Question

Consider the following payoff table in which D1 through D3 represent decision alternatives, S1 through S4 represent states of nature, and the values in the cells represent return on investments in millions.

    S1 S2 S3 S4

D1 30 20 -50 100

D2 60 150 40 -80

D3 40 10 80 80

1. What are the decision alternatives, and what are the chance events for this problem?

2. Construct a decision tree

3. What is the preferred alternative when the decision maker is optimistic?

4. What is the preferred alternative when the decision maker is conservative?

5. What is the preferred alternative when using the minimax regret criterion?

6. Suppose that each state of nature is equally likely to occur, what is the expected value of the

   payoff?

7. Assume the probabilities remain the same under state of natures in the preceding question (part

   f), what is the expected value of perfect information. (show your work to receive full credit)

8. Suppose P(s1) = 0.4, P(s2) = 0.3, and P(s3) = 0.2, what is the best decision using the expected

   value approach?

9. Perform sensitivity analysis on the best decision alternative payoffs from the preceding question (part h). Assuming the probabilities remain the same (as in part h), find the range of payoffs under state of nature s1 that will keep the solution found in the preceding question (part h) optimal. (show your work to receive full credit)

Note: You need to be able to interpret the results for parts c-e of this problem.
For part f, you need to know how to construct a risk profile for the optimal decision For parts g and i, you need to show your work to receive full credit.

Answer 1

The decision alternatives are D1,D2 and D3. The chance events are S1,S2,S3,S4

under optimist decision, the decision maker which choose the alternative which maximizes the profit ie 100 here under D1.

Conservative approach is minimax criteria. Ie choosing the alternative which maximizes the minimum gain. Here it is 10 under D3.

Equally likely means each state of nature occurs with equal probability ie 0.25

Expected value under D1 then: (30+20-50+100)/4 = 25

Under D2: (60+150+40-80)/4 = 42.5

Under D3: (40+10+80+80)/4 = 52.5

So we can see that under expected value criteria D3 is the perfect choice.

f)EVPI= EV with Pi - EV without PI

EV with PI= maximum payoff under each state of nature*probability= (60+150+80+100)/4 = 97.5

EV without PI= max emv = 52.5

So EVPI= 97.5-52.5 = 45

g)When the probabilities have changed,

Expected value under D1: 30*0.4 + 20*0.3 - 50*0.2 + 100*0.1= 18

Similarly under D2: 69

Under D3: 43

So now Expected value criteria is D2