Problems

1. How would our optimal bid change if there were 12 competitors?

2. Suppose we are bidding for an oil well that we believe will yield $40 million (including the cost of developing and mining the oil) in profits. Three competitors are bidding against us, and each competitor’s bid is assumed to follow a normal random variable with a mean of $30 million and a standard deviation of $4 million. What should we bid (within $1 million)?

3. A commonly used continuous random variable is the uniform random variable. A uniform random variable-written as U(a,b)-is equally likely to assume any value between two given numbers a and b. Explain why the formula a+(b–a)*RAND() can be used to simulate U(a,b).

4. Investor Peter Fischer is bidding to take over a biotech company. The company is equally likely to be worth any amount between $0 and $200 per share. Only the company itself knows its true value. Peter is such a good investor that the market will immediately estimate the firm’s value at 50 percent more than its true value. What should Peter bid per share for this company?

5. Ichiro is asking for salary arbitration on his baseball contract. Salary arbitration in Major League Baseball works as follows. The player submits a salary that he thinks he should be paid, as does the team. The arbitrator (without seeing the salaries submitted by the player or the team) estimates a fair salary. The player is then paid the submitted salary that is closer to the arbitrator’s estimate. For example, suppose Ichiro submits a $12 million offer and the Seattle Mariners submit a $7 million offer. If the arbitrator says a fair salary is $10 million, then Ichiro will be paid $12 million, whereas if the arbitrator says a fair salary is $9 million, Ichiro will be paid $7 million. Assume that the arbitrator’s estimate is equally likely to be anywhere between $8 and $11 million and the team’s offer is equally likely to be anywhere between $6 million and $9 million. Within $1 million, what salary should Ichiro submit?