If you did the previous problem correctly, you found that if April believes that Bart will bid exactly as much as the rug is worth to him, then she will bid only half as much as the rug is worth to her. If this is the case, it doesn’t seem reasonable for April to believe that Bart will bid his full value. Let’s see what would the best thing for April to do if she believed that Bart would bid only half as much as the rug is worth to him.
(a) If Bart always bids half of what the rug is worth to him, what is the highest amount that Bart would ever bid? ______. Why would it never pay for April to bid more than $500.01? ______.
(b) Suppose that the rug is worth $800 to April and she bids $300 for it. April will only get the rug if the value of the rug to Bart is less than ______. What is the probability that she will get the rug if she bids $300 for it? ______. What is her profit if she bids $300 and gets the rug? ______. What is her expected profit if she bids $300? ______.
(c) Suppose that the rug is worth $800 to April. What is the probability that she will get it if she bids $x where $x < $500? ______. Write a formula for her expected profit as a function of her bid $x when the rug is worth ______. What bid maximizes her expected profit in this case? ______.
(d) Suppose that April values the rug at $V and she believes that Bart will bid half of his true value. Show that the best thing for April is to bid half of her own true value.
(e) Suppose that April believes that Bart will bid half of his actual value and Bart believes that April will bid half of her actual value. Suppose also that they both act to maximize their expected profit given these beliefs. Will these beliefs be self-confirming in the sense that given these beliefs, each will take the action that the other expects?