No Slide Title · 2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall12-7 Table 12.3 Payoff table illustrating a maximax decision In the maximax criterion the

1

12-1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis

Chapter 12


■ Components of Decision Making

■ Decision Making without Probabilities

■ Decision Making with Probabilities

■ Decision Analysis with Additional Information

■ Utility

Chapter Topics


Previous chapters used an assumption of certainty with regards to

problem parameters.

This chapter relaxes the certainty assumption

Two categories of decision situations:

Probabilities can be assigned to future occurrences

Probabilities cannot be assigned to future occurrences

Decision Analysis

Overview


Table 12.1 Payoff table

■ A state of nature is an actual event that may occur in the future.

■ A payoff table is a means of organizing a decision situation,

presenting the payoffs from different decisions given the various

states of nature.

Decision Analysis

Components of Decision Making


Decision Analysis

Decision Making Without Probabilities

Figure 12.1 Decision

situation with real estate

investment alternatives


Decision-Making Criteria

maximax maximin minimax

minimax regret Hurwicz equal likelihood

Decision Analysis

Decision Making without Probabilities

Table 12.2 Payoff table for the real estate investments

2


Table 12.3 Payoff table illustrating a maximax decision

In the maximax criterion the decision maker selects the decision

that will result in the maximum of maximum payoffs; an

optimistic criterion.


Maximax Criterion


Table 12.4 Payoff table illustrating a maximin decision

In the maximin criterion the decision maker selects the decision

that will reflect the maximum of the minimum payoffs; a

pessimistic criterion.


Maximin Criterion


Table 12.5 Regret table

Regret is the difference between the payoff from the best

decision and all other decision payoffs.

Example: under the Good Economic Conditions state of nature,

the best payoff is $100,000. The manager’s regret for choosing

the Warehouse alternative is $100,000-$30,000=$70,000


Minimax Regret Criterion


Table 12.6 Regret table illustrating the minimax regret decision

The manager calculates regrets for all alternatives under each

state of nature. Then the manager identifies the maximum

regret for each alternative.

Finally, the manager attempts to avoid regret by selecting the

decision alternative that minimizes the maximum regret.


Minimax Regret Criterion


The Hurwicz criterion is a compromise between the maximax

and maximin criteria.

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. Here, = 0.4.


Hurwicz Criterion

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


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.


Equal Likelihood Criterion

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

3


■ 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 likelihood Apartment building


Summary of Criteria Results


Exhibit 12.1


Solution with QM for Windows (1 of 3)


Exhibit 12.2



Equal likelihood weight


Exhibit 12.3





Solution with Excel

Exhibit 12.4

=MIN(C7,D7)

=MAX(E7,E9)

=MAX(C18,D18)

=MAX(F7:F9)

=MAX(C7:C9)-C9

=C7*C25+D7*C26

=C7*0.5+D7*0.5


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

Decision Making with Probabilities

Expected Value

4


■ The expected opportunity loss is the expected value of the regret for each decision.

■ The expected value and expected opportunity loss criterion result in the same decision.

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 table with probabilities for states of nature


Expected Opportunity Loss


Exhibit 12.5

Expected Value Problems

Solution with QM for Windows

Expected values


Exhibit 12.6


Solution with Excel and Excel QM (1 of 2)

Expected value for

apartment building



Solution with Excel and Excel QM (2 of 2)

Exhibit 12.7

Click on “Add-Ins” to access

the “Excel QM” menu


■ The expected value of perfect information (EVPI) is the

maximum amount a decision maker would pay for additional

information.

■ EVPI equals the expected value given perfect information

minus the expected value without perfect information.

■ EVPI equals the expected opportunity loss (EOL) for the best

decision.


Expected Value of Perfect Information


Table 12.9 Payoff table with decisions, given perfect information


EVPI Example (1 of 2)

5


■ 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


EVPI Example (2 of 2)


Exhibit 12.8


EVPI with QM for Windows

The expected value, given

perfect information, in Cell F12

=MAX(E7:E9)

=F12-F11


A decision tree is a diagram consisting of decision nodes

(represented as squares), probability nodes (circles), and

decision alternatives (branches).

Table 12.10 Payoff table for real estate investment example


Decision Trees (1 of 4)


Figure 12.2 Decision tree for real estate investment example




■ 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

■ Branches with the greatest expected value are selected.




Figure 12.3 Decision tree with expected value at probability nodes



6



Decision Trees with QM for Windows

Exhibit 12.9

Select node to add from

Number of branches

from node 1

Add branches from node

1 to 2, 3, and 4



Decision Trees with Excel and TreePlan (1 of 4)

Exhibit 12.10


Exhibit 12.11



To create another branch, click

“B5,” then the “Decision Tree”

menu, and select “Add Branch”

Invoke TreePlan from

the “Add Ins” menu


Exhibit 12.12



Click on cell “F3,”

then “Decision Tree”

Select “Change to

Event Node” and add

two new branches




Exhibit 12.13

Add numerical dollar and probability

values in these cells in column H

These cells contain decision tree

formulas; do not type in these

cells in columns E and I


Exhibit 12.14

Sequential Decision Tree Analysis

Solution with QM for Windows

Cell A16 contains

the expected value

of $44,000

7



Sequential Decision Trees (1 of 4)

■ A sequential decision tree is used to illustrate a situation

requiring a series of decisions.

■ Used where a payoff table, limited to a single decision, cannot

be used.

■ The next slide shows the real estate investment example

modified to encompass a ten-year period in which several

decisions must be made.


Figure 12.4 Sequential decision tree






■ Expected value of apartment building is:

$1,290,000-800,000 = $490,000

■ Expected value if land is purchased is:

$1,360,000-200,000 = $1,160,000

■ The decision is to purchase land; it has the highest net expected

value of $1,160,000.


Figure 12.5 Sequential decision tree with nodal expected values



12-41 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 12.15


Solution with Excel QM

12-42 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 12.16


Solution with TreePlan

8


■ Bayesian analysis uses additional information to alter the marginal probability of the occurrence of an event.

■ In the real estate investment example, using the expected value criterion, the best decision was to purchase the office building with an expected value of $444,000, and EVPI of $28,000.

Table 12.11 Payoff table for real estate investment

Decision Analysis with Additional Information

Bayesian Analysis (1 of 3)


■ A conditional probability is the probability that an event will

occur given that another event has already occurred.

■ An economic analyst provides additional information for the

real estate investment decision, forming conditional

probabilities:

g = good economic conditions

p = poor economic conditions

P = positive economic report

N = negative economic report

P(Pg) = .80 P(NG) = .20

P(Pp) = .10 P(Np) = .90




■ A posterior probability is the altered marginal probability of an

event based on additional information.

■ Prior probabilities for good or poor economic conditions in the

real estate decision:

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

■ Posterior probabilities by Bayes’ rule:

(gP) = P(PG)P(g)/[P(Pg)P(g) + P(Pp)P(p)]

= (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923

■ Posterior (revised) probabilities for decision:

P(gN) = .250 P(pP) = .077 P(pN) = .750





Decision Trees with Posterior Probabilities (1 of 4)

Decision trees with posterior probabilities differ from earlier

versions in that:

■ Two new branches at the beginning of the tree represent

report outcomes.

■ Probabilities of each state of nature are posterior

probabilities from Bayes’ rule.


Figure 12.6 Decision tree with posterior probabilities






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

= $48,460

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

9


Figure 12.7 Decision tree analysis for real estate investment




Table 12.12 Computation of posterior probabilities


Computing Posterior Probabilities with Tables



Computing Posterior Probabilities with Excel

Exhibit 12.17


■ The expected value of sample information (EVSI) is the

difference between the expected value with and without

information:

For example problem, EVSI = $63,194 - 44,000 = $19,194

■ 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 = .68


Expected Value of Sample Information


Table 12.13 Payoff table for auto insurance example


Utility (1 of 2)


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

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

The 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.


Utility (2 of 2)

10


States of Nature

Decision Good Foreign Competitive

Conditions Poor Foreign Competitive

Conditions

Expand Maintain Status Quo Sell now

$ 800,000 1,300,000 320,000

$ 500,000 -150,000 320,000

Decision Analysis

Example Problem Solution (1 of 9)

A corporate raider contemplates the future of a recent acquisition.

Three alternatives are being considered in two states of nature. The

payoff table is below.


Decision Analysis


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 the following, P(Pg) = .70, P(Ng) = .30, P(Pp) = 20,

P(Np) = .80, determine posterior probabilities using Bayes’

rule.

f. Perform a decision tree analysis using the posterior probability

obtained in part e.


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

Decision Analysis



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

Decision Analysis



Equal Likelihood 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

Decision Analysis



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

Decision Analysis


11


Step 4 (part d): Develop a decision tree.

Decision Analysis



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(pP) = .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

Decision Analysis



Step 6 (part f): Decision tree analysis.

Decision Analysis



All rights reserved. No part of this publication may be reproduced, stored in a retrieval

system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,

recording, or otherwise, without the prior written permission of the publisher.

Printed in the United States of America.

Documents

No Slide Title · 2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall12-7 Table 12.3 Payoff table illustrating a maximax decision In the maximax criterion the