4. (20 points) A seller is selling one unit of an indivisible commodity through first price sealed bid auction. There are 2 bidders indexed by i = 1, 2. Simultaneously each bidder i submits a sealed bid bi 0. The highest bidder wins and pays her bid. If bidder i is the winner, and her bid is bi then her net payoff is vi bi . If bidder i does not win, she gets a payoff of 0. Each bidder i knows her value, vi for the object ( her maximal willingness to pay) but does not know the value of the other bidder vj . It is commonly known that v1, v2 are distributed identically and independently over the interval [0, 1] according to the distribution function,

F(x) = x

In the symmetric equilibrium of this game find the equilibrium bid function for a bidder.