2420 Lecture 09

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Decision Analysis

What is it?

What is the objective?

More exampleTutorial: 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” (to p3)

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Payoff TableA payoff table consists of

Columns: state of nature (ie events)Rows: choice of decision

It corresponding entries are “payoff values”

Example (to p4)

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Payoff Table

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Objective

Our objective here is to determine whichwhich 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 .

How to derive this decision?

Example:

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Procedural Steps:Step1: For each row (or decision), select its max valuesStep 2: Choose the decision that has the max value of

Step1 Example:

Max

50,000100,00030,000

100,000max

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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

How to obtain it?

Example

Select this action(to p10)

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Procedural steps:Step 1: For each row (or decision), obtain its min valuesStep 2: Choose the decision that has the max value from step 1

Min

30,000-40,00010,000

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The Minimax: Regret CriterionRegret 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.

How to obtain it?Solution

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Procedural steps:Step 1: Find out the max value for each columnStep 2: Subtract value of step 1 to all entry of that columnStep 3: Find out the max value for each rowStep 4: Choose the decision that has smallest value of step 3

Step 1:

Max 100,000 30,000

Example

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(100,000 - 50,000)

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

(30,000 - 30,000)Note: this is a regret table.

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Max50,00070,00070,000

Min

Step 3: Find out the max value for each rowStep 4: Choose the decision that has smallest value of step 3

Select this decision (to p3)

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Decision Making without Probabilities

The 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

Select this,max

Example:

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The 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.

Decision Values

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

Select this, Max

Example:

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Summary 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(to p3)

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

Exhibit 12.1

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

Exhibit 12.2

Exhibit 12.3

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Posterior analysis with Bayesian rulesHere, we need to understand the following concepts:

1) Expected values2) Expected Opportunity loss 3) Expected values of Perfect Information

(Ex. 18 and 26)4) Decision Tree

a) basic tree b) sequential tree5) Posterior analysis with Bayesian rules

(Ex. 33, 35 and 37)

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Expected ValueExpected 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

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Decision Making with ProbabilitiesSolution of Expected Value Problems with Excel and Excel QM

(1 of 2)

Exhibit 12.5

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Decision Making with ProbabilitiesSolution of Expected Value Problems with Excel and Excel QM

(2 of 2)

Exhibit 12.6

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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 withwith perfect information) –(expected value withoutwithout perfect information)

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

How to compute them? (to p27)

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Decision Making with ProbabilitiesEVPI Example

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

Table 12.9 Payoff Table with Decisions, Given Perfect Information

Best outcome for each column Best outcome Worst outcome

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Decision Making with ProbabilitiesEVPI with QM for Windows

Exhibit 12.7

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Decision Trees (1 of 2)A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches).

Table 12.10Payoff Table for Real Estate Investment Example

Figure 12.1Decision 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:

Use to denote not chosen path(s)

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Decision Making with ProbabilitiesDecision Trees with QM for Windows

Exhibit 12.8

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Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan

(1 of 4)

Exhibit 12.9

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(2 of 4)

Exhibit 12.10

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(3 of 4)

Exhibit 12.11

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(4 of 4)

Exhibit 12.12

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Sequential Decision Trees(1 of 2)

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

.

How to make use of them? (to p37)

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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.

0.6*2M+0.4*0.225M

Extra information

Max (1.29M-0.8M, 1.36M-0.2M)=1.16M

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Sequential Decision Tree Analysis with QM for Windows

Exhibit 12.13

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Sequential Decision Tree Analysis with Excel and TreePlan

Exhibit 12.14

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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, whatposterior informationcan be obtained here?

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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?

Total = 0.8+0.2 = 1.0

NewNew

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(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? (to p44)

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Form this table to compute them

Table 12.12Computation 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 (to p42)

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 12.5Decision tree with posterior

probabilities

Old oneMaking 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

What good are these values?

The derivative is obtained from slide44

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• We can make use these information to determine how much should we pay for, say, sampling a survey

How? Example (to p47)

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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?

From the decision tree From withoutperfect 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

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.

Table 12.13 Payoff Table for Auto Insurance Example

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Example Problem Solution(1 of 7)

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.

States of Nature

DecisionGood ForeignCompetitiveConditions

Poor ForeignCompetitiveConditions

ExpandMaintain Status QuoSell now

$800,0001,300,000320,000

$500,000-150,000320,000

Consider:

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Step 1 (part a): Determine Decisions Without Probabilities

Maximax Decision: Maintain status quo

Decisions Maximum Payoffs

Expand $800,000

Status quo 1,300,000 (maximum)

Sell 320,000

Maximin Decision: Expand

Decisions Minimum Payoffs

Expand $500,000 (maximum)

Status quo -150,000

Sell 320,000

States of Nature




$800,0001,300,000320,000

$500,000-150,000320,000

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Minimax Regret Decision: Expand

Decisions Maximum 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

States of Nature




$800,0001,300,000320,000

$500,000-150,000320,000

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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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States of Nature




$800,0001,300,000320,000

$500,000-150,000320,000

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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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States of Nature




$800,0001,300,000320,000

$500,000-150,000320,000

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Step 4 (part d): Develop a Decision Tree

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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) = .109P(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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2420 Lecture 09

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