Chapter 3:
DECISION ANALYSIS
Part 2
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Decision Making Under Risk
 Probabilistic decision situation
 States of nature have probabilities of occurrence.
 The probability estimate for the occurrence of each
state of nature( if available) can be incorporated in
the search for the optimal decision.
 For each decision calculate its expected payoff by
S
(Probability)(Payoff)
Expected Payoff =
Over States of Nature
2
Decision Making Under Risk (cont.)
 Select the decision with the best expected payoff
3
TOM BROWN - continued
(0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130
4
Decision Making Criteria (cont.)
 When to Use the Expected Value Approach
 The Expected Value Criterion is useful in cases
where long run planning is appropriate, and
decision situations repeat themselves.
 One problem with this criterion is that it does not
consider attitude toward possible losses.
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Expected Value of Perfect Information
 The gain in Expected Return obtained from knowing
with certainty the future state of nature is called:
Expected Value of Perfect Information
(EVPI)
 It is also the Smallest Expect Regret of any decision
alternative.
Therefore, the EVPI is the expected regret
corresponding to the decision selected
using the expected value criterion
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Expected Value of Perfect Information
(cont.)
 EVPI = ERPI - EREV
 EREV: Expected Return of the EV criterion .
 Expected Return with Perfect Information ERPI=
(best outcome of 1st state of nature)*(Probability of
1st state of nature) + ….. +(best outcome of last
state of nature)*(Probability of last state of nature)
7
TOM BROWN - continued
If it were known with certainty
that there will be a “Large Rise” in the market
-100 rise
Large
250
Stock
500
60
... the optimal decision would be to invest in...
Similarly,
Expected Return with Perfect information =
0.2(500)+0.3(250)+0.3(200)+0.1(300)+0.1(60) = $271
EVPI = ERPI - EV = $271 - $130 = $141
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Expected Value of Perfect Information
(cont.)
 Another way to determine EVPI as follows
If Tom knows the market will show a large rise, he
should buy the “stock”, within profit $500, or a
gain of $250 over what he would earn from the
“bond” (optimal decision without the additional
information).
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Expected Value of Perfect Information
(cont.)
If Tom knows in advance
the market would undergo
His optimal decision
With gain of payoff
A large rise
stock
500-250= $250
A small rise
stock
250-200= $ 50
No change
gold
200-150= $ 50
A small fall
gold
300-(-100)=$400
A large fall
C/D
60-(-150)= 210
EVPI= 0.2(250) + 0.3(50) +0.3(50)+ 0.1(400)+ 0.1(210)= 141
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Baysian Analysis - Decision Making with
Imperfect Information
 Baysian Statistic play a role in assessing additional
information obtained from various sources.
 This additional information may assist in refining
original probability estimates, and help improve
decision making.
11
TOM BROWN - continued
 Tom can purchase econometric forecast results for
$50.
 The forecast predicts “negative” or “positive”
econometric growth.
 Statistics regarding the forecast.
The Forecast
predicted
Positive econ. growth
Negative econ. growth
When the stock market showed a...
Large Rise
80%
20%
Small Rise No Change
70%
30%
50%
50%
Small Fall
40%
60%
When the stock market showed a large rise the forecast was
“positive growth” 80% of the time.
Large Fall
0%
100%
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TOM BROWN - continued
 P(forecast predicts “positive” | small rise in
market) = 0.7
 P(forecast predicts “ negative” | small rise in
market) = 0.3
Should Tom purchase the Forecast ?
13
SOLUTION
 Tom should determine his optimal decisions when
the forecast is “positive” and “negative”.
 If his decisions change because of the forecast, he
should compare the expected payoff with and
without the forecast.
 If the expected gain resulting from the decisions
made with the forecast exceeds $50, he should
purchase the forecast.
14
SOLUTION
 To find Expected payoff with forecast Tom
should determine what to do when:
 The forecast is “positive growth”
 The forecast is “negative growth”
15
SOLUTION
 Tom needs to know the following probabilities





P(Large rise | The forecast predicted “Positive”)
P(Small rise | The forecast predicted “Positive”)
P(No change | The forecast predicted “Positive ”)
P(Small fall | The forecast predicted “Positive”)
P(Large Fall | The forecast predicted “Positive”)
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SOLUTION





P(Large rise | The forecast predicted “Negative ”)
P(Small rise | The forecast predicted “Negative”)
P(No change | The forecast predicted “Negative”)
P(Small fall | The forecast predicted “Negative”)
P(Large Fall) | The forecast predicted “Negative”)
Bayes’ Theorem provides a procedure to calculate these probabilities
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Bayes’ Theorem
 P(A|B) =
 Proof:
p(A|B)= P (A and B) / P(B) (1)
P(B|A)= P(A and B)/P(A)
 P(A and B) = P(B|A)*P(A)
(1) P(A|B)=P(B|A)*P(A)/P(B)
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Bayes’ Theorem (cont.)
 Often we begin probability analysis with initial or
prior probabilities.
 Then, from a sample , special report, or product
test we obtain some additional information.
 Given this information, we calculate revised or
posterior probability.
Prior
probabilities
New
information
Posterior
probabilities
19
Bayes’ Theorem(cont.)
P(A i
|
P(B A i)P(A i)
| B) =
[ P(B
|A
1)P(A 1)+
P(B | A 2)P(A 2)+…+ P(B
|A
Posterior probabilities
Prior probabilities
Probabilities determined
after the additional info
becomes available
Probabilities estimated
Determined based on
Current info, before
New info becomes available
n)P(A n)
20
]
 The tabular approach to calculating posterior
probabilities for positive economical forecast
States of
Nature
Large Rise
Small Rise
No Change
Small Fall
Large Fall
Prior
Probab.
0.2
0.3
0.3
0.1
0.1
Conditnal
Probab.
X
0.8
0.7
0.5
0.4
0
Joint
Probab.
0.16
0.21
0.15
0.04
0
Ai: large rise
B: forecast positive
P(Bi |Ai )P(Ai)
P(forecast= Positive| large rise)P( large rise)
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No Change
Small Fall
Large Fall
0.3
0.1
0.1
X
The Probability that the forecast is
“positive” and the stock market
shows “Large Rise”.
0.5
0.4 =
0
Sum =
0.15
0.04
0
0.56
0.268
0.071
0.000
0.16/ 0.56
The probability that the stock
market shows “Large Rise”
given
that the forecast
predicted “Positive”
Probability( forecast= positive) = 0.16+ 0.21+0.15+ 0.04+ 0.0 = 0.56
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 The tabular approach to calculating posterior
probabilities for “negative” ecnomical forecast
Nature
Large Rise
Small Rise
No Change
Small Fall
Large Fall
Probab.
0.2
0.3
0.3
0.1
0.1
Probab.
0.2
0.3 =
X
0.5
0.6
1
Sum =
Probab.
0.04
0.09
0.15
0.06
0.1
0.44
Probab.
0.091
0.205
0.341
0.136
0.227
 Probability (forecast= negative) = 0.44
23
WINQSB printout for the calculation of
the Posterior probabilities
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Expected Value of Sample Information
EVSI
 gain from making decisions based on Sample Information.
 With the forecast available, the Expected Value of Return
is revised.
 Calculate Revised Expected Values for a given forecast
as follows.
Gold
Bond
EV(Invest in……. |“Positive” forecast) =
-100
100
200
300
250
200
200
300
=.286( -100
)+.375( 100
)+.268( 150
)+.071( -100
)+0(
-100
100
200
300
-100
Gold 100
Bond
200
300
00
-150
)
0
0
$84
=$180
EV(Invest
in …….100
| “Negative”
forecast)-100
=
-100
250
200
200
150
300
-150
0
$120
$ 65
36
=.091(
)+.205(
)+.341(
)+.136(
)+.227(
)=
EREV
Expected
Value
Sampling
= 130
 =The
rest of
theWithout
revised
EV sInformation
are calculated
in a
similar manner.
Expected Value of Sample Information - Excel
Prior
Decisions/
large rise
small rise
no change
small fall
large fall EV
Gold
-100
100
200
300
0
100
Bond
250
200
150 -100 -150
130
Stock
500
250
100 -200 -600
125
C/D Account60
60
60
60
60
60
Prior Probability
0.2
0.3
0.3
0.1
0.1
Pos. Economic
0.29 Forecast
0.38
0.27
0.07
0
Neg. Economic
0.09 Forecast
0.21
0.34
0.14
0.23
Revised EV
Pos
Neg
84
120
180
65
250
-37
60
60
0.56
0.44
Invest=inExpected
Stock when
the with
Forecast
is “Positive”
ERSI
Return
sample
Information =
(0.56)(250) + (0.44)(120) = $193
Invest in Gold when the forecast is “Negative”
So, Should Tom purchase the Forecast ?
37
 EVSI = Expected Value of Sampling Information
= ERSI - EREV = 193 - 130 = $63.
Yes, Tom should purchase the Forecast.
His expected return is greater than the Forecast cost.
 Efficiency = EVSI / EVPI = 63 / 141 = 0.45
38
Game Theory

Game theory can be used to determine optimal
decision in face of other decision making players.
 All the players are seeking to maximize their return.
 The payoff is based on the actions taken by all the
decision making players.
39
Game Theory (cont.)
 Classification of Games
 Number of Players
 Two players - Chess
 Multiplayer - More than two competitors (Poker)
 Total return
 Zero Sum - The amount won and amount lost by all
competitors are equal (Poker among friends)
 Nonzero Sum -The amount won and the amount lost
by all competitors are not equal (Poker In A Casino)
40
Game Theory (cont.)
 Sequence of Moves
 Sequential - Each player gets a play in a given
sequence.
 Simultaneous - All players play simultaneously.
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