Question

Problem 13-29 (Algorithmic)

Three decision makers have assessed utilities for the following decision problem (payoff in dollars):

	State of Nature
Decision Alternative	S1	S2	S3
d1	10	40	-30
d2	90	110	-80

The indifference probabilities are as follows:

	Indifference Probability (p)
Payoff	Decision maker A	Decision maker B	Decision maker C
110	1.00	1.00	1.00
90	0.95	0.80	0.85
40	0.85	0.70	0.75
10	0.75	0.55	0.60
-30	0.60	0.25	0.50
-80	0.00	0.00	0.00

Find a recommended decision for each of the three decision makers, if P(s₁) = 0.25, P(s₂) = 0.60, and P(s₃) = 0.15. (Note: For the same decision problem, different utilities can lead to different decisions.) If required, round your answers to two decimal places.

Decision maker A

EU(d1) =

EU(d2) =

Recommended decision:

Decision maker B

EU(d1) =

EU(d2) =

Recommended decision:

Decision maker C

EU(d1) =

EU(d2) =

Recommended decision:

Answer 1

Ans. Expected value / utility here is a weighted average of the payoffs for a decision alternative.

EU (something) = Probablities * payoff values

In the second indifference probability table payoffs are the same as the first table & given by the combination of payoff between s1,s2,s3 and given ascendingly. The payoff of decision 1 in first table which exactly be given in the second table & corresponding probabilities , those will be using to calculate the EU(d1) for decision maker A,B,C and same process will be following to calculate the EU(d2) for decision maker A,B,C.

Decision maker A

EU(d1) = 23.5

EU(d2) = 195.5 EU(d2) - EU(d1) = 172

Recommended decision: d2

Decision maker B

EU(d1) = 24

EU(d2) = 182 EU(d2) - EU(d1) = 182 - 24 = 158

Recommended decision: d2

Decision maker C

EU(d1) = 34.5

EU(d2) = 186.5 EU(d2) - EU(d1) = 186.5 - 34.5 = 152

Recommended decision: d2 Now expected utility of EU(d1) & EU(d2) across state of nature : Since P(s1) = 0.25 P(s2) = 0.60 P(s3) = 0.15 So, EU(d1) = 10*0.25 + 40*0.60 - 30*0.15 = 22 EU(d2) = 90*0.25 + 110*0.60 - 80*0.15 = 76.5 Now EU(d2) - EU(d1) = 54.5 So if EU(d2) - EU(d1) <=> 54.5 , for three decision makers then decision makers will decide d1, indifferent between two , d2 accordingly.