(12pts)Second-priceauction.

There are 10 bidders. Each bidderivalues the object atvi>0.The indices are chosen in a way so

thatv1>v2>v10>0.

Each bidderican submit a bidbi0.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 bidderi, gets a payoffvip, wherepis the highest bid made byotherbidders. 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 andbi=v6,fori=3) a Nash equilibrium? Explain your answer.

(c) (5pts) Can you find and verify all Nash equilibria in which player3wins the object?

(Hint: Need to find and verify all Nash AND show that other strategy profiles are not Nash. Look at the example we did in class where Player 2 was the winner: How high must Player 2's bid be? How low must other players' bids be? Use those observations and Part (b) to answer this question.)