Consider a second-price auction with 4 bidders, indexed by i = 1, 2, ..., 4. There is one object that is being auctioned. The bidders simultaneously submit bids (nonnegative numbers). The bidder with the highest bid gets the object and pays the second-highest bid. If k 2 bidders tie for highest bid with a bid p, each of them gets the object, and pays p, with probability 1 k . Each bidder i has valuation vi , where v1 > v2 > v3 > v4 > 0. A bidder's payoff is vi p if she gets the object and pays a price p, and is 0 if she does not get the object.

For each of the following strategy profiles, state whether it is a Nash equilibrium or not, and give a complete justification of your answer.

i. (v1, v2, v3, v4) (each bidder bids her own valuation).

ii. (v2, 0, 0, 0)

iii. (0, v2, 0, 0)

b. Now consider a first-price auction with 3 bidders. In this type of auction, the winner pays his own bid, and if there is a tie, the bidder with the lowest index among the highest bidders is the winner. A bidder's payoff is vi p if he gets the object and pays the price p, and is 0 if he does not get the object. Assume that v1 > v2 > v3 > 0.

i. Show that a strategy profile is not a Nash equilibrium if the two highest bids are not the same.

ii. Show that the strategy profile (v2, v2, v3) is a Nash equilibrium.