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Decision Analysis

(to p2)

What is it?

What is the objective?

More example

Tutorial:

8th ed:: 5, 18, 26, 37

9th ed: 3, 12, 17, 24

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What is it?

It concerns with making a decision or an action based on a series of possible outcomes

The possible outcomes are normally presented in a table format called

“ Payoff Table”

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Payoff Table

A payoff table consists of

Columns: state of nature (ie events)

Rows: choice of decision

It corresponding entries are “payoff values”

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Example

Payoff Table

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Objective

Our objective here is to determine which decision should be chosen based on the payoff table

How to achieve it?

Simple two-dimensional cases

Posterior analysis with Bayesian rules

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How to achieve it?

- The final decision could be based on one of the following criteria:

1) maximax,

2) maximin,

3) minimax regret,

4) Hurwicz,

5) equal likelihood

- Each one is better one?
- Ex 4 and 7
- QM programs

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The Maximax Criterion

Known as “optimistic” person, he/she will pick the decision that has the maximum of maximum payoffs .

Example:

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How to derive this decision?

Step1: For each row (or decision), select its max values

Step 2: Choose the decision that has the max value of

Step1

Example:

Max

50,000

100,000

30,000

max

100,000

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The Maximin Criterion

Known as the “pessimistic” person, she/he will select the decision that has the maximum of the minimum from the payoff table

Example

Select this action

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How to obtain it?

Step 1: For each row (or decision), obtain its min values

Step 2: Choose the decision that has the max value from step 1

Min

30,000

-40,000

10,000

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The Minimax: Regret Criterion

Regret is the difference between the payoff from the best decision and all other decision payoffs.

The decision maker attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret.

Solution

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How to obtain it?

Step 1: Find out the max value for each column

Step 2: Subtract value of step 1 to all entry of that column

Step 3: Find out the max value for each row

Step 4: Choose the decision that has smallest value of step 3

Example

Step 1:

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Max 100,000 30,000

Step 2:Subtract value of step 1 to all entry of that column

(100,000 - 50,000)

(30,000 - 30,000)

Note: this is a regret table.

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Step 3: Find out the max value for each row

Step 4: Choose the decision that has smallest value of step 3

Max

50,000

70,000

70,000

Min

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Select this decision

Decision Making without ProbabilitiesThe Hurwicz Criterion

- The Hurwicz criterion is a compromise between the maximax and maximin criterion.

- A coefficient of optimism, , is a measure of the decision maker’s optimism.

- The Hurwicz criterion multiplies the best payoff by and the worst payoff by 1- ., for each decision, and the best result is selected.

Decision Values

Apartment building $50,000(.4) + 30,000(.6) = 38,000

Office building $100,000(.4) - 40,000(.6) = 16,000

Warehouse $30,000(.4) + 10,000(.6) = 18,000

Example:

Select this,max

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Decision Making without ProbabilitiesThe Equal Likelihood Criterion

- The equal likelihood ( or Laplace) criterion multiplies the decision payoff for each state of nature by an equal weight, thus assuming that the states of nature are equally likely to occur.

DecisionValues

Apartment building $50,000(.5) + 30,000(.5) = 40,000

Office building $100,000(.5) - 40,000(.5) = 30,000

Warehouse $30,000(.5) + 10,000(.5) = 20,000

Example:

Select this, Max

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Decision Making without ProbabilitiesSummary of Criteria Results

A dominant decision is one that has a better payoff than another decision under each state of nature.

The appropriate criterion is dependent on the “risk” personality and philosophy of the decision maker.

Criterion Decision (Purchase)

Maximax Office building

Maximin Apartment building

Minimax regret Apartment building

Hurwicz Apartment building

Equal liklihood Apartment building

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Decision Making without ProbabilitiesSolutions with QM for Windows (1 of 2)

Exhibit 12.1

Posterior analysis with Bayesian rules

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Here, we need to understand the following concepts:

1) Expected values

2) Expected Opportunity loss

3) Expected values of Perfect Information

(Ex. 18 and 26)

4) Decision Tree

a) basic tree

b) sequential tree

5) Posterior analysis with Bayesian rules

(Ex. 33, 35 and 37)

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Expected Value

- Expected value is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence.
- EV(Apartment) = $50,000(.6) + 30,000(.4) = 42,000
- EV(Office) = $100,000(.6) - 40,000(.4) = 44,000
- EV(Warehouse) = $30,000(.6) + 10,000(.4) = 22,000

Table 12.7 Payoff table with Probabilities for States of Nature

Select this, Max

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Decision Making with ProbabilitiesExpected Opportunity Loss

The expected opportunity loss is the expected value of the regret for each decision (Minimax)

EOL(Apartment) = $50,000(.6) + 0(.4) = 30,000

EOL(Office) = $0(.6) + 70,000(.4) = 28,000

EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000

Table 12.8 Regret (Opportunity Loss) Table with Probabilities for States of Nature

Select this, Min

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Decision Making with ProbabilitiesSolution of Expected Value Problems with QM for Windows

Exhibit 12.4

Decision Making with ProbabilitiesSolution of Expected Value Problems with Excel and Excel QM(1 of 2)

Exhibit 12.5

Decision Making with ProbabilitiesSolution of Expected Value Problems with Excel and Excel QM(2 of 2)

Exhibit 12.6

Expected Value of Perfect Information

The expected value of perfect information (EVPI) is the maximum amount a decision maker would pay for additional information.

EVPI = (expected value with perfect information) –(expected value without perfect information)

EVPI = the expected opportunity loss (EOL) for the best decision.

How to compute them?

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Decision Making with ProbabilitiesEVPI Example

Table 12.9 Payoff Table with Decisions, Given Perfect Information

Best outcome Worst outcome

Best outcome for each column

Decision with perfect information: $100,000(.60) + 30,000(.40) = $72,000 Decision without perfect information: EV(office) = $100,000(.60) - 40,000(.40) = $44,000 EVPI = $72,000 - 44,000 = $28,000 EOL(office) = $0(.60) + 70,000(.4) = $28,000

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A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches).

Decision Trees (1 of 2)Table 12.10

Payoff Table for Real Estate Investment Example

Figure 12.1

Decision tree for real estate investment example

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Decision Trees (2 of 2)

The expected value is computed at each probability node:

EV(node 2) = .60($50,000) + .40(30,000) = $42,000

EV(node 3) = .60($100,000) + .40(-40,000) = $44,000

EV(node 4) = .60($30,000) + .40(10,000) = $22,000

And

Choose branch with max value:

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Use to denote not chosen path(s)

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Decision Making with ProbabilitiesDecision Trees with QM for Windows

Exhibit 12.8

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan(1 of 4)

Exhibit 12.9

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan(2 of 4)

Exhibit 12.10

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan(3 of 4)

Exhibit 12.11

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan(4 of 4)

Exhibit 12.12

Sequential Decision Trees(1 of 2)

A sequential decision tree uses to illustrate a series of decisions.

.

How to make use of them?

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Sequential Decision Trees(2 of 2)

Compute the expected values for each stage

Then choose action that produces the highest value.

Extra information

0.6*2M+0.4*0.225M

Max (1.29M-0.8M, 1.36M-0.2M)

=1.16M

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Bayesian Analysis(1 of 3)

Bayesian analysis deals with posterior information, and it can be computed based on their conditional probabilities

Example: Consider the following example. Using expected value criterion, best decision was to purchase office building with expected value of $444,000, and EVPI of $28,000.

Now, what

posterior information

can be obtained here?

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Bayesian Analysis(2 of 3)

Let consider the following information:

g = good economic conditions

p = poor economic conditions

P = positive economic report

N = negative economic report

From Economic analyst,we can obtain the following “conditional probability”, the probability that an event will occur given that another event has already occurred.

P(Pg) = .80

P(Ng) = .20

P(Pp) = .10

P(Np) = .90

From about information, we can further compute the posterior information such as:

P(gP)

How it works?

New

Total = 0.8+0.2 = 1.0

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Bayesian Analysis(3 of 3)

(A posterior probability is the altered marginal probability of an event based on additional

information)

Example for its application:

Let P(g) = .60; P(p) = .40

Then, the posterior probabilities is:

P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)] = (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923

Systematic way to compute it?

Other posterior (revised) probabilities are:

P(gN) = .250

P(pP) = .077

P(pN) = .750

Where are these values located/appeared in our tree diagram?

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Form this table to compute them

0.52

Table 12.12

Computation of Posterior Probabilities

P(P) = P(P/g)P(g) +P(P/p) P(p) =0.8* 0.6 +0.1* 0.4 = 0.52

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Decision Trees with Posterior Probabilities (1 of 2)

- Decision tree below differs from earlier versions in that :

1. Two new branches at beginning of tree represent report outcomes;

2. Newly introduced

Figure 12.5

Decision tree with posterior probabilities

Old one

Making Decision

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Decision Trees with Posterior Probabilities (2 of 2)

- EV (apartment building) = $50,000(.923) + 30,000(.077) = $48,460

- EV (strategy) = $89,220(.52) + 35,000(.48) = $63,194

The derivative is obtained

from slide44

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What good are these values?

We can make use these information to determine

how much should we pay for, say, sampling a survey

How? Example

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The Expected Value of Sample Information

- The expected value of sample information (EVSI)
- = expected value with information – expect value without information.:
- For example problem, EVSI = $63,194 - 44,000 = $19,194
- How efficiency is this value?

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From the decision tree

From without

perfect information

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Efficiency

- The efficiency of sample information is the ratio of the expected value of sample information to the expected value of perfect information:
- efficiency = EVSI /EVPI = $19,194/ 28,000 = 0.68

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Decision Analysis with Additional Information Utility

Table 12.13 Payoff Table for Auto Insurance Example

Expected Cost (insurance) = .992($500) + .008(500) = $500

Expected Cost (no insurance) = .992($0) + .008(10,000) = $80

- Decision should be do not purchase insurance, but people almost always do purchase insurance.

- Utility is a measure of personal satisfaction derived from money.

- Utiles are units of subjective measures of utility.

- Risk averters forgo a high expected value to avoid a low-probability disaster.

- Risk takers take a chance for a bonanza on a very low-probability event in lieu of a sure thing.

Example Problem Solution(1 of 7)

Consider:

a. Determine the best decision without probabilities using the 5 criteria of the chapter.

b. Determine best decision with probabilities assuming .70 probability of good conditions, .30 of poor conditions. Use expected value and expected opportunity loss criteria.

c. Compute expected value of perfect information.

d. Develop a decision tree with expected value at the nodes.

e. Given following, P(Pg) = .70, P(Ng) = .30, P(Pp) = 20, P(Np) = .80, determine posterior probabilities using Bayesians rule.

f. Perform a decision tree analysis using the posterior probability obtained in part e.

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Example Problem Solution(2 of 7)

Step 1 (part a): Determine Decisions Without Probabilities

Maximax Decision: Maintain status quo

DecisionsMaximum Payoffs

Expand $800,000

Status quo 1,300,000 (maximum)

Sell 320,000

Maximin Decision: Expand

DecisionsMinimum Payoffs

Expand $500,000 (maximum)

Status quo -150,000

Sell 320,000

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Example Problem Solution(3 of 7)

Minimax Regret Decision: Expand

DecisionsMaximum Regrets

Expand $500,000 (minimum)

Status quo 650,000

Sell 980,000

Hurwicz ( = .3) Decision: Expand

Expand $800,000(.3) + 500,000(.7) = $590,000

Status quo $1,300,000(.3) - 150,000(.7) = $285,000

Sell $320,000(.3) + 320,000(.7) = $320,000

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Example Problem Solution(4 of 7)

Equal Liklihood Decision: Expand

Expand $800,000(.5) + 500,000(.5) = $650,000

Status quo $1,300,000(.5) - 150,000(.5) = $575,000

Sell $320,000(.5) + 320,000(.5) = $320,000

Step 2 (part b): Determine Decisions with EV and EOL

Expected value decision: Maintain status quo

Expand $800,000(.7) + 500,000(.3) = $710,000

Status quo $1,300,000(.7) - 150,000(.3) = $865,000

Sell $320,000(.7) + 320,000(.3) = $320,000

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Example Problem Solution(5 of 7)

Expected opportunity loss decision: Maintain status quo

Expand $500,000(.7) + 0(.3) = $350,000

Status quo 0(.7) + 650,000(.3) = $195,000

Sell $980,000(.7) + 180,000(.3) = $740,000

Step 3 (part c): Compute EVPI

EV given perfect information = 1,300,000(.7) + 500,000(.3) = $1,060,000

EV without perfect information = $1,300,000(.7) - 150,000(.3) = $865,000

EVPI = $1.060,000 - 865,000 = $195,000

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Example Problem Solution(7 of 7)

Step 5 (part e): Determine Posterior Probabilities

P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)] = (.70)(.70)/[(.70)(.70) + (.20)(.30)] = .891 P(p P) = .109

P(gN) = P(Ng)P(g)/[P(Ng)P(g) + P(Np)P(p)] = (.30)(.70)/[(.30)(.70) + (.80)(.30)] = .467 P(pN) = .533 Step 6 (part f): Perform Decision tree Analysis with Posterior Probabilities

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