. A seller will run a second-price, sealed-bid auction for an
object. There are two bidders, a and b, who have independent,
private values vi which are either 0 or 1. For both bidders the
probabilities of vi = 0 and vi = 1 are each 1/2. Both bidders
understand the auction, but bidder b sometimes makes a mistake
about his value for the object. Half of the time his value is 1 and
he is aware that it is 1; the other half of the time his value is 0
but occasionally he mistakenly believes that his value is 1. Let’s
suppose that when b’s value is 0 he acts as if it is 1 with
probability 21 and as if it is 0 with probability 21 . So in effect
bidder b sees value 0 with probability 41 and value 1 with
probability 43 . Bidder a never makes mistakes about his value for
the object, but he is aware of the mistakes that bidder b makes.
Both bidders bid optimally given their perceptions of the value of
the object. Assume that if there is a tie at a bid of x for the
highest bid the winner is selected at random from among the highest
bidders and the price is x.
(a) Is bidding his true value still a dominant strategy for
bidder a? Explain briefly
(b) What is the seller’s expected revenue? Explain briefly.