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It is ECON course of the GAME THEORY.

Lowest-Price-Auction Consider the sealed-bid lowest-price- auction, an auction where the highest bidder wins but he pays the lowest bidders bid. There are n > 2 bidders. The values for the object of the bidders are ordered by v1 > v2 > ... > vn > 0 and known to all bidders. If several bidders bid the highest bid, then the bidder with the lowest name gets the object. (E.g. if both bidders 3 and 6 have the highest bid, then bidder 3 will get the object because his name, 3, is lower than 6.) Show that for any bidder i, the bid of vi weakly dominates any lower bid. Show further that for any bidder i, the bid of vi does not weakly dominate any higher bid. Next, show that the action profile in which each player bids her valuation is not a Nash equilibrium. Finally, find a symmetric Nash equilibrium.