In this problem we will ask how irrational behavior on the part of one bidder aects 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 rst 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 bidders values)?

(b) Suppose now that bidder 3 irrationally bids more than his true value for the object; in particular, bidder 3s bid is (v3 + 1)/2. All other bidders know that bidder 3 is irrational in this way, although they do not know bidder 3s actual value for the object. How does this aect the behavior of the other bidders?

(c) What eect does bidder 3s irrational behavior have on the expected payos 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 eect is ne. [Remember a bidders payo is the bidders value for the object minus the price, if the bidder wins the auction; or 0, if the bidder does not win the auction.]