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Chapter 14Decision Analysis

- Problem Formulation
- Decision Making without Probabilities
- Decision Making with Probabilities
- Risk Analysis and Sensitivity Analysis
- Decision Analysis with Sample Information
- Computing Branch Probabilities
- Utility and Decision Making

Problem Formulation

- A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs.
- The decision alternatives are the different possible strategies the decision maker can employ.
- The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive.

Payoff Tables

- The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff.
- A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table.
- Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure.

Decision Trees

- A decision tree is a chronological representation of the decision problem.
- Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives.
- The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives.
- At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.

Decision Making without Probabilities

- Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are:
- the optimistic approach
- the conservative approach
- the minimax regret approach.

Optimistic Approach (MaxiMax)

- The optimistic approach would be used by an optimistic decision maker.
- The decision with the largest possible payoff is chosen.
- If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.

Conservative Approach (MaxiMin)

- The conservative approach would be used by a conservative decision maker.
- For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.)
- If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.)

Minimax Regret Approach

- The minimax regret approach requires the construction of a regret table or an opportunity loss table.
- This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature.
- Then, using this regret table, the maximum regret for each possible decision is listed.
- The decision chosen is the one corresponding to the minimum of the maximum regrets.

Example

Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits:

States of Nature

s1s2s3

d1 4 4 -2

Decisionsd2 0 3 -1

d3 1 5 -3

Maximax

payoff

Example- Optimistic Approach

An optimistic decision maker would use the optimistic (maximax) approach. We choose the decision that has the largest single value in the payoff table.

Maximum

DecisionPayoff

d1 4

d2 3

d3 5

Example

- Minimax Regret Approach

For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. In this example, in the first column subtract 4, 0, and 1 from 4; etc. The resulting regret table is:

s1s2s3

d1 0 1 1

d2 4 2 0

d3 3 0 2

Decision Making with Probabilities

- Expected Value Approach
- If probabilistic information regarding the states of nature is available, one may use the expected value (EV) approach.
- Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring.
- The decision yielding the best expected return is chosen.

Recall Probability Concepts

- Probability of any event is between 0 and 1.

An event with probability = 0 is an impossible event.

An event with probability = 1 is a sure thing.

- Probability of sample space = 1. That is, probability of one of the events from the sample space occurs is 1.

Which is random – State of Nature or Decisions?

Recall Probability Concepts

Joint, Marginal and Conditional Probabilities

Dem Rep Total

Rich 100 700 800

Poor 1500 900 2400

Total 1600 1600 3200

Marginal Probability: P(Rich) = 800/3200

P(Democrat) = 1600/3200

Joint Probability: P(Rich and Rep) = 700/3200

P(Poor and Dem) = 1500/3200

Recall Probability Concepts

Joint, Marginal and Conditional Probabilities

Dem Rep Total

Rich 100 700 800

Poor 1500 900 2400

Total 1600 1600 3200

Conditional Probability: | read as given.

P( Dem | Rich) = 100/800

P (Poor | Rep) = 900/1600

P(Rep | Poor) = 900/2400

Expected Value of a Decision Alternative

- The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative.
- The expected value (EV) of decision alternative di is defined as:

where: N = the number of states of nature

P(sj ) = the probability of state of nature sj

Vij = the payoff corresponding to decision alternative di and state of nature sj

Summation Notation

Ó is used to indicate adding a number of items.

If we have a list of 20 items, each item can be indexed by i and the list is represented by a variable x.

represents the first item and represents 15th item.

To add up all the items in the list . i is the index of

summation. This says add up numbers 1 thru 20.

To add number 4 thru 18 write

If what is added up is clear, for example, all the numbers, the index may be omitted. , for example says add all the numbers in the list

Example: Burger Prince

Burger Prince Restaurant is contemplating opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120. The payoff table for the three models is on the next slide.

Decision: Model.

State of Nature: Customers per hour

What does the decision maker choose?

Example: Burger Prince

- Payoff Table
- 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

Example: Burger Prince

- Expected Value Approach

Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d1, d2, d3 represent the decision alternatives of models A, B, C, and s1, s2, s3 represent the states of nature of 80, 100, and 120.

Example: Burger Prince

- Decision Tree

Payoffs

.4

s1

10,000

s2

.2

2

15,000

s3

.4

d1

14,000

.4

s1

8,000

d2

.2

1

3

s2

18,000

s3

d3

.4

12,000

.4

s1

6,000

4

s2

.2

16,000

s3

.4

21,000

- Expected Value For Each Decision
- Choose the model with largest EV, Model C.

EMV = .4(10,000) + .2(15,000) + .4(14,000)

= $12,600

2

d1

Model A

EMV = .4(8,000) + .2(18,000) + .4(12,000)

= $11,600

d2

Model B

1

3

d3

EMV = .4(6,000) + .2(16,000) + .4(21,000)

= $14,000

Model C

4

Expected Value of Perfect Information

- Frequently information is available which can improve the probability estimates for the states of nature.
- 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.

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 these optimal returns.

- Step 3:

Subtract the EV of the optimal decision from the amount determined in step (2).

Example: Burger Prince

- Expected Value of Perfect Information

Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision.

EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,000

Risk Analysis

- Risk analysis helps the decision maker recognize the difference between:
- the expected value of a decision alternative, and
- the payoff that might actually occur
- The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities.

Example: Burger Prince

- Risk Profile for the Model C Decision Alternative

.50

.40

.30

Probability

.20

.10

5 10 15 20 25

Profit ($thousands)

Sensitivity Analysis

- Sensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative:
- probabilities for the states of nature
- values of the payoffs
- If a small change in the value of one of the inputs causes a change in the recommended decision alternative, extra effort and care should be taken in estimating the input value.

Bayes’ Theorem and Posterior Probabilities

- Knowledge of sample or survey information can be used to revise the probability estimates for the states of nature.
- Prior to obtaining this information, the 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 outcomes of this analysis are called posterior probabilities or branch probabilities for decision trees.

Computing Branch Probabilities

- Branch (Posterior) Probabilities Calculation
- Step 1:

For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator.

- Step 2:

Sum these joint probabilities over all states -- this gives the marginal probability for the indicator.

- Step 3:

For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution.

Expected Value of Sample Information

- The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information.

Expected Value of Sample Information

- EVSI Calculation
- Step 1:

Determine the optimal decision and its expected return 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 returns.

- Step 3:

Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2).

Efficiency of Sample Information

- Efficiency of sample information is the ratio of EVSI to EVPI.
- As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.

Example: Burger Prince

- Sample Information

Burger Prince must decide whether or not to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable" or "unfavorable". The conditional probabilities are:

P(favorable | 80 customers per hour) = .2

P(favorable | 100 customers per hour) = .5

P(favorable | 120 customers per hour) = .9

Should Burger Prince have the survey performed by Stanton Marketing?

Example: Burger Prince

- Influence Diagram

Decision

Chance

Consequence

Market

Survey

Results

Avg. Number

of Customers

Per Hour

Market

Survey

Restaurant

Size

Profit

Example: Burger Prince

- Posterior Probabilities

Favorable

StatePriorConditionalJointPosterior

80 .4 .2 .08 .148

100 .2 .5 .10 .185

120 .4 .9 .36.667

Total .54 1.000

P(favorable) = .54

Example: Burger Prince

- Posterior Probabilities

Unfavorable

StatePriorConditionalJointPosterior

80 .4 .8 .32 .696

100 .2 .5 .10 .217

120 .4 .1 .04.087

Total .46 1.000

P(unfavorable) = .46

Example: Burger Prince

- Decision Tree (top half)

s1 (.148)

$10,000

s2 (.185)

4

d1

$15,000

s3 (.667)

$14,000

s1 (.148)

$8,000

d2

s2 (.185)

5

2

$18,000

s3 (.667)

I1

(.54)

d3

$12,000

s1 (.148)

$6,000

s2 (.185)

6

$16,000

s3 (.667)

1

$21,000

Example: Burger Prince

- Decision Tree (bottom half)

1

s1 (.696)

$10,000

s2 (.217)

I2

(.46)

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

Example: Burger Prince

- 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

Example: Burger Prince

- 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 this is less than the cost of the survey, the survey should not be purchased.

Example: Burger Prince

- Efficiency of Sample Information
- The efficiency of the survey:

EVSI/EVPI = ($900.88)/($2000) = .4504

Example: Weiss Advisors

Weiss Advisors have analyzed the profit potential of five different investments. The probabilities of the gains on $1000 are as follows:

Gain

Investment$0$200$500$1000

A .9 0 0 .1

B 0 .8 .2 0

C .05 .9 0 .05

D 0 .8 .1 .1

E .6 0 .3 .1

One of Weiss’ investors informs Weiss that he is indifferent between investments A, B, and C.

Example: Weiss Advisors

- Developing Utilities for Payoffs
- Assign a utility of 10 to a $1000 gain and a utility of 0 to a gain of $0. Let x = the utility of a $200 gain and y = the utility of a $500 gain. The expected utility on investment A is then .9(0) + .1(10) = 1.
- Since the investor is indifferent between investments A and C, this must mean the expected utility of investment C = the expected utility of investment A = 1. But the expected utility of investment C = .05(0) +.90x + .05(10). Since this must equal 1, solving for x, gives x = 5/9.

Example: Weiss Advisors

- Developing Utilities for Payoffs (continue)
- Also since the investor is indifferent between A, B, and C, the expected utility of investment B must be 1. Thus, 0(0) + .8(5/9) + .2y + 0(10) = 1. Solving for y, gives y = 25/9.
- Thus the utility values for gains of 0, 200, 500, and 1000 are 0, 5/9, 25/9, and 10, respectively.

Expected Utility Approach

- Once a utility function has been determined, the optimal decision can be chosen using the expected utility approach.
- Here, the expected utility for each decision alternative is computed as:
- The decision alternative with the highest expected utility is chosen.

Example: Risk Avoider

Consider a three-state, three-decision problem with the following payoff table in dollars:

s1s2s3

d1 +100,000 +40,000 -60,000

d2 +50,000 +20,000 -30,000

d3 +20,000 +20,000 -10,000

The probabilities for the three states of nature are: P(s1) = .1, P(s2) = .3, and P(s3) = .6.

Example: Risk Avoider

- Utility Table for Decision Maker
- s1s2s3
- d1 100 90 0
- d2 94 80 40
- d3 80 80 60

Example: Risk Avoider

- Expected Utility

Expected

s1s2s3 Utility

d1 100 90 0 37.0

d2 94 80 40 57.4

d3 80 80 60 68.0

Probability .1 .3 .6

Decision maker should choose decision d3.

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