This task explores auction theory mathematically. In auction theory,????bidders (????is a positive integer) have valuations which represent how much they value an item; we will make the simplifying assumption that the valuations are i.i.d. (independent identically distributed) with continuous density????. In the first-price auction, the bidder who makes the highest bid wins the item and pays his/her bid. In the second-price auction, the bidder who makes the highest bid wins the auction, and pays an amount equal to the second-highest bid. A strategy for the auction is a bidding function????, which is a function of the bidder's valuation. The bidding function determines how much to bid as a function of the bidder's valuation, and the goal is to find a bidding function????() which maximizes your expected utility (0 if you do not win, and your valuation minus the amount of money you bid if you do win).

1. a)For the first-price auction, consider the following scenario: each person draws his/her valuation uniformly from the interval(0, 1)(so????(????) = 1for???? (0, 1)). Suppose that the other bidders bid their own valuations(they use ????(????) =????,the identity bidding function). Consider the case where there is only one other bidder. What is your optimal bidding function? Try to prove it.
2. b)Consider the same situation as the previous part, but now assume that there are????other bidders. Again, what is your optimal bidding function? Try to prove it.