The following is a payoff table giving profits for various situations. The probabilities for...

Question:

The following is a payoff table giving profits for various situations.

Alternatives A B C
Alternative 1 100 120 180
Alternative 2 120 140 120
Alternative 3 200 100 50
Do Nothing 0 0 0

The probabilities for states of nature A, B, and C are 0.3, 0.5, and 0.2, respectively.

a. What are the expected values for each alternative?

b. What decision should be made under expected value?

c. What is the EVPI?

Decision-Making Process:

In business, the decisions ought to be made with the objective of maximizing the profits or minimizing the costs at any particular point. This leads to the criteria for solving the payoff matrix in various ways.

In this question, Hurwicz Criterion is used for making the decision according to the probabilities given.

(a)

The expected value of any alternative is calculated as :

{eq}expected~value=\alpha\times (maximum ~payoff ~value ~corresponding~ to~ a ~course~of~ action)+(1-\alpha)\times (minimum ~payoff ~value~ corresponding ~to ~the ~same~ course~ of ~action) {/eq}

Where, {eq}\alpha {/eq} corresponds to the probability of that course of action.

The probability for alternative {eq}A_{1} {/eq} is given to be 0.3

Thus, {eq}\alpha=0.3\\ (1-\alpha)=(1-0.3)=0.7 {/eq}

Thus, the expected value for alternative {eq}A_{1} {/eq} will be given by:

{eq}E[A_{1} ]=0.3\times 180+0.7\times 100=124 {/eq}

The probability for alternative {eq}A_{2} {/eq} is given to be 0.5

Thus, {eq}\alpha=0.5\\ (1-\alpha)=(1-0.5)=0.5 {/eq}

Thus, the expected value for alternative {eq}A_{2} {/eq} will be given by:

{eq}E[A_{2} ]=0.5\times 140+0.5\times 120=130 {/eq}

The probability for alternative {eq}A_{3} {/eq} is given to be 0.2

Thus, {eq}\alpha=0.2\\ (1-\alpha)=(1-0.2)=0.8 {/eq}

Thus, the expected value for alternative {eq}A_{3} {/eq} will be given by:

{eq}E[A_{3} ]=0.2\times 200+0.8\times 50=80 {/eq}

Where, {eq}A_{1},A_{2} and A_{3} {/eq} are Alternative 1, Alternative 2 and Alternative 3 respectively.

(b)

Now, we find the maximum of all calculated expected values.

{eq}max(124,130,80)=130 {/eq} which corresponds to Alternative 2.

Hence, according to the expected values, Alternative 2 must be chosen as it gives the maximum profit.

(c)

EVPI is the Expected Value of Perfect Information which is calculated by subtracting Expected Monetary Value(EMV) from the Expected Profit with Perfect Information (EPPI).

That is, {eq}EVPI=EPPI-EMV^{*} {/eq}

First calculating EMV for the courses of actions {eq}A_{1}, A_{2} ~and ~A_{3} {/eq}

EMV for {eq}A_{1}=0.3\times 100+0.5\times 120+0.2\times 180=126 {/eq}

EMV for {eq}A_{2}=0.3\times 120+0.5\times 140+0.2\times 120=130 {/eq}

EMV for {eq}A_{3}=0.3\times 200+0.5\times 100+0.2\times 50=120 {/eq}

{eq}EMV*=max(126, 130, 120)=130 {/eq}

EPPI is calculated as :

{eq}EPPI=\sum_{i}P(states~of~nature)\times (payoff~value~of~optimum~course~of~action~for ~that~particular~state~of~nature) {/eq}

The table below shows the calculations of Expected Profit with Perfect Information (EPPI):

States of Nature Probability Optimum Course of Action Payoff Value for Optimum Course of Action Weighted Opportunity Profit
A 0.3 A1 200 0.3X200=60
B 0.5 A2 140 0.5X140=70
C 0.2 A3 180 0.2X180=36

{eq}EPPI=60+70+36=166 {/eq}

Thus,

{eq}\begin{align} EVPI&= EPPI-EMV\\ &=166-130\\ &=36 \end{align} {/eq}