Decision Analysis

Chapter 12 - Decision Analysis 1

Chapter 12 - Decision Analysis

Chapter Topics

• Components of Decision Making

• Decision Making without Probabilities

• Decision Making with Probabilities

• Decision Analysis with Additional Information

• Utility


Decision Analysis

Components of Decision Making

• 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.

Table 12.1 Payoff Table


Decision Analysis

Decision Making without Probabilities

Decision situation:

Decision-Making Criteria:

maximax, maximin, minimax, minimax regret, Hurwicz, equal likelihood

Table 12.2

Payoff Table for the Real Estate Investments



The Maximax Criterion - In the maximax criterion the decision maker selects the decision that will result in the

maximum of maximum payoffs; an optimistic criterion.

Table 12.3

Payoff Table Illustrating a Maximax Decision



The Maximin Criterion

- In the maximin criterion the decision maker selects the decision that will reflect the maximum of

the minimum payoffs; a pessimistic criterion.

Table 12.4

Payoff Table Illustrating a Maximin Decision



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.

Table 12.6

Regret Table Illustrating the Minimax Regret Decision



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



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



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



Solutions with QM for Windows (1 of 2)

Exhibit 12.1



Solutions with QM for Windows (2 of 2)

Exhibit 12.2

Exhibit 12.3


Decision Making with Probabilities

Expected Value -Expected value is computed by multiplying each decision outcome under each state of nature by

the probability of its occurance.

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



Expected Opportunity Loss - 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 (Opportunity Loss) Table with Probabilities for States of Nature



Solution of Expected Value Problems with QM for Windows

Exhibit 12.4



Solution of Expected Value Problems with Excel and Excel QM

(1 of 2)

Exhibit 12.5



Solution 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 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.



EVPI 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



EVPI with QM for Windows

Exhibit 12.7



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 12.10

Payoff Table for Real

Estate Investment

Example

Figure 12.1

Decision tree for

real estate

investment example



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

- Branches with the greartest expected value are selected :

Figure 12.2

Decision tree with

expected value at

probability nodes



Decision Trees with QM for Windows

Exhibit 12.8



Decision Trees with Excel and TreePlan

(1 of 4)

Exhibit 12.9




(2 of 4)

Exhibit 12.10




(3 of 4)

Exhibit 12.11




(4 of 4)

Exhibit 12.12



Sequential Decision Trees

(1 of 2) - 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.

- Real estate investment example modified to encompass a ten-year period in which several

decisions must be made:

Figure 12.3

Sequential decision tree



Sequential Decision Trees

(2 of 2) - Decision is to purchase land; highest net expected value ($1,160,000).

- Payoff of the decision is $1,160,000.

Figure 12.4

Sequential decision tree

with nodal expected values


Sequential Decision Tree Analysis with QM for Windows

Exhibit 12.13


Sequential Decision Tree Analysis with Excel and TreePlan

Exhibit 12.14


Decision Analysis with Additional Information

Bayesian Analysis

(1 of 3) - Bayesian analysis uses additional information to alter the marginal probability of the

occurence of an event.

- In real estate investment example, using expected value criterion, best decision was to

purchase office building with expected value of $444,000, and EVPI of $28,000.

Table 12.11 Payoff Table for the Real Estate Investment Example



Bayesian Analysis

(2 of 3)

- A conditional probability is the probability that an event will occur given that another

event has already occurred.

- Economic analyst provides additional information for 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



Bayesian Analysis

(3 of 3)

- A posteria probability is the altered marginal probability of an event based on additional

information.

-Prior probabilities for good or poor economic conditions in real estate decision:

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

- Posteria probabilities by Bayes’s rule:

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

- Posteria (revised) probabilities for decision:

P(gN) = .250

P(pP) = .077

P(pN) = .750



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. Probabilities of each state of nature are posterior probabilities from Bayes’s rule.

Figure 12.5

Decision tree with posterior

probabilities



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

Figure 12.6

Decision tree analysis



Computing Posterior Probabilities with Tables

Table 12.12

Computation of Posterior Probabilities



The Expected Value of Sample Information

• 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


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


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 probabilites 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. Develp 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 posteria

probabilities using Bayes’s rule.

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

States of Nature

Decision

Good Foreign

CompetitiveConditions

Poor Foreign

CompetitiveConditions

Expand

Maintain Status QuoSell now

$800,000

1,300,000320,000

$500,000

-150,000320,000



(2 of 7)

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



(3 of 7)

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



(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



(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



(6 of 7) Step 4 (part d): Develop a Decision Tree



(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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Decision Analysis

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