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- Overview

Expected Value of Perfect Information

- Expected Value of Perfect Information

Expected Value of Perfect Information

Overview

Expected Value of Perfect Information is the increase in the expected profit that would result if one knew with certainty which state of nature would occur. It provides an upper bound on the expected value of any sample or survey information that better estimates the probability estimates for the states of nature.

Expected Value of Perfect Information

- Frequently, information is available that can better estimate the probabilities for the states of nature. (For example, you can take the time to listen to the radio to find out about the weather.)
- In the extreme, you can find the state of nature with certainty. (For example, you can call the Pepperdine hotline to see if PCH is open.)
- The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur.
- The EVPI provides an upper bound on the expected value of any sample or survey information that better estimates the probability estimates for the states of nature.

Expected Value of Perfect Information

- EVPI Calculation
- Step 1:

Determine the optimal return corresponding to each state of nature.

- Step 2:

Compute the expected value of those optimal returns.

- Step 3:

Subtract the EV of the optimal decision without knowing the state of nature from the amount determined in Step 2.

Expected Value of Perfect Information

- How much should Medieval Times be willing to pay to determine the average number of customers per hour (the state of nature)?
- The probabilities of states s1, s2, s3 were .4, .2, and .4:
- Maximum expected profitwas EV(C) = .4($6,000) + .2($16,000) + .4($21,000) = $14,000

Average Number of Customers Per Hour

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

Model C$ 6,000 $16,000 $21,000

Expected Value of Perfect Information

- If you knew the state were s1, choose A and earn $10,000.
- If you knew the state were s2, choose B and earn $18,000
- If you knew the state were s3, choose C and earn $21,000
- EV = .4(10,000) + .2(18,000) + .4(21,000) = $16,000
- EV(perfect information) is $2,000 than EV(no info).

Average Number of Customers Per Hour

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

Model C$ 6,000 $16,000 $21,000

- Bayes’ Rule

Overview

Bayes’ Rule revises prior probability estimates for the states of nature into posterior probabilities. The rule uses conditional probabilities for the outcomes or indicators of the sample or survey information.

- Knowledge of sample (survey) information can better estimate the probabilities for the states of nature.
- Before getting this information, probability estimates for the states of nature are called prior probabilities.
- With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem. (The accuracy of a sample or survey is measured by its conditional probabilities.)
- The revised probabilities are called posterior probabilities or branch probabilities for decision trees.

- For example, suppose a certain drug test will correctly identify a drug user as testing positive 99% of the time, and will correctly identify a non-user as testing negative 99% of the time.
- Suppose a corporation decides to test its employees for opium use, and the corporation believes 0.5% of the employees use the drug. We want to know the probability that, given a positive drug test, an employee is actually a drug user.

- Let "D" be the event of being a drug user; and "N“, being a non-user. Let "+" be the event of a positive drug test.
- The population is defined by P(D), or the prior probability that the employee is a drug user. This is 0.005, since 0.5% of the employees are drug users. Hence, P(N), or the probability that the employee is not a drug user, is 1-P(D), or 0.995.
- The accuracy of the drug test is defined by two numbers:
- P(+|D), or the probability that the test is positive, given that the employee is a drug user. This is 0.99, since the test will correctly identify a drug user as testing positive 99% of the time.
- P(+|N), or the probability that the test is positive, given the employee is not a user. This is 0.01, since the test produces a false positive for 1% of non-users.

Hence, follow a sequence of steps to compute the probability a person is a drug user given a positive test:

- The probability a person is a drug user and tests positive is

P(D&+) = P(+|D) x P(D) = 0.99 x 0.005 = 0.00495 = 0.495%

- The probability a person is a non-drug user but tests positive is

P(N&+) = P(+|N) x P(N) = 0.01 x 0.995 = 0.00995 = 0.995%

- The probability a person tests positive

P(+) = P(D&+) + P(N&+) = 0.0149 = 1.49%.

- The probability a person is a drug user given a positive test

P(D|+) = P(D&+)/P(+) = 0.00495/0.0149 = .3322.

- Even though the test is 99% accurate, P(D|+) = .3322, only 33.22% of those testing positive are drug users!

Expected Value of Sample Information

- Expected Value of Sample Information

Expected Value of Sample Information

Overview

Expected Value of Sample Information is the additional expected profit possible through knowledge of the sample or survey information. It is less than the Expected Value of Perfect Information when samples and surveys are imperfect.

Expected Value of Sample Information

- In general, the expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information.
- EVSI calculation
- Step 1: Determine the optimal decision and its expected profit for the possible outcomes of the sample using the posterior probabilities for the states of nature.
- Step 2:Compute the expected value of these optimal profits.
- Step 3: Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in Step 2.

Question: Medieval Times Dinner Theater is considering opening a new theater on Main Street. It has three different models, each with a different seating capacity. Medieval Times estimates that the average number of customers per show will be 800, 1000, or 1200. Here is the payoff table for the three models:

Average Number of Customers Per Hour

s1 = 800 s2 = 1000 s3 = 1200

Model A $10,000 $15,000 $14,000

Model B $ 8,000 $18,000 $12,000

Model C $ 6,000 $16,000 $21,000

Expected Value of Sample Information

Medieval Times Dinner Theater must decide whether or not to spend $1,000 for a marketing survey from Stanton Marketing. The two possible results of the survey are "favorable" or "unfavorable". The conditional probabilities are:

P(favorable | 800 customers per show) = .2

P(favorable | 1000 customers per show) = .5

P(favorable | 1200 customers per show) = .9

Should Medieval Times spend $1,000 for the survey from Stanton Marketing if the prior probabilities of states s1, s2, and s3 are .4, .2, and .4?

Expected Value of Sample Information

Answer: Compare $1000 to EVSI. Compute EVSI in three steps.

- Step 1: Determine the optimal decision and its expected profit for the possible outcomes of the sample using the posterior probabilities for the states of nature.
- Step 2:Compute the expected value of these optimal profits.
- Step 3: Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in Step 2.

Expected Value of Sample Information

Favorable survey result

StatePriorConditionalJointPosterior

800 .4 .2 .08 .148

1000 .2 .5 .10 .185

1200 .4 .9 .36.667

Total .54 1.000

P(favorable) = .54

- For the first state (80 customers),
- Joint probability = 0.08 = 0.4 x 0.2 = Prior x Conditional
- Marginal P(favorable) = 0.54 = 0.08+0.10+0.36 = SJoints
- Posterior probability = 0.148 = 0.08/0.54 = Joint/Marginal

Expected Value of Sample Information

Unfavorable survey result

StatePriorConditionalJointPosterior

800 .4 .8 .32 .696

1000 .2 .5 .10 .217

1200 .4 .1 .04.087

Total .46 1.000

P(unfavorable) = .46

- For the first state (80 customers),
- Joint probability = 0.32 = 0.4 x 0.8 = Prior x Conditional
- Marginal P(unfavorable) = 0.46 = 0.32+0.10+0.04 = SJoints
- Posterior probability = 0.696 = 0.32/0.46 = Joint/Marginal

Expected Value of Sample Information

- Top half (favorable survey result)

s1 (.148)

$10,000

s2 (.185)

4

$15,000

d1

s3 (.667)

$14,000

s1 (.148)

$8,000

d2

s2 (.185)

5

2

$18,000

s3 (.667)

I1

(.54)

$12,000

d3

s1 (.148)

$6,000

s2 (.185)

6

$16,000

s3 (.667)

1

$21,000

Expected Value of Sample Information

- Bottom half (unfavorable survey result)

1

s1 (.696)

$10,000

I2

(.46)

s2 (.217)

7

$15,000

d1

s3 (.087)

$14,000

s1 (.696)

$8,000

d2

s2 (.217)

8

3

$18,000

s3 (.087)

$12,000

d3

s1 (.696)

$6,000

s2 (.217)

9

$16,000

s3 (.087)

$21,000

Expected Value of Sample Information

- EMV = .148(10,000) + .185(15,000)
- + .667(14,000) = $13,593

d1

4

- $17,855

d2

- EMV = .148 (8,000) + .185(18,000)
- + .667(12,000) = $12,518

5

2

I1

(.54)

d3

- EMV = .148(6,000) + .185(16,000)
- +.667(21,000) = $17,855

6

1

- EMV = .696(10,000) + .217(15,000)
- +.087(14,000)= $11,433

7

d1

I2

(.46)

d2

- EMV = .696(8,000) + .217(18,000)
- + .087(12,000) = $10,554

8

3

d3

- $11,433

- EMV = .696(6,000) + .217(16,000)
- +.087(21,000) = $9,475

9

Expected Value of Sample Information

If the outcome of the survey is "favorable”, choose Model C. If it is “unfavorable”, choose Model A.

EVSI = .54($17,855) + .46($11,433) - $14,000 = $900.88

Since that is less than the cost of the survey, the survey should not be purchased.

- Review Questions
- You should try to answer some of the following questions before the next class.
- You will not turn in your answers, but students may request to discuss their answers to begin the next class.
- Your upcoming Final Exam will contain some similar questions, so you should eventually consider every review question before taking your exams.

Review 1: Expected Value of Sample Information

- Review 1: Expected Value of Sample Information

Review 1: Expected Value of Sample Information

The Gorman Manufacturing Company must decide whether to make a component part at its Milan, Michigan, plan or buy the part from a supplier. The profit depends on the demand for the product. The following table shows the profit (in thousands of dollars):

States of Demand

s1(low) s2(med) s3 (high)

Manufacture, d1 -20 40 100

Purchase,d2 10 45 70

Review 1: Expected Value of Sample Information

States of Demand

s1(low) s2(med) s3 (high)

Manufacture, d1 -20 40 100

Purchase,d2 10 45 70

Compute the optimal decision (which maximizes expected value) given priors

P(s1) = 0.35, P(s2) = 0.35, P(s3) = 0.30.

EV(d1) = 0.35 x (-20) + 0.35 x (40) + 0.30 x (100) = 37.

EV(d2) = 0.35 x (10) + 0.35 x (45) + 0.30 x (70) = 40.25.

So, choose Purchase, d2.

Review 1: Expected Value of Sample Information

States of Demand

s1 (low) s2 (med) s3 (high)

Manufacture, d1 -20 40 100

Purchase, d2 10 45 70

Use EVPI (expected value of perfect information) to determine whether Gorman should try to better estimate demand.

If you had perfect information, then in state s1, pick d2 for payoff 10; in state s2, pick d2 for payoff 45; and in state s3, pick d1 for payoff 100. Hence, the expected payoff is 0.35 x (10) + 0.35 x (45) + 0.30 x (100) = 49.25, which is 9 more than the expected payoff EV(d2) = 40.25 from the optimum given the priors. EVPI thus = 9 (that is, 9 thousand), which is positive so information is valuable.

Review 1: Expected Value of Sample Information

States of Demand

s1 (low) s2 (med) s3 (high)

Manufacture, d1 -20 40 100

Purchase, d2 10 45 70

What is Gorman’s optimal strategy (which maximizes expected value) if it has access to a market study that concludes a F = Favorable or U = Unfavorable outcome with the following conditional probabilities?

P(favorable | low state s1) = 0.1

P(favorable | med state s2) = 0.4

P(favorable | high state s3) = 0.6

Review 1: Expected Value of Sample Information

States of Demand

s1 (low) s2 (med) s3 (high)

Manufacture, d1 -20 40 100

Purchase, d2 10 45 70

Favorable survey result

StatePrior P(si)Cond. P(F|si)Joint P(si&F)Post P(si|F)

s1 0.35 0.1 0.035 0.0986

s2 0.35 0.4 0.140 0.3944

s3 0.30 0.6 0.1800.5070

P(favorable) = 0.355

- EV(d1) = 0.0986 x (-20) + 0.3944 x (40) + 0.5070 x (100) = 64.51.
- EV(d2) = 0.0986 x (10) + 0.3944 x (45) + 0.5070 x (70) = 54.23.
- So, if the survey is Favorable, choose Manufacture, d1.

Review 1: Expected Value of Sample Information

States of Demand

s1 (low) s2 (med) s3 (high)

Manufacture, d1 -20 40 100

Purchase, d2 10 45 70

Unfavorable survey result

StatePrior P(si)Cond. P(U|si)Joint P(si&U)Post P(si|U)

s1 0.35 0.9 0.315 0.4884

s2 0.35 0.6 0.210 0.3256

s3 0.30 0.4 0.1200.1860

P(Unfavorable) = 0.645

- EV(d1) = 0.4884 x (-20) + 0.3256 x (40) + 0.1860 x (100) = 21.86.
- EV(d2) = 0.4884 x (10) + 0.3256 x (45) + 0.1860 x (70) = 32.56.
- So, if the survey is Unfavorable, choose Purchase, d2.

Review 1: Expected Value of Sample Information

States of Demand

s1 (low) s2 (med) s3 (high)

Manufacture, d1 -20 40 100

Purchase, d2 10 45 70

- The survey is Favorable with probability P(Favorable) = 0.355. And contingent on Favorable, the expected payoff is 64.51.
- The survey is Unfavorable with probability P(Unfavorable) = 0.645. And contingent on Unfavorable, the expected payoff is 32.56.
- Therefore, the un-contingent expected payoff is

0.355 x 64.51 + 0.645 x 32.56 = 43.90.

- The expected payoff with no information (priors) is 40.25.
- Therefore, the EVSI = 43.90-40.25 = 3.65 ($3,650).

Review 2: Expected Value of Sample Information

- Review 2: Expected Value of Sample Information

- Dollar Department Stores has received an offer from Harris Diamonds to purchase Dollar’s store on Grove Street for $120,000. Dollar is an expected-value maximizer. Dollar has determined probability estimates of the store's future profitability, based on economic outcomes, as:
- P($80,000) = .2, P($100,000) = .3, P($120,000) = .1, and P($140,000) = .4.
- Should Dollar sell the store on Grove Street?
- What is the EVPI?
- Dollar can have an economic forecast performed. The forecast indicates either G = Good business conditions or B = Bad business conditions. Probabilities of the indicators conditional on future profitability are P(G|$80,000) = .1; P(G|$100,000) = .2; P(G|$120,000) = .6; P(G|$140,000) = .3. Should Dollar purchase that forecast for $10,000? For $1,000?

- This problem can be solved like the previous problem by first converting the profit data and selling price into a payoff table:
- P(s1 = $80,000) = .2, P(s2 = $100,000) = .3, P(s3 = $120,000) = .1, and P(s4 = $140,000) = .4
- Offer to sell for $120,000.

States of Demand

- s1s2s3 s4
- Do not sell, d180,000 100,000 120,000 140,000
- Sell, d2120,000120,000 120,000 120,000

- Consider priors
- P($80,000) = .2, P($100,000) = .3, P($120,000) = .1, and P($140,000) = .4.
- Should Dollar sell the store on Grove Street?
- Expected profit (in thousands of $) = .2 x 80 + .3 x 100 + .1 x 120 + .4 x 140 = 114, so selling for $120,000 increases value from $114,000.
- Here is a more elaborate way to say the same thing:
- EV(d1=Not sell) = .2x80 + .3x100 + .1x120 + .4x140 = 114.
- EV(d2=Sell) = .2x120 + .3x120 + .1x120 + .4x120 = 120.
- So, choose Sell, d2.

- Consider priors
- P($80,000) = .2, P($100,000) = .3, P($120,000) = .1, and P($140,000) = .4.
- What is the EVPI? Expected value of perfect information is what you can expect if you know the store’s future profitability before you have to decide whether to sell.
- If future profit is either $80,000 or $100,000, then you sell for $120,000 (choose d2); if future profit is $120,000, then it does not matter whether you sell or not (choose d1 or d2); and if future profit is $140,000, then you do not sell (choose d1).
- So, with perfect information, you earn $120,000, except in the probability P($140,000) = .4 state 4 event, when you earn $140,000. So you can expect .6 x $120,000 + .4 x $140,000 = $128,000, which is $8,000 more than selling for $120,000.
- Therefore, EVPI = $8,000.

- Consider priors
- P($80,000) = .2, P($100,000) = .3, P($120,000) = .1, and P($140,000) = .4.
- EVPI = $8,000.
- Should Dollar purchase the forecast for $10,000?
- No, because the value of the forecast is at most EVPI, which is less than $10,000.

- Should Dollar purchase the forecast for $1,000?
- First, compute posterior probabilities under the two possible forecasts: G = Good business conditions or B = Bad business conditions.
- If G, then
- Priors Conditional Joint Posterior
- P(s1) = .2 P(G|s1) = .1 P(G&s1) = .02 P(s1|G) = .02/.26 = .077
- P(s2) = .3 P(G|s2) = .2 P(G&s2) = .06 P(s2|G) = .06/.26 = .231
- P(s3) = .1 P(G|s3) = .6 P(G&s3) = .06 P(s3|G) = .06/.26 = .231
- P(s4) = .4 P(G|s4) = .3 P(G&s4) = .12 P(s4|G) = .12/.26 = .462

P(G) = .26

- Expected Value of Store Given G =

.077x$80,000 + .231x$100,000+.231x$120,000+.462x$140,000=$121,660

- Since that value is greater than $120,000, you do not sell the store if the report is G = Good business conditions.

- Should Dollar purchase the forecast for $1,000?
- If B, then
- Priors Conditional Joint Posterior
- P(s1) = .2 P(B|s1) = .9 P(B&s1) = .18 P(s1|B) = .18/.74 = .243
- P(s2) = .3 P(B|s2) = .8 P(B&s2) = .24 P(s2|B) = .24/.74 = .324
- P(s3) = .1 P(B|s3) = .4 P(B&s3) = .04 P(s3|B) = .04/.74 = .054
- P(s4) = .4 P(B|s4) = .7 P(B&s4) = .28 P(s4|B) = .28/.74 = .378

P(B) = .74

- Expected Value of Store Given B =

.243x$80,000 + .324x$100,000+.054x$120,000+.378x$140,000=$111,240

- Since that value is less than $120,000, you do sell the store if the report is B = Bad business conditions.

- Putting it all together,
- P(G) = .26 of a Good report and expected profit =$121,660.
- P(B) = .74 of a Bad report and expected profit $120,000.
- Overall, expected profit is .26x$121,660 + .74$120,000 = $120,431.
- Therefore, EVSI = $120,431-$120,000 = $431, and you should not pay $1,000 for the economic forecast.

BA 452 Quantitative Analysis

End of Lesson C.4
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