You are competing for a rent with one rival. Your valuation and your competitor's valuation are private information. You believe that the other bidder's valuation is equally likely to lie anywhere in the interval between 0 and $5,000. Your own valuation is $2,000. Suppose that you expect your rival will submit a bid that is exactly one-half of his valuation. Thus you believe that your rival is equally likely to bid anywhere between 0 and $2,500 (depending on the realized valuation between 0 and $5,000).
a. Show that if you submit a bid of B, the probability that you win the contest is the probability that your bid B will exceed your rival's bid, and that this probability of winning is B/250.
b. Your expected profit from bidding B is [200 - B] x Probability of winning. Show that the profit-maximizing strategy consists of bidding half your valuation.