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Chapter 3

To accompany
Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna

Power Point slides created by Brian Peterson

Decision Analysis

Learning Objectives

List the steps of the decision-making process.

Describe the types of decision-making environments.

Make decisions under uncertainty.

Use probability values to make decisions under risk.

After completing this chapter, students will be able to:

Learning Objectives

Develop accurate and useful decision trees.

Revise probabilities using Bayesian analysis.

Use computers to solve basic decision-making problems.

Understand the importance and use of utility theory in decision making.

After completing this chapter, students will be able to:

Chapter Outline

3.1 Introduction

3.2 The Six Steps in Decision Making

3.3 Types of Decision-Making Environments

3.4 Decision Making under Uncertainty

3.5 Decision Making under Risk

3.6 Decision Trees

3.7 How Probability Values Are Estimated by Bayesian Analysis

3.8 Utility Theory

Introduction

What is involved in making a good decision?
Decision theory is an analytic and systematic approach to the study of decision making.
A good decision is one that is based on logic, considers all available data and possible alternatives, and the quantitative approach described here.

The Six Steps in Decision Making

Clearly define the problem at hand.

List the possible alternatives.

Identify the possible outcomes or states of nature.

List the payoff (typically profit) of each combination of alternatives and outcomes.

Select one of the mathematical decision theory models.

Apply the model and make your decision.

Thompson Lumber Company

Step 1 – Define the problem.

The company is considering expanding by manufacturing and marketing a new product – backyard storage sheds.

Step 2 – List alternatives.

Construct a large new plant.
Construct a small new plant.
Do not develop the new product line at all.

Step 3 – Identify possible outcomes.

The market could be favorable or unfavorable.

Thompson Lumber Company

Step 4 – List the payoffs.

Identify conditional values for the profits for large plant, small plant, and no development for the two possible market conditions.

Step 5 – Select the decision model.

This depends on the environment and amount of risk and uncertainty.

Step 6 – Apply the model to the data.

Solution and analysis are then used to aid in decision-making.

Thompson Lumber Company

Table 3.1

Decision Table with Conditional Values for Thompson Lumber

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)
Construct a large plant	200,000	–180,000
Construct a small plant	100,000	–20,000
Do nothing	0	0

Types of Decision-Making Environments

Type 1: Decision making under certainty

The decision maker knows with certainty the consequences of every alternative or decision choice.

Type 2: Decision making under uncertainty

The decision maker does not know the probabilities of the various outcomes.

Type 3: Decision making under risk

The decision maker knows the probabilities of the various outcomes.

Decision Making Under Uncertainty

Maximax (optimistic)

Maximin (pessimistic)

Criterion of realism (Hurwicz)

Equally likely (Laplace)

Minimax regret

There are several criteria for making decisions under uncertainty:

Maximax

Used to find the alternative that maximizes the maximum payoff.

Locate the maximum payoff for each alternative.
Select the alternative with the maximum number.

Table 3.2

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	MAXIMUM IN A ROW ($)
Construct a large plant	200,000	–180,000	200,000
Construct a small plant	100,000	–20,000	100,000
Do nothing	0	0	0

Maximax

Maximin

Used to find the alternative that maximizes the minimum payoff.

Locate the minimum payoff for each alternative.
Select the alternative with the maximum number.

Table 3.3

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	MINIMUM IN A ROW ($)
Construct a large plant	200,000	–180,000	–180,000
Construct a small plant	100,000	–20,000	–20,000
Do nothing	0	0	0

Maximin

Criterion of Realism (Hurwicz)

This is a weighted average compromise between optimism and pessimism.

Select a coefficient of realism , with 0≤α≤1.
A value of 1 is perfectly optimistic, while a value of 0 is perfectly pessimistic.
Compute the weighted averages for each alternative.
Select the alternative with the highest value.

Weighted average = (maximum in row)

+ (1 – )(minimum in row)

Criterion of Realism (Hurwicz)

For the large plant alternative using  = 0.8:
(0.8)(200,000) + (1 – 0.8)(–180,000) = 124,000
For the small plant alternative using  = 0.8:
(0.8)(100,000) + (1 – 0.8)(–20,000) = 76,000

Table 3.4

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	CRITERION OF REALISM ( = 0.8) $
Construct a large plant	200,000	–180,000	124,000
Construct a small plant	100,000	–20,000	76,000
Do nothing	0	0	0

Realism

Equally Likely (Laplace)

Considers all the payoffs for each alternative

Find the average payoff for each alternative.
Select the alternative with the highest average.

Table 3.5

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	ROW AVERAGE ($)
Construct a large plant	200,000	–180,000	10,000
Construct a small plant	100,000	–20,000	40,000
Do nothing	0	0	0

Equally likely

Minimax Regret

Based on opportunity loss or regret, this is the difference between the optimal profit and actual payoff for a decision.

Create an opportunity loss table by determining the opportunity loss from not choosing the best alternative.
Opportunity loss is calculated by subtracting each payoff in the column from the best payoff in the column.
Find the maximum opportunity loss for each alternative and pick the alternative with the minimum number.

Minimax Regret

Table 3.6

Determining Opportunity Losses for Thompson Lumber

STATE OF NATURE
FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)
200,000 – 200,000	0 – (–180,000)
200,000 – 100,000	0 – (–20,000)
200,000 – 0	0 – 0

Minimax Regret

Table 3.7

Opportunity Loss Table for Thompson Lumber

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)
Construct a large plant	0	180,000
Construct a small plant	100,000	20,000
Do nothing	200,000	0

Minimax Regret

Table 3.8

Thompson’s Minimax Decision Using Opportunity Loss

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	MAXIMUM IN A ROW ($)
Construct a large plant	0	180,000	180,000
Construct a small plant	100,000	20,000	100,000
Do nothing	200,000	0	200,000

Minimax

Decision Making Under Risk

This is decision making when there are several possible states of nature, and the probabilities associated with each possible state are known.
The most popular method is to choose the alternative with the highest expected monetary value (EMV).
This is very similar to the expected value calculated in the last chapter.

EMV (alternative i) = (payoff of first state of nature)

x (probability of first state of nature)

+ (payoff of second state of nature)

x (probability of second state of nature)

+ … + (payoff of last state of nature)

x (probability of last state of nature)

EMV for Thompson Lumber

Suppose each market outcome has a probability of occurrence of 0.50.
Which alternative would give the highest EMV?
The calculations are:

EMV (large plant) = ($200,000)(0.5) + (–$180,000)(0.5)

= $10,000

EMV (small plant) = ($100,000)(0.5) + (–$20,000)(0.5)

= $40,000

EMV (do nothing) = ($0)(0.5) + ($0)(0.5)

= $0

EMV for Thompson Lumber

Table 3.9

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	EMV ($)
Construct a large plant	200,000	–180,000	10,000
Construct a small plant	100,000	–20,000	40,000
Do nothing	0	0	0
Probabilities	0.50	0.50

Largest EMV

Expected Value of Perfect Information (EVPI)

EVPI places an upper bound on what you should pay for additional information.

EVPI = EVwPI – Maximum EMV

EVwPI is the long run average return if we have perfect information before a decision is made.

EVwPI = (best payoff for first state of nature)

x (probability of first state of nature)

+ (best payoff for second state of nature)

x (probability of second state of nature)

+ … + (best payoff for last state of nature)

x (probability of last state of nature)

Expected Value of Perfect Information (EVPI)

Suppose Scientific Marketing, Inc. offers analysis that will provide certainty about market conditions (favorable).
Additional information will cost $65,000.
Should Thompson Lumber purchase the information?

Expected Value of Perfect Information (EVPI)

Table 3.10

Decision Table with Perfect Information

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	EMV ($)
Construct a large plant	200,000	-180,000	10,000
Construct a small plant	100,000	-20,000	40,000
Do nothing	0	0	0
With perfect information	200,000	0	100,000
Probabilities	0.5	0.5

EVwPI

Expected Value of Perfect Information (EVPI)

The maximum EMV without additional information is $40,000.

EVPI = EVwPI – Maximum EMV

= $100,000 - $40,000

= $60,000

So the maximum Thompson should pay for the additional information is $60,000.

Expected Value of Perfect Information (EVPI)

The maximum EMV without additional information is $40,000.

EVPI = EVwPI – Maximum EMV

= $100,000 - $40,000

= $60,000

Therefore, Thompson should not pay $65,000 for this information.

So the maximum Thompson should pay for the additional information is $60,000.

Expected Opportunity Loss

Expected opportunity loss (EOL) is the cost of not picking the best solution.
First construct an opportunity loss table.
For each alternative, multiply the opportunity loss by the probability of that loss for each possible outcome and add these together.
Minimum EOL will always result in the same decision as maximum EMV.
Minimum EOL will always equal EVPI.

Expected Opportunity Loss

EOL (large plant) = (0.50)($0) + (0.50)($180,000)

= $90,000

EOL (small plant) = (0.50)($100,000) + (0.50)($20,000)

= $60,000

EOL (do nothing) = (0.50)($200,000) + (0.50)($0)

= $100,000

Table 3.11

STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	EOL
Construct a large plant	0	180,000	90,000
Construct a small plant	100,000	20,000	60,000
Do nothing	200,000	0	100,000
Probabilities	0.50	0.50

Minimum EOL

Sensitivity Analysis

Sensitivity analysis examines how the decision might change with different input data.
For the Thompson Lumber example:

P = probability of a favorable market

(1 – P) = probability of an unfavorable market

Sensitivity Analysis

EMV(Large Plant) = $200,000P – $180,000)(1 – P)

= $200,000P – $180,000 + $180,000P

= $380,000P – $180,000

EMV(Small Plant) = $100,000P – $20,000)(1 – P)

= $100,000P – $20,000 + $20,000P

= $120,000P – $20,000

EMV(Do Nothing) = $0P + 0(1 – P)

= $0

Sensitivity Analysis

EMV (do nothing)

Figure 3.1

$300,000

$200,000

$100,000

–$100,000

–$200,000

EMV Values

EMV (large plant)

EMV (small plant)

Point 1

Point 2

.167

.615

Values of P

Sensitivity Analysis

Point 1:

EMV(do nothing) = EMV(small plant)

Point 2:

EMV(small plant) = EMV(large plant)

Sensitivity Analysis

Figure 3.1

BEST ALTERNATIVE	RANGE OF P VALUES
Do nothing	Less than 0.167
Construct a small plant	0.167 – 0.615
Construct a large plant	Greater than 0.615

$300,000

$200,000

$100,000

–$100,000

–$200,000

EMV Values

EMV (large plant)

EMV (small plant)

EMV (do nothing)

Point 1

Point 2

.167

.615

Values of P

Using Excel

Program 3.1A

Input Data for the Thompson Lumber Problem Using Excel QM

Using Excel

Program 3.1B

Output Results for the Thompson Lumber Problem Using Excel QM

Decision Trees

Any problem that can be presented in a decision table can also be graphically represented in a decision tree.
Decision trees are most beneficial when a sequence of decisions must be made.
All decision trees contain decision points or nodes, from which one of several alternatives may be chosen.
All decision trees contain state-of-nature points or nodes, out of which one state of nature will occur.

Five Steps of
Decision Tree Analysis

Define the problem.

Structure or draw the decision tree.

Assign probabilities to the states of nature.

Estimate payoffs for each possible combination of alternatives and states of nature.

Solve the problem by computing expected monetary values (EMVs) for each state of nature node.

Structure of Decision Trees

Trees start from left to right.
Trees represent decisions and outcomes in sequential order.
Squares represent decision nodes.
Circles represent states of nature nodes.
Lines or branches connect the decisions nodes and the states of nature.

Thompson’s Decision Tree

Figure 3.2

Favorable Market

Unfavorable Market

Favorable Market

Unfavorable Market

Do Nothing

Construct Large Plant

Construct Small Plant

A Decision Node

A State-of-Nature Node

Thompson’s Decision Tree

Favorable Market

Unfavorable Market

Favorable Market

Unfavorable Market

Do Nothing

Construct Large Plant

Construct Small Plant

Figure 3.3

Alternative with best EMV is selected

EMV for Node 1 = $10,000

= (0.5)($200,000) + (0.5)(–$180,000)

EMV for Node 2 = $40,000

= (0.5)($100,000)
+ (0.5)(–$20,000)

Payoffs

$200,000

–$180,000

$100,000

–$20,000

(0.5)

Thompson’s Complex Decision Tree

Figure 3.4

First Decision Point

Second Decision Point

Favorable Market (0.78)

Unfavorable Market (0.22)

Favorable Market (0.78)

Unfavorable Market (0.22)

Favorable Market (0.27)

Unfavorable Market (0.73)

Favorable Market (0.27)

Unfavorable Market (0.73)

Favorable Market (0.50)

Unfavorable Market (0.50)

Favorable Market (0.50)

Unfavorable Market (0.50)

Large Plant

Small Plant

No Plant

Conduct Market Survey

Do Not Conduct Survey

Large Plant

Small Plant

No Plant

Large Plant

Small Plant

No Plant

Results

Favorable

Results

Negative

Survey (0.45)

Survey (0.55)

–$190,000

$190,000

$90,000

–$30,000

–$10,000

–$180,000

$200,000

$100,000

–$20,000

–$190,000

$190,000

$90,000

–$30,000

–$10,000

Payoffs

Thompson’s Complex Decision Tree

1. Given favorable survey results,

EMV(node 2) = EMV(large plant | positive survey)

= (0.78)($190,000) + (0.22)(–$190,000) = $106,400

EMV(node 3) = EMV(small plant | positive survey)

= (0.78)($90,000) + (0.22)(–$30,000) = $63,600

EMV for no plant = –$10,000

2. Given negative survey results,

EMV(node 4) = EMV(large plant | negative survey)

= (0.27)($190,000) + (0.73)(–$190,000) = –$87,400

EMV(node 5) = EMV(small plant | negative survey)

= (0.27)($90,000) + (0.73)(–$30,000) = $2,400

EMV for no plant = –$10,000

Thompson’s Complex Decision Tree

3. Compute the expected value of the market survey,

EMV(node 1) = EMV(conduct survey)

= (0.45)($106,400) + (0.55)($2,400)

= $47,880 + $1,320 = $49,200

4. If the market survey is not conducted,

EMV(node 6) = EMV(large plant)

= (0.50)($200,000) + (0.50)(–$180,000) = $10,000

EMV(node 7) = EMV(small plant)

= (0.50)($100,000) + (0.50)(–$20,000) = $40,000

EMV for no plant = $0

5. The best choice is to seek marketing information.

Thompson’s Complex Decision Tree

Figure 3.5

First Decision Point

Second Decision Point

Favorable Market (0.78)

Unfavorable Market (0.22)

Favorable Market (0.78)

Unfavorable Market (0.22)

Favorable Market (0.27)

Unfavorable Market (0.73)

Favorable Market (0.27)

Unfavorable Market (0.73)

Favorable Market (0.50)

Unfavorable Market (0.50)

Favorable Market (0.50)

Unfavorable Market (0.50)

Large Plant

Small Plant

No Plant

Large Plant

Small Plant

No Plant

Large Plant

Small Plant

No Plant

–$190,000

$190,000

$90,000

–$30,000

–$10,000

–$180,000

$200,000

$100,000

–$20,000

–$190,000

$190,000

$90,000

–$30,000

–$10,000

Payoffs

Conduct Market Survey

Do Not Conduct Survey

Results

Favorable

Results

Negative

Survey (0.45)

Survey (0.55)

$40,000

$2,400

$106,400

$49,200

$106,400

$63,600

–$87,400

$2,400

$10,000

$40,000

Expected Value of Sample Information

Suppose Thompson wants to know the actual value of doing the survey.

= (EV with sample information + cost)

– (EV without sample information)

EVSI = ($49,200 + $10,000) – $40,000 = $19,200

Expected value
with sample
information, assuming
no cost to gather it

Expected value
of best decision
without sample
information

EVSI = –

Sensitivity Analysis

How sensitive are the decisions to changes in the probabilities?
How sensitive is our decision to the probability of a favorable survey result?
That is, if the probability of a favorable result (p = .45) where to change, would we make the same decision?
How much could it change before we would make a different decision?

Sensitivity Analysis

p = probability of a favorable survey result

(1 – p) = probability of a negative survey result

EMV(node 1) = ($106,400)p +($2,400)(1 – p)

= $104,000p + $2,400

We are indifferent when the EMV of node 1 is the same as the EMV of not conducting the survey, $40,000

$104,000p + $2,400 = $40,000

$104,000p = $37,600

p = $37,600/$104,000 = 0.36

If p<0.36, do not conduct the survey. If p>0.36, conduct the survey.

Bayesian Analysis

There are many ways of getting probability data. It can be based on:
Management’s experience and intuition.
Historical data.
Computed from other data using Bayes’ theorem.
Bayes’ theorem incorporates initial estimates and information about the accuracy of the sources.
It also allows the revision of initial estimates based on new information.

Calculating Revised Probabilities

In the Thompson Lumber case we used these four conditional probabilities:

P (favorable market(FM) | survey results positive) = 0.78

P (unfavorable market(UM) | survey results positive) = 0.22

P (favorable market(FM) | survey results negative) = 0.27

P (unfavorable market(UM) | survey results negative) = 0.73

But how were these calculated?
The prior probabilities of these markets are:

P (FM) = 0.50

P (UM) = 0.50

Calculating Revised Probabilities

Through discussions with experts Thompson has learned the information in the table below.
He can use this information and Bayes’ theorem to calculate posterior probabilities.

Table 3.12

STATE OF NATURE
RESULT OF SURVEY	FAVORABLE MARKET (FM)	UNFAVORABLE MARKET (UM)
Positive (predicts favorable market for product)	P (survey positive \| FM) = 0.70	P (survey positive \| UM) = 0.20
Negative (predicts unfavorable market for product)	P (survey negative \| FM) = 0.30	P (survey negative \| UM) = 0.80

Calculating Revised Probabilities

Recall Bayes’ theorem:

For this example, A will represent a favorable market and B will represent a positive survey.

where

Calculating Revised Probabilities

P (FM | survey positive)

P (UM | survey positive)

Calculating Revised Probabilities

Table 3.13

Probability Revisions Given a Positive Survey

POSTERIOR PROBABILITY
STATE OF NATURE	CONDITIONAL PROBABILITY P(SURVEY POSITIVE \| STATE OF NATURE)	PRIOR PROBABILITY	JOINT PROBABILITY	P(STATE OF NATURE \| SURVEY POSITIVE)
FM	0.70	X 0.50	=	0.35	0.35/0.45 =	0.78
UM	0.20	X 0.50	=	0.10	0.10/0.45 =	0.22
P(survey results positive) =	0.45	1.00

Calculating Revised Probabilities

P (FM | survey negative)

P (UM | survey negative)

Calculating Revised Probabilities

Table 3.14

Probability Revisions Given a Negative Survey

POSTERIOR PROBABILITY
STATE OF NATURE	CONDITIONAL PROBABILITY P(SURVEY NEGATIVE \| STATE OF NATURE)	PRIOR PROBABILITY	JOINT PROBABILITY	P(STATE OF NATURE \| SURVEY NEGATIVE)
FM	0.30	X 0.50	=	0.15	0.15/0.55 =	0.27
UM	0.80	X 0.50	=	0.40	0.40/0.55 =	0.73
P(survey results positive) =	0.55	1.00

Using Excel

Program 3.2A

Formulas Used for Bayes’ Calculations in Excel

Using Excel

Program 3.2B

Results of Bayes’ Calculations in Excel

Potential Problems Using Survey Results

We can not always get the necessary data for analysis.
Survey results may be based on cases where an action was taken.
Conditional probability information may not be as accurate as we would like.

Utility Theory

Monetary value is not always a true indicator of the overall value of the result of a decision.
The overall value of a decision is called utility.
Economists assume that rational people make decisions to maximize their utility.

Utility Theory

Figure 3.6

Your Decision Tree for the Lottery Ticket

Heads (0.5)

Tails (0.5)

$5,000,000

Accept Offer

Reject Offer

$2,000,000

EMV = $2,500,000

Utility Theory

Utility assessment assigns the worst outcome a utility of 0, and the best outcome, a utility of 1.
A standard gamble is used to determine utility values.
When you are indifferent, your utility values are equal.

Expected utility of alternative 2 = Expected utility of alternative 1

Utility of other outcome = (p)(utility of best outcome, which is 1)

+ (1 – p)(utility of the worst outcome, which is 0)

Utility of other outcome = (p)(1) + (1 – p)(0) = p

Standard Gamble for Utility Assessment

Figure 3.7

Best Outcome

Utility = 1

Worst Outcome

Utility = 0

Other Outcome

Utility = ?

(p)

(1 – p)

Alternative 1

Alternative 2

Investment Example

Jane Dickson wants to construct a utility curve revealing her preference for money between $0 and $10,000.
A utility curve plots the utility value versus the monetary value.
An investment in a bank will result in $5,000.
An investment in real estate will result in $0 or $10,000.
Unless there is an 80% chance of getting $10,000 from the real estate deal, Jane would prefer to have her money in the bank.
So if p = 0.80, Jane is indifferent between the bank or the real estate investment.

Investment Example

Figure 3.8

Utility for $5,000 = U($5,000) = pU($10,000) + (1 – p)U($0)

= (0.8)(1) + (0.2)(0) = 0.8

p = 0.80

(1 – p) = 0.20

Invest in

Real Estate

Invest in Bank

$10,000

U($10,000) = 1.0

U($0.00) = 0.0

$5,000

U($5,000) = p = 0.80

Investment Example

Utility for $7,000 = 0.90

Utility for $3,000 = 0.50

We can assess other utility values in the same way.
For Jane these are:

Using the three utilities for different dollar amounts, she can construct a utility curve.

Utility Curve

Figure 3.9

U ($7,000) = 0.90

U ($5,000) = 0.80

U ($3,000) = 0.50

U ($0) = 0

1.0 –

0.9 –

0.8 –

0.7 –

0.6 –

0.5 –

0.4 –

0.3 –

0.2 –

0.1 –

| | | | | | | | | | |

$0 $1,000 $3,000 $5,000 $7,000 $10,000

Monetary Value

Utility

U ($10,000) = 1.0

Utility Curve

Jane’s utility curve is typical of a risk avoider.
She gets less utility from greater risk.
She avoids situations where high losses might occur.
As monetary value increases, her utility curve increases at a slower rate.

A risk seeker gets more utility from greater risk
As monetary value increases, the utility curve increases at a faster rate.

Someone with risk indifference will have a linear utility curve.

Preferences for Risk

Figure 3.10

Monetary Outcome

Utility

Risk Avoider

Risk Indifference

Risk Seeker

Utility as a
Decision-Making Criteria

Once a utility curve has been developed it can be used in making decisions.
This replaces monetary outcomes with utility values.
The expected utility is computed instead of the EMV.

Utility as a
Decision-Making Criteria

Mark Simkin loves to gamble.
He plays a game tossing thumbtacks in the air.
If the thumbtack lands point up, Mark wins $10,000.
If the thumbtack lands point down, Mark loses $10,000.
Mark believes that there is a 45% chance the thumbtack will land point up.
Should Mark play the game (alternative 1)?

Utility as a
Decision-Making Criteria

Figure 3.11

Decision Facing Mark Simkin

Tack Lands Point Up (0.45)

Alternative 1

Mark Plays the Game

Alternative 2

$10,000

–$10,000

Tack Lands
Point Down (0.55)

Mark Does Not Play the Game

Utility as a
Decision-Making Criteria

Step 1– Define Mark’s utilities.

U (–$10,000) = 0.05

U ($0) = 0.15

U ($10,000) = 0.30

Step 2 – Replace monetary values with
utility values.

E(alternative 1: play the game) = (0.45)(0.30) + (0.55)(0.05)

= 0.135 + 0.027 = 0.162

E(alternative 2: don’t play the game) = 0.15

Utility Curve for Mark Simkin

Figure 3.12

1.00 –

0.75 –

0.50 –

0.30 –

0.25 –

0.15 –

0.05 –

0 –

| | | | |

–$20,000 –$10,000 $0 $10,000 $20,000

Monetary Outcome

Utility

Utility as a
Decision-Making Criteria

Figure 3.13

Using Expected Utilities in Decision Making

Tack Lands Point Up (0.45)

Alternative 1

Mark Plays the Game

Alternative 2

0.30

0.05

0.15

Tack Lands
Point Down (0.55)

Don’t Play

Utility

E = 0.162

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