1. Consider two lotteries:
L1: (0.5, 1000, 0.2,500, 0.2,300, 0.1, 200)
L2: (0.6, 800, 0.1,600, 0.3, 100)
a) Using an exponential utility function with R=250 determine which lottery is preferred on the basis of expected utility.
b) Transform the utility values using a linear equation u’(x) =au(x) +b so that the largest and smallest payoffs among the two lotteries have transformed utility values of 1 and 0 respectively. Which lottery is now preferred using the transformed utility values?
2. Mark Harris, Production Manager of Medical Electronics Inc. is preparing for the delivery of one of his company’s new blood analyzers to the Hershey Medical Center. All that remains is to subject the unit to a test procedure to determine whether it meets its design specifications. They will earn a profit of $3100 if the analyzer meets its specifications. If testing reveals a failure to meet specs the equipment will be completely reworked before delivery thereby guaranteeing that the analyzer will be satisfactory. In the event that rework is done, the profit will be only $1600. Should it happen that the analyzer passes the test, is delivered to Hershey and then is found to be unsatisfactory rework costs and a heavy penalty clause in the sales contract will force Medical electronics to take a $900 loss on the deal. Harris will always act in accordance with the test result-deliver if the unit passes the test and rework if it fails.
Harris must choose which of two test procedures to use. If the unit actually meets specification test 1 will indicate it to be satisfactory 80% of the time while test 2 will give a satisfactory result 60% of the time. When the unit fails to meet specifications test 1 will indicate satisfactory 30% of the time while for test 2 the figure is 10%.
Given that Harris assesses the prior probability that the unit meets specifications as 60%, draw a separate decision tree incorporating each of the two tests and compute the EVSI for each.
3. A manufacturer must decide whether to extend credit to a retailer who would like to open an account with the firm. Past experience with new accounts indicates that 45% are high risk, 35% are moderate risk and 20% are low risk customers. If credit is extended the manufacturer can expect to lose $60,000 with a high risk customer, make $50,000 with a moderate risk customer and make $100,000 with a low-risk customer. If the manufacturer decides not to extend credit to a customer, the manufacturer neither makes nor loses money. Prior to making a credit extension decision the manufacturer
can obtain a credit report on the customer at a cost of $2000. The credit agency will rate the retailer as belonging to low, medium or high risk categories. The credit agency admits that its ratings are not perfect. In particular they will rate a low risk customer as moderate risk with probability 0.10 and as a high risk customer with probability 0.05. Furthermore they will rate a moderate risk customer as low risk with probability 0.06 and as a high risk customer with probability 0.07. Finally the rating procedure will rate a high risk customer as low risk with probability of 0.01 and as medium risk with probability 0.05.
Find the strategy that maximizes the manufacturers expected net earnings. Compute the EVSI and EVPI for this decision problem
4. Peter Brown’s utility for total wealth x can be represented by the utility function u(x) =ln(x). He currently has $1000 in cash. A business deal of interest to him yields a reward of $100 with probability 0.5 and $0 with probability 0.5.
a) If he owns this business deal in addition to the $1000, what is the smallest amount for which he would sell the deal?
b) Suppose he does not own the deal what is the largest amount he would pay for the deal?