You are bidding against one other bidder in a first-price sealed-bid auction with private values. You believe that the other bidder's valuation is equally likely to lie anywhere in the interval between $0 and $500. Your own valuation is $200. Suppose you expect your rival to submit a bid that is exactly one half of its valuation. Thus, you believe that your rival's bids are equally likely to fall anywhere between 0 and $250. Given this, if you submit a bid of Q, the probability that you win the auction is the probability that your bid Q will exceed your rival's bid. It turns out that this probability is equal to Q/250.
Your profit from winning the auction is equal to (200-bid)*probability of winning. Show that your profit maximizing strategy is bidding half of your valuation.