Late in the day at an antique rug auction there are only two bidders left, April and Bart. The last rug is brought out and each bidder takes a look at it. The seller says that she will accept sealed bids from each bidder and will sell the rug to the highest bidder at the highest bidder’s bid. Each bidder believes that the other is equally likely to value the rug at any amount between 0 and $1,000. Therefore for any number X between 0 and 1,000, each bidder believes that the probability that the other bidder values the rug at less than X is X/1, 000. The rug is actually worth $800 to April. If she gets the rug, her profit will be the difference between $800 and what she pays for it, and if she doesn’t get the rug, her profit will be zero. She wants to make her bid in such a way as to maximize her expected profit.
(a) Suppose that April thinks that Bart will bid exactly what the rug is worth to him. If she bids $700 for the rug, what is the probability that she will get the rug? _______. If she gets the rug for $700, what is her profit? _______. What is her expected profit if she bids $700? _______.
(b) Suppose that Bart will pay exactly what the rug is worth to him. If April bids $600 for the rug, what is the probability that she will get the rug? _______. What is her profit if she gets the rug for $600? _______. What is her expected profit if she bids $600? _______.
(c) Again suppose that Bart will bid exactly what the rug is worth to him. If April bids $x for the rug (where x is a number between 0 and 1,000) what is the probability that she will get the rug? _______ What is her profit if she gets the rug? ______Write a formula for her expected profit if she bids $x._______. Find the bid x that maximizes her expected profit. (Hint: Take a derivative.) _______.
(d) Now let us go a little further toward finding a general answer. Suppose that the value of the rug to April is $V and she believes that Bart will bid exactly what the rug is worth to him. Write a formula that expresses her expected profit in terms of the variables V and x if she bids $x. _______. Now calculate the bid $x that will maximize her expected profit. (Same hint: Take a derivative.) _______.