Question

The payoff table below provides the profits (in thousands of dollars) for each of four alternatives in each of three supplies.

	Supplies
Alternative	S1	S2	S3
A1	112	67	-26
A2	82	85	101
A3	85	72	80
A4	-50	90	110

Suppose that the probabilities for the supplies above are P(S₁) = 0.6, P(S₂) = 0.2, and P(S₃) = 0.1.

1. Which alternative should be selected under Bayes’ Rule?
2. What is the expected value of perfect information for this decision?

Answer 1

[Note: Usually the probabilities given must sum to 1 but 0.6+0.2+0.1 =0.9. So, please check the probabilities. However, the following solution is given by ignoring it].

a.

Alternative	S1	S2	S3	Expected payoff (in 1000's of dollars)
A1	112	67	-26	112(0.6)+67(0.2)-26(0.1) =67.2+13.4-2.6 =78
A2	82	85	101	82(0.6)+85(0.2)+101(0.1) =49.2+17+10.1 =76.3
A3	85	72	80	85(0.6)+72(0.2)+80(0.1) =51+14.4+8 =73.4
A4	-50	90	110	-50(0.6)+90(0.2)+110(0.1) = -30+18+11 = -1
Prior probability	0.6	0.2	0.1	Maximum expected payoff =78

The maximum expected payoff of 78 is obtained at A1. Thus, Alternative 1 should be selected under Bayes’ Rule.

b.

EVPI =Expected Value of Perfect Information

EVwPI =Expected Value with Perfect Information =112(0.6)+90(0.2)+110(0.1) =67.2+18+11 =96.2

EVwoPI =Expected Value without Perfect Information =Maximum expected payoff =Maximum(78, 76.3, 73.4, -1) =78

Thus, EVPI =EVwPI - EVwoPI =96.2 - 78 =18.2

Therefore, the expected value of perfect information for this decision =18.2 thousand dollars =$18200