Consider the following version of nth price auction, n ≥ 1. There are n or more bidders, and an house to be sold. Bidder i's valuation for the house is vi > 0, and valuations are common knowledge among the bidders. Bidders submit bids simultaneously, and the highest submitted bid wins (when there are more than one highest bidders, then the winner is chosen randomly among the highest bidders with equal probability).
However, his payment is now the nth highest of the submitted bids (when more than one, say k, bidders bids the same amount, we count them as k separate bids). If bidder i wins and pays p, his payoff is vi - p. If he does not win, his payoff is 0.
(a) Suppose n = 1. Is it a strictly dominant strategy to bid your true valuation in this auction? Is it a weakly dominant strategy to bid your true valuation in this auction? If the answer is yes, provide an argument. If the answer is no, provide an example and explain.
(b) Suppose n = 3. Is it a strictly dominant strategy to bid your true valuation in this auction? Is it a weakly dominant strategy to bid your true valuation in this auction? If the answer is yes, provide an argument. If the answer is no, provide an example and explain.