Chapter 19:
Decision
Analysis
Learning Objectives
LO1
Make decisions under certainty by constructing a
decision table.
LO2
Make decisions under uncertainty using the maximax
criterion, the maximum criterion, the Hurwicz criterion,
and the minimax regret.
LO3
Make decisions under risk by constructing decision
trees, calculating expected monetary value and
expected value of perfect information, and analyzing
utility.
LO4
Revise probabilities in light of sample information by
using Bayesian analysis and calculating the expected
value of sample information.
Decision-Making Scenarios
• Decision-making under certainty
• Decision-making under uncertainty
• Decision-making under risk
LO1
Three Variables
in Decision Analysis Model
• Many decision analysis problems can be viewed as having
variables
– Decision Alternatives are the various choices or options available to
the decision maker in any given problem situation (actions or
strategies)
– States of nature are the occurrences of nature that can happen after a
decision is made that can affect the outcome of the decision and over
which the decision maker has little or no control.
• States of nature can be environmental, business climate, political, or any
condition or state of affairs.
– Payoffs are the benefits or rewards (positive or negative) that result
from selecting a particular decision alternative. They are often
expressed in dollars, but may be stated in other units, such as market
share.
LO1
Construction of the
Decision or Payoff Table
• The concepts of decision alternatives, states of nature, and
payoffs can be examined jointly by using a decision table or
payoff table.
• The table a cross tabular table with “states of nature” and
“decision alternatives” as classification variables. The
associated outcomes in the cells are the payoffs or benefits
resulting from making certain choices and a certain state of
nature occurring.
• Table 19.1 on the next slide illustrates the structure of the
problem.
LO1
Table 19.1: The Decision or
Payoff Table
Decision
Alternatives
d
d
d
1
1
2
3

d
s
P
P
P
1,1
2 ,1
3,1

m
P
m ,1
States of Nature
s2 s3 
P1,2 P1,3 
P2,2 P2,3 
P3,2 P3,3 



Pm,2 Pm,3 
where: sj = state of nature
dj = decision alternative
Pi,j = payoff for decision i under state j
LO1
s
P
P
P
n
1, n
2, n
3, n

P
m, n
Yearly Payoffs on an Investment of
$10 000: Description of Problem
• An investor is faced with the decision of where and how
to invest $10,000 under several possible states of nature
• States of Nature
– A stagnant economy
– A slow-growth economy
– A rapid-growth economy
• Decision Alternatives being considered
–
–
–
–
Invest in the stock market
Invest in the Bond market
Invest in GICs
Invest in a mixture of stocks and bonds
• The payoffs are presented in Table 19.2 on the next slide.
LO1
Table 19.2: Decision Table
for an Investor
Stagnant
Stocks
$ (500)
Bonds
$ (100)
GICs
$ 300
Mixture $ (200)
Slow
Growth
$ 700
$ 600
$ 500
$ 650
Annual payoffs for an investment of $10,000
LO1
Rapid
Growth
$ 2,200
$
900
$
750
$ 1,300
Rule for Decision Making
Under Certainty
• In making decisions under certainty the
states of nature are known.
• The decision maker needs merely to
examine the payoffs under different decision
alternatives and select the alternative with
the highest with the largest payoff
LO1
Decision Making Under Certainty
The states of nature are known.
The Greatest Possible Payoff
Slow Rapid
Stagnant Growth Growth
Stocks
$ (500) $ 700 $ 2,200
Bonds
$ (100) $ 600 $ 900
GICs
$ 300 $ 500 $ 750
Mixture $ (200) $ 650 $ 1,300
Annual payoffs for an investment of $10,000
LO1
The economy
will grow
rapidly.
Invest in stocks.
Criteria for Decision Making
Under Uncertainty
• Maximax payoff: Choose the best of the
best (An optimist’s choice)
• Maximin payoff: Choose the best of the
worst (A pessimist’s choice)
• Hurwicz payoff: Use a weighted average of
the extremes (optimist and pessimist)
• Minimax regret: Minimize the maximum
opportunity loss
LO2
Maximax Criterion
1. Identify the maximum payoff for each alternative.
2. Choose the alternative with the largest maximum.
Stocks
Bonds
GICs
Mixture
LO2
Stagnant
$ (500)
$ (100)
$ 300
$ (200)
Slow
Growth
$ 700
$ 600
$ 500
$ 650
Rapid
Growth
$ 2,200
$
900
$
750
$ 1,300
Maximum
$ 2,200
$
900
$
750
$ 1,300
Maximin Criterion
1. Identify the minimum payoff for each alternative.
2. Choose the alternative with the largest minimum.
Stocks
Bonds
GICs
Mixture
LO2
Stagnant
$ (500)
$ (100)
$ 300
$ (200)
Slow
Rapid
Growth Growth
Minimum
$ 700 $ 2,200 $
(500)
$ 600 $
900 $
(100)
$ 500 $
750 $
300
$ 650 $ 1,300 $
(200)
Hurwicz Criterion
1. Identify the maximum payoff for each alternative.
2. Identify the minimum payoff for each alternative.
3. Calculate a weighted average of the maximum and the minimum using
and (1 - ) for weights.
4. The size of α is between 0 and 1 and will depend on how optimistic or
pessimistic the decision-maker is.
4. Choose the alternative with the largest weighted average.
Stagnant
Stocks
$ (500)
Bonds
$ (100)
GICs
$ 300
Mixture $ (200)
LO2
 =.7
 =.3
Slow
Rapid
Growth Growth
Maximum Minimum
$ 700 $ 2,200 $ 2,200 $ (500)
$ 600 $
900 $
900 $ (100)
$ 500 $
750 $
750 $
300
$ 650 $ 1,300 $ 1,300 $ (200)
Weighted
Average
$ 1,390
$
600
$
615
$
850
Decision Alternatives
for Various Values of 

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
LO2
1-
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Stocks
Max
Min
2,200
-500
-500
-230
40
310
580
850
1120
1390
1660
1930
2200
Bonds
Max
Min
900
-100
-100
0
100
200
300
400
500
600
700
800
900
GICs
Max
Min
750
300
300
345
390
435
480
525
570
615
660
705
750
Mixture
Max
Min
1,300
-200
-200
-50
100
250
400
550
700
850
1000
1150
1300
Graph of Hurwicz Criterion
Selections for Various Values of 
LO2
Investment Example:
Selected Regrets
Stagnant
Stocks $ (500)
Bonds $ (100)
GICs
$
300
Mixture $ (200)
Slow
Rapid
Growth Growth
$ 700 $ 2,200
$ 600 $ 900
$ 500 $ 750
$ 650 $ 1,300
I invested in GICs.
Then the economy
grew rapidly. I am
out $1,450.
LO2
I invested in stocks, and
the economy grew slowly.
I have no regrets.
I invested in stocks.
Then the economy
stagnated. I regret not
investing in GICs. I am
$800 down from where
I could have been.
Investment Example:
Opportunity Loss Table
Stocks
Bonds
GICs
Mixture
LO2
Stagnant
800
400
0
500
Slow
Rapid
Growth Growth
0
0
100
1,300
200
1,450
50
900
Investment Example:
Calculating Opportunity Loss
Payoff Table
Stagnant
Stocks $ (500)
Bonds $ (100)
GICs
$
300
Mixture $ (200)
Slow
Rapid
Growth Growth
$ 700 $ 2,200
$ 600 $ 900
$ 500 $ 750
$ 650 $ 1,300
Opportunity Loss Table
Stocks
Bonds
GICs
Mixture
Slow
Rapid
Stagnant Growth Growth
800
0
0
400
100
1,300
0
200
1,450
500
50
900
OLi,j = Max(column j) - Pi,j
LO2
Minimax Regret
1. Identify the maximum regret for each alternative.
2. Choose the alternative with the least maximum regret.
Stocks
Bonds
GICs
Mixture
LO2
Stagnant
800
400
0
500
Slow Rapid
Growth Growth Maximum
0
0
800
100
1,300
1,300
200
1,450
1,450
50
900
900
Decision Making under Risk
• Probabilities of the states of nature have been
determined
– Decision making under uncertainty: probabilities of the states
of nature are unknown
– Decision making under risk: probabilities of the states of nature
are known (have been estimated)
• Decision Trees
• Expected Monetary Value of Alternatives
LO3
Decision Table with States of Nature
Probabilities for Investment Example
LO3
Decision Tree
for the Investment Example
Stagnant (.25)
Chance
Node
Decision
Node Stocks
Bonds
GICs
-$500
Slow growth (.45)
$700
Rapid Growth (.30)
$2,200
Stagnant (.25)
-$100
Slow growth (.45)
Rapid Growth (.30) $600
$900
Stagnant (.25)
$300
Slow growth (.45)
$500
Rapid Growth (.30)
Mixture
$750
Stagnant (.25)
-$200
Slow growth (.45)
$650
Rapid Growth (.30)
$1,300
LO3
Expected Monetary Value Criterion
LO3
EMV Calculations
for the Investment Example
LO3
Decision Tree with Expected Monetary Values
for the Investment Example
Stagnant (.25)
$850
Stocks
$515
Bonds
GICs
$525
-$500
Slow growth (.45)
$700
Rapid Growth (.30)
$2,200
Stagnant (.25)
-$100
Slow growth (.45)
Rapid Growth (.30) $600
$900
Stagnant (.25)
$300
Slow growth (.45)
$500
Rapid Growth (.30)
Mixture
$750
$623.50
Stagnant (.25)
-$200
Slow growth (.45)
$650
Rapid Growth (.30)
$1,300
LO3
Decision Tree with Expected Monetary Values
for the Investment Example
LO3
EMV Criterion
for the Investment Example
1. Calculate the expected monetary value of each alternative.
2. Choose the alternative with the largest EMV: $850
LO3
Definition of Expected Value
of Perfect Information
• What is the value of knowing which state of
nature will occur and when? What is the value of
sampling information or undertaking the
prediction of an event?
• The concept of the expected value of perfect
information answers these questions and provide
some insight into how much the decision maker
should pay for market research.
LO3
Definition of Expected Value
of Perfect Information
• The expected value of perfect information
– the difference between the payoff that would occur if the
decision maker knew which state of nature would occur and
the expected monetary payoff from the best decision
alternative when there is no information about the occurrence
about the states of nature
• Expected Value of Perfect Information = Expected
Monetary Payoff with Perfect Information – Expected
Monetary Payoff with Information
Choice Criterion Under Perfect Information: Choose
the Maximum Payoff for any Given State of Nature
M
A
X
I
M
U
M
MAXIMUM
LO3
M
A
X
I
M
U
M
Expected Monetary Payoff with Perfect
Information for the Investment Example
• The investment of stocks was selected under the EMV
strategy because it resulted in the maximum payoff of
$850. This decision was made with no information about
the states of nature (Refer to slide above Perfect
Information)
• Maximum Payoffs for each state of nature under perfect
information: Stagnant Economy = $300; Slow Growth =
$700;
Rapid Growth = $2,200 (refer to slide above: Perfect
Information Criterion )
• The expected Monetary Value with perfect information
= (300)(0.25) + ($700)(0.45) + ($2,200)(0.30)
= $1,050
LO3
Expected Value of Perfect Information
for the Investment Example
Expected Value of Perfect Information
= Expected Monetary Payoff with Perfect Information - Max(EMV[di])
= $1050 - $850
= $200
It would not be economically wise to spend more than $200 to obtain perfect
Information about these states of nature. The cost of collecting and
processing the information is very high relative to the benefits.
LO3
Utility
• Utility is the degree of pleasure or displeasure a decision
maker has in being involved in the outcome selection
process given the risks and opportunities available.
• The degree of pleasure will depend on the individual
tolerance of risk. An investor may be classified as
– Risk-Avoider
– Risk-Neutral
– Risk-Taker
LO3
Measurement of Utility:
Standard Gamble Method
• A person has the chance to enter a contest with a 50-50
chance of winning $100,000
• If the person wins the contest, he or she wins $100,000.
• If the person loses, he or she receives $0.
• Cost of entering the game is zero dollars.
• The Expected value of the game is :
– ($100,000)*(.5)+($0)*(.5) = $50,000. But the person betting will not
get this unless he or she continues to bet indefinitely on the game.
• Would a person take an offer of $30,000 for certain, in the
condition that he or she drops out of the game. The answer
to this depends on the person’s assets and whether the
person is risk neutral, a risk avoider, or a risk taker.
LO3
Utility Curves for Three Types of Game
Players
•The straight line is where the expected value of the game is equal the payment offered to drop out
of the game (Risk Neutral) rather than continue the gamble
•For the risk avoider the expectation of winning must be higher than the long run probability that
makes EMV = the equivalent certainty value: the utility curve is above the Risk Neutral line
•The risk taker will bet on the gamble even if the chances of winning is below that required to make
EMV = to the equivalent certainty value . The utility curve is below the Risk Neutral line
Risk-Avoider
Chance of
Winning
the Contest
Risk
Neutral
Risk-Taker
LO3
Monetary Payoff
Risk Neutral Game Player in a Standard Gamble
Game: Indifferent to Owning “a” or “b”
• The game player decides to take the $50,000 and not continue
gambling. The amount is equal to the expected value of the game
at probability of winning =0.5
.5
$100 000
a
.5
b
-$0
$50 000
LO3
Risk Avoider in a Standard Gamble Game:
Indifferent to Owning “a” or “b”
• Game player decides to take the $20,000 for certain, rather
than continue to play, even though the expected value of
the game is much higher ($50,000)
.5
$100 000
a
.5
b
LO3
-$0
$20 000
Risk Taker in a Standard Gamble Game:
Indifferent to Owning “a” or “b”
• Game player decides not to take the offer of $70,000 to leave
the game, despite the fact that the expected value of the
gamble is much less ($50,000).
.5
$100 000
a
.5
b
LO3
-$0
$70 000
Risk Curves For Three Game Players
LO3
Revising Probabilities
in Light of Sample Information
• Bayes’ Rule
• Expected Value of Sample Information
LO4
Interpretation
• X represents the gamble responses of a risk-avoider
• X makes decision based on a utility the segment of
parabolic function above the risk-neutral line
• Y represents the gamble responses of a risk-taker
• Y makes decisions based on an exponential utility
function below the risk –neutral.
LO4
Interpretation
• Let Z represent the responses of a risk-neutral game
player
• Z is indifferent between a certain guarantee amount, and
gambling or not gambling. He remains on the EMV line
• The gamble is $10,000. Probability of winning is p= 0.5,
EMV = $50,000.
• The risk curve shows that for a guarantee of $50,000 to
drop gambling in the game , the risk avoider (X) will only
gamble if the probability of winning is p=0.8. On the other
hand the risk-taker will gamble even if the guarantee is
just under$80,000, approximately $30,000 more than the
EMV at p= 0.5.
LO4
Decision Table
for Investment Problem
Bonds
Stocks
LO4
No
Rapid
Growth
Growth
(.65)
(.35)
$
500 $
100
$ (200) $ 1,100
Expected Monetary Value Criterion for the
Investment Example
Bonds
Stocks
LO4
No
Rapid
Growth
Growth
0.65
0.35
$
500 $
100
$ (200) $ 1,100
Expected
Monetary
Value
$ 360.00
$ 255.00
Revising Probabilities
in the Light of Sample Information
• In this section we address the revision of prior probabilities
using Bayes’ rule with sampling information in the context of
the $10,000 case discussed above.
• The probabilities of the various states of nature are frequently
not fixed or known in an exact way. Thus prior subjective
probabilities (or probabilities based on our best guess) may be
used initially to obtain the EMV. These probabilities can be
updated by introducing information obtained from samples.
The updated probabilities can be incorporated into the decision
process to hopefully help make better decisions.
LO4
Simplified Version of the $10,000 Investment
Decision Problem: Table 19.6
LO4
Decision Tree
for the Investment Example: Figure 19.5
No Growth (.65)
$500
EMV=$360
Rapid Growth (.35)
Bonds
$100
($360)
No Growth (.65)
-$200
Stocks
EMV=$255
Rapid Growth (.35)
$1,100
LO4
The Success and Failure Rates of the Forecaster in
Forecasting the Two States of the Economy
Actual State of Economy
No Growth
Rapid Growth
(s1)
(s2)
.80
.30
.20
.70
Forecaster Predicts
No Growth (F1 )
Forecaster Predicts
Rapid Growth (F2 )
P(Fi|sj)
LO4
Bayes’ Rule
LO4
Revision Based on a Forecast
of No Growth (F1)
State of
Prior
Conditional
Economy Probabilities Probabilities
No
Growth
(s 1)
Rapid
Growth
(s 2)
LO4
Joint
Probabilities
Revised
Probabilities
P(sj|F1)
P(s 1) = .65
P(F1| s 1) = .80 P(F1  s 1) = .520 .520/.625 = .832
P(s 2) =.35
P(F1| s 2) = .30 P(F1  s 2) = .105 .105/.625 = .168
P(F1) = .625
Revision Based on a Forecast
of Rapid Growth (F2)
State of
Economy
No
Growth
(s 1)
Rapid
Growth
(s 2)
LO4
Prior
Probabilities
Conditional
Probabilities
Joint
Probabilities
P(s 1) = .65
P(F 2| s 1) = .20 P(F 2  s 1) = .130 .130/.375 = .347
P(s 2) =.35
P(F 2| s 2) = .70 P(F 2  s 2) = .245 .245/.375 = .653
P(F2) = .375
Revised
Probabilities
P(sj|F2)
Decision Tree for the Investment Example
After Revision of Probabilities: Figure 19.6
No Growth (.832)
$500
$432.80
Rapid Growth (.168)
Bonds
$432.80
Forecast
No Growth
(.625)
$100
No Growth (.832)
-$200
Stocks
Rapid Growth (.168)
Buy
Forecast
$18.40
$513.84
$1,100
No Growth (.347)
$500
$238.80
Forecast
Rapid Growth
(.375)
Rapid Growth (.653)
Bonds
$648.90
$100
No Growth (.347)
-$200
Stocks
Rapid Growth (.653)
$648.90
$1,100
LO4
Expected Value of Sample Information for the
Investment Example
 In general, the expected value of sample information
= expected monetary value with information
- expected monetary value without information
= $513.84 - $360
= $153.84
But what if the decision maker had to pay $100 for the
forecaster’s prediction?
 This would reduce the value of getting perfect information from
$513.84 shown in Figure 19.6 in the previous slide to $413.84.
 Note that this is still superior to the $360 without sample
information
LO4
Decision Tree Investment Example
All Options Included
Figure 19.7 is constructed
by combining Figures 19.5
and 19.6.
This is the Investment Tree
for the investment
information with the
options of buying the
information or not buying
the information included .
It includes a cost of buying
Information ($100) and the
EMV with this purchased
information ($413.84)
LO4
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