There are N>=2 collectors who engage in the auction of an antique. The collectors have a common valuation of the antique, denoted by v, which is known to all. The collectors make a simultaneous bid. Let pn denote the bid by collector n = 1,....,N. The one with the highest bid wins the antique. The winner receives payoff v-pi. The other(s) receive zero payoff. If more than one collectors make the same highest bid, then they have an equal chance of winning the item. Prove that: A) It is not a Nash Equilibrium (NE) if the highest bid is v and only one collector bids this price. (b) It is not a NE if the highest bid is less than v. (c) It is a NE that the highest bid is v and more than one collector bids this price

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