In a number of countries, government contracts are awarded by means auctions in which the winner is the bidder whose bid is closest to the average bid. Consider such an auction with three bidders competing to win a single object. Bidder i has a valuation vi for the object, where v1>v2>v3>0. Bidders simultaneously submit bids bi0. The bidder whose bid is closest to the average of b1,b2, and b3 wins the object and pays her own bid (thus if i is the winning bidder, her payoff is vibi ). In case of a tie, the bidder with the highest valuation wins the object. Losing bidders receive a payoff of 0 . (a) Find all (pure strategy) Nash equilibria of this game. Solution: Given b2 and b3, note that 1 can win the auction by bidding min{b2,b3}. If min{b2,b3}>v1, then only losing bids are best responses for 1. Hence either 2 or 3 is winning and paying more than her valuation, which cannot be a best response. Hence we must have min{b2,b3}v1. Similarly, min{b2,b3}=v1 can only be a best response for 2 and 3 if 1 is winning, which can only be a best response for 1 if b1=v1. The profiles (v1,b2,b3) such that min{b2,b3}=v1 are indeed Nash equilibria. If min{b2,b3}