Question

Suppose that a car-rental agency offers insurance for a week that costs $100. A minor fender bender will cost $3,500, whereas a major accident might cost $16,000 in repairs. Without the insurance, you would be personally liable for any damages. What should you do? Clearly, there are two decision alter-natives: take the insurance, or do not take the insurance. The uncertain consequences, or events that might occur, are that you would not be involved in an accident, that you would be involved in a fender bender, or that you would be involved in a major accident. Develop a payoff table for this situation. What decision should you make using each of the following strategies?

a. aggressive strategy

b. conservative strategy

c. opportunity-loss strategy

Part B

For the car-rental situation described, assume that you researched insurance industry statistics and found out that the probability of a major accident is 0.05% and that the probability of a fender bender is 0.16%. What is the expected value decision? Would you choose this? Why or why not? Also calculate the expected value of perfect information.

Answer 1

(a)

Aggressive strategy (Maximax)

Take maximum possible payoff for each alternative. Then select the alternative which shows the maximum of payoff out of these maxima.

Payoff matrix	States of nature			Max
Alternatives	No accident	Minor accident	Major accident	payoff
Take insurance	-100	-100	-100	-100
Don't take insurance	0	-3,500	-16,000	0

Note that the payoffs have been written as negative of costs.

So, based on the aggressive strategy, the best decision is to not to buy the insurance.

(b)

Conservative strategy (Maximin)

Take minimum possible payoff for each alternative. Then select the alternative which shows the maximum of payoff out of these minima.

Payoff matrix	States of nature			Min
Alternatives	No accident	Minor accident	Major accident	payoff
Take insurance	-100	-100	-100	-100
Don't take insurance	0	-3,500	-16,000	-16,000

So, based on the conservative strategy, the best decision is to buy the insurance.

(c)

Opportunity-loss strategy (Minimax Regret)

First, develop the regret matrix by subtracting each payoff from the corresponding column maxima.

Regret matrix	States of nature
Alternatives	No accident	Minor accident	Major accident
Take insurance	100	0	0
Don't take insurance	0	3,400	15,900

Then take maximum possible regrets for each alternative and select the alternative which shows the minimum of regrets out of these maxima.

Regret matrix	States of nature			Max
Alternatives	No accident	Minor accident	Major accident	regret
Take insurance	100	0	0	100
Don't take insurance	0	3,400	15,900	15,900

So, based on the opportunity-loss strategy, the best decision is to buy the insurance.

Part-B

Payoff matrix	States of nature			Expected
Alternatives	No accident	Minor accident	Major accident	payoff
Take insurance	-100	-100	-100	-100
Don't take insurance	0	-3,500	-16,000	-13.6
Probability	99.79%	0.16%	0.05%

The max. EMV = -13.6 and the best decision is to not to take the insurance.

Expected value with perfect information = Max(-100,0)*99.79% + Max(-100,-3500)*0.16% + Max(-100,-16000)*0.05% = -0.21

Expected value of perfect information = Expected value with perfect information - Max. EMV

= -0.21 - (-13.6)

= 13.39