1. In first price auction of two bidders, the bidder with higher bid wins the auction and pays his own bid. Suppose valuations aredistributed in the interval [0,1] with cumulative distribution function F(v) = v^n. ( n is a positive int) Suppose also that the valuations are independently drawn.

(a)Suppose that bidder i assumes that bidder j bids according to the function bj = kvj + s for some constants k and s. What's the expected payoff to bidder i as a function of his bid and his valuation.

(b)Maximize this payoff in order to derive bidder i's optimal bid as a function of his valuation. Use this to find a symmetric Bayes Nash equilibrium of the game