There are 10 bidders. Each bidder i values the object at vi > 0. The indices are chosen in a way so that v1 > v2 > v10 > 0. Each bidder i can submit a bid bi 0. The bidder whose bid is the highest wins the object. If there are multiple highest bids, then the winner is the bidder whose valuation is the highest (or whose index is the smallest) among the highest bidders. (For example, if bidder 3 and bidder 9 have the highest bid, then bidder 3 is the winner.) The winner, say bidder i, gets a payoff vi p, where p is the highest bid made by other bidders. Losing players all receive zero payoff.

(a) (3pts) Use the definition of Nash equilibrium to explain why (b1 = v1 + 1, b2 = v2,b3 = v3,...,b10 = v10), is a Nash equilibrium,

(b) (4pts) Is the profile (b3 = v1 + 1 and bi = v6, for i 6= 3) a Nash equilibrium? Explain your answer.

(c) (5pts) Find and verify all Nash equilibria in which player 3 wins the object.