decision theory

Dr. C. Lightner Fayetteville State University

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Chapter 14Decision Analysis

Problem FormulationDecision Making without ProbabilitiesDecision Making with ProbabilitiesRisk Analysis and Sensitivity AnalysisDecision Analysis with Sample InformationComputing Branch ProbabilitiesUtility and Decision Making


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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 will ultimately affect decision results. States of nature should be defined so that they are mutually exclusive and contain all possible future events that could affect the results of all potential decisions.


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Decision Theory Models

Decision theory problems are generally represented as one of the following:

– Influence Diagram– Payoff Table– Decision Tree


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Influence Diagrams

An influence diagram is a graphical device showing the relationships among the decisions, the chance events, and the consequences.Squares or rectangles depict decision nodes.Circles or ovals depict chance nodes.Diamonds depict consequence nodes.Lines or arcs connecting the nodes show the direction of influence.


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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.


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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.


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Example: CAL Condominium Complex

A developer must decide how large a luxury condominium complex to build – small, medium, or large. The profitability of this complex depends upon the future level of demand for the complex’s condominiums.


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CAL Condos: Elements of Decision Theory

States of nature: The states of nature could be defined as low demand and high demand.Alternatives: CAL could decide to build a small, medium, or large condominium complex.Payoffs: The profit for each alternative under each

potential state of nature is going to be determined.

We develop different models for this problem on the following slides.


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CAL Condos: Payoff Table

Alternatives Low HighSmall 8 8Medium 5 15Large -11 22

States of Nature

(payoffs in millions of dollars)

THIS IS A PROFIT PAYOFF TABLE


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CAL Condos: Decision Tree

Small Complex

Medium Complex

Large Complex

Low demand

Low demand

Low demand

High demand

High demand

High demand

8

8

5

15

22

-11


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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.


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Optimistic Approach

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


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Conservative Approach

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


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Minimax Regret Approach

1. 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 best payoff for that state of nature.

2. Then, using this regret table, the maximum regret for each possible decision is listed.

3. The decision chosen is the one corresponding to the minimum of the maximum regrets.


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Solving CAL Condominiums Problem

Suppose that information regarding the probability (or likelihood) that there will be a high or low demand is unavailable.– A conservative or pessimistic decision maker would select the

decision alternative determined by the conservative approach.– An optimistic decision maker would select the decision

alternative rendered by the optimistic approach. – The minimax regret approach is generally selected by a

decision maker who reflects on decisions “after the fact”, and complains about or “regrets” their decisions based upon the profits that they could have made (or cheaper costs that they could have spent) had a different decision been selected.


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CAL Condos: Optimistic Decision

If the optimistic approach is selected:STATES OF NATURE BEST

Alternatives Low High PROFITSmall 8 8 8Medium 5 15 15Large -11 22 22

MaximaMaximax payoffx payoff

MaximaMaximax x

decisiondecision


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CAL Condos: Conservative Decision

If the conservative approach is selected:STATES OF NATURE WORST

Alternatives Low High PROFIT Small 8 8 8 Medium 5 15 5 Large -11 22 -11

Maximin Maximin payoffpayoff

MaximiMaximin n

decisiodecisionn

The decision with the best profit from the column of worst profits is selected.


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CAL Condos: Minimax Regret Decision

If the minimax regret approach is selected:Step 1: Determine the best payoff for each state of nature and create a regret table.

STATES OF NATURE Alternatives Low High Small 8 8 Medium 5 15 Large -11 22

Best Profit for Low 8

Best Profit for High 22


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If the minimax regret approach is selected:Step 1: Create a regret table (continued).

STATES OF NATURE Alternatives Low High Small 0 14 Medium 3 7 Large 19 0

For a profit payoff table, entries in the regret table represent profits that could have been earned.

If they knew in advanced that the demand would be low, they would have built a small complex. Without this “psychic insight”, if they decided to build a medium facility and the demand turned out to be low, they would regret building a medium complex because they only made 5 million dollars instead of 8 million had they built a small facility instead. They regret their decision by 3 million dollars.


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If the minimax regret approach is selected:Step 2: Create a regret table (continued). Step 3: Determine the maximum regret for each decision.

STATES OF NATURE Max Alternatives Low High Regret Small 0 14 14 Medium 3 7 7 Large 19 0 19

Regret not getting a profitof 19 more than not makinga profit of 0.


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If the minimax regret approach is selected:Step 4: Select the decision with the minimum value from the column of max regrets.

STATES OF NATURE Max Alternatives Low High Regret Small 0 14 14 Medium 3 7 7 Large 19 0 19

MinimaMinimaxx

RegretRegretpayoffpayoff

MinimaMinimaxx

Regret Regret decisiondecision


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Generic Example

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

States of Nature s1 s2 s3

d1 4.5 3 2

Decisions d2 0.5 4 1

d3 1 5 3

COST PAYOFF TABLE


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Generic Example : Optimistic Decision

Optimistic Approach An optimistic decision maker would use the optimistic

(maximax) approach. We choose the decision that has the best single value in the payoff table.

Best Decision Cost

d1 2 d2 0.5

d3 1

MaximaxMaximaxpayoffpayoffMaximaxMaximax

decisiondecision


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Generic Example: Conservative Approach

Conservative ApproachA conservative decision maker would use the conservative

(maximin) approach. List the worst payoff for each decision. Choose the decision with the best of these worst payoffs.

Worst Decision Payoff

d1 4.5 d2 4

d3 5

MaximinMaximindecisiondecision

MaximinMaximinpayoffpayoff


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Minimax Regret Approach

States of Nature s1 s2 s3

d1 4.5 3 2

Decisions d2 0.5 4 1 d3 1 5 3

Best cost for each state of nature.

Generic Example: Minimax Regret Decision

For a cost payoff table, entries in the regret table represent overpayments (i.e. higher costs incurred).


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Example

Minimax Regret Approach (continued)For each decision list the maximum regret. Choose the

decision with the minimum of these values.

States of Nature Max s1 s2 s3 Regret

d1 4 0 1 4 Decisions d2 0 1 0 1 d3 0.5 2 2 2

MinimaxMinimaxdecisiondecision

MinimaxMinimaxregretregret


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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.


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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

EV( ) ( )d P s Vi j ijj

N

1

EV( ) ( )d P s Vi j ijj

N

1


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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 (profits) for the three models is on the next slide.


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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


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Expected Value ApproachCalculate 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.


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Decision Tree

1

.2

.4

.4

.4

.2

.4

.4

.2

.4

d1

d2

d3

s1

s1

s1

s2s3

s2

s2s3

s3

Payoffs10,00015,00014,0008,000

18,00012,0006,000

16,00021,000

2

3

4


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Expected Value For Each Decision

Choose the model with largest EV, Model C.

3

d1

d2

d3

EMV = .4(10,000) + .2(15,000) + .4(14,000) = $12,600

EMV = .4(8,000) + .2(18,000) + .4(12,000) = $11,600

EMV = .4(6,000) + .2(16,000) + .4(21,000) = $14,000

Model A

Model B

Model C

2

1

4


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CAL Condos Revisited

Suppose market research was conducted in the community where the complex will be built. This research allowed the company to estimate that the probability of low demand will be 0.35, and the probability of high demand will be 0.65. Which decision alternative should they select.


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STATES OF NATURE Alternatives Low (0.35) High (0.65) Small 8 8 Medium 5 15 Large -11 22


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STATES OF NATUREAlternatives Low High

(0.35) (0.65) Expected value (EV)Small 8 8 8(0.35) + 8(0.65) = 8Medium 5 15 5(0.35) + 15(0.65) = 11.5Large -11 22 -11(0.35) + 22(0.65) =

10.45

Recall that this is a profit payoff table. Thus since the decision to build a medium complex has the highest expected profit, this is our best decision.


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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.


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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).


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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


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Sensitivity Analysis

Some of the quantities in a decision analysis, particularly the probabilities, are often intelligent guesses at best.It is important to accompany any decision analysis with a 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.


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Sensitivity Analysis

One approach to sensitivity analysis is to arbitrarily assign different values to the probabilities of the states of nature and/or the payoffs and resolve the problem. If the recommended decision changes, then you know that the solution is sensitive to the changes.For the special case of two states of nature, a graphical technique can be used to determine how sensitive the solution is to the probabilities associated with the states of nature.


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CAL Condos: Sensitivity Analysis

This problem has two states of nature. Previously, we stated that CAL Condominiums estimated that the probability of future low demand is 0.35 and 0.65 is the probability of high demand. These probabilities yielded the recommended decision to build the medium complex.In order to see how sensitive this recommendation is to changing probability values, we will let p equal the probability of low demand. Thus (1-p) is the probability of high demand. ThereforeEV( small) = 8*p + 8*(1-p)= 8EV( medium) = 5*p + 15*(1-p) = 15 – 10pEV( large) = -11*p + 22*(1-p) = 22 – 33p


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Next we will plot the expected value lines for each decision by plotting p on the x axis and EV on the y axis. EV( small) = 8EV( medium) = 15 – 10pEV( large) = 22 – 33p


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0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

EV( medium)

EV( large)

EV( small)


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Since CAL condominiums list payoffs are in terms of profits, we know that the highest profits is desirable.Look over the entire range of p (p=0 to p=1) and determine the range over which each decision yields the highest profits.


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0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

EV( medium)

EV( large)

EV( small)

B1 B2


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Do not estimate the values of B1 or B2 (the points where the intersection of lines occur). Determine the exact intersection points.B1 is the point where the EV( large) line intersects with the EV( medium) line:

To find this point set these two lines equal to each other and solve for p. 22-33p= 15-10p

7= 23pp=7/23= 0.3403

B2 is the point where the EV( medium) line intersects with the EV( small) line:

15-10p = 8 7 = 10p

p = 0.7

So B1 equals 0.3403

So B2 equals 0.7


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0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

EV( medium)

EV( large)

EV( small)

0.3403 0.7


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From the graph we see that if the probability of low demand (p) is between 0 and 0.3403, we recommend building a large complex.From the graph we see that if the probability of low demand (p) is between 0.3403 and 0.7, we recommend building a medium complex.From the graph we see that if the probability of low demand (p) is between 0.7 and 1, we recommend building a large complex.

From this sensitivity analysis we see that if CAL Condos estimate of 0.35 for the probability of low demand was slightly lower, the recommended decision would change.


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End of Chapter 14

See your textbook for more examples and detailed explanations

of all topics discussed in these notes.

decision theory

Documents

decision nodes

decision treesa decision

decision maker

decision results

probabilities decision

potential state of nature

different states of

square nodes