Introduction to Decision Analysis
• The field of decision analysis provides
framework for making important decisions.
• Decision analysis allows us to select a decision
from a set of possible decision alternatives when
uncertainties regarding the future exist.
• The goal is to optimized the resulting payoff in
terms of a decision criterion.
Components of a Decision Model
• A set of possible decisions
– discrete, continuous
– finite, infinite
• A set of possible events that could occur
(states of nature)
– discrete, continuous
– finite, infinite
• Payoffs (returns or costs) associated with
each decision/state of nature pair
Payoff Table Analysis
• Payoff Tables
– Payoff Table analysis can be applied when • There is a finite set of discrete decision alternatives.
• The outcome of a decision is a function of a single future event.
– In a Payoff Table • The rows correspond to the possible decision alternatives.
• The columns correspond to the possible future events.
• Events (States of Nature) are mutually exclusive and collectively
exhaustive.
• The body of the table contains the payoffs.
EXAMPLE
TOM BROWN INVESTMENT DECISION
• Tom Brown has inherited $1000.
• He has decided to invest the money for one year.
• A broker has suggested five potential investments.
–
–
–
–
–
Gold.
(Junk) Bond.
Growth Stock.
Certificate of Deposit.
Stock Option Hedge.
• Tom has to decide how much to invest in each
investment.
• Constructing a Payoff Table
– Determine the set of possible decision alternatives.
• for Tom this is the set of five investment opportunities.
– Defined the states of nature.
• Tom considers several stock market states (expressed by
changes in the DJA)
State of Nature
DJA Correspondence
S.1: A large rise in the stock market
Increase over 1000 points
S.2: A small rise in the stock market
Increase between 300 and 1000
S.3: No change in the stock market
Change between -300 and +300
S.4: A small fall in stock market
Decrease between 300 and 800
S5: A large fall in the stock market
Decrease of more than 800
Determining Payoffs
• Our broker reasons:
– Stocks and bonds generally move in the same
direction of the market
– Gold is an investment hedge that seems to move
opposite to the market direction
– A C/D account pays the same interest irrespective of
market conditions
• Specifically our broker’s analysis leads to the
following payoff table:
Payoff Table
States of Nature
Decision Alternatives
Large Rise Small Rise No Change Small Fall Large Fall
Gold
-100
100
200
300
0
Bond
250
200
150
-100
-150
Stock
500
250
100
-200
-600
C/D Account
60
60
60
60
60
Stock Option Hedge200
150
150
-200
-150
Dominated Decision Alternatives
States of Nature
Decision Alternatives
Large Rise Small Rise No Change Small Fall Large Fall
Gold
-100
100
200
300
0
Bond
250
200
150
-100
-150
Stock
500
250
100
-200
-600
C/D Account
60
60
60
60
60
Stock Option Hedge200
150
150
-200
-150
The Stock Option hedge is dominated by the Bond Alternative,
because the payoff for each state of nature for the stock option
 the payoff for the bond option. Thus the stock option hedge
can be eliminated from any consideration.
Decision Making Criteria
• Classifying Decision Making Criteria
– Decision making under certainty
• The future state of nature is assumed known
– Decision making under risk
• There is some knowledge of the probability of the states of
nature occurring.
– Decision making under uncertainty.
• There is no knowledge about the probability of the states of
nature occurring.
Decision Making Under Certainty
• Given the “certain” state of nature, select the
best possible decision for that state of nature
Decisions for
Decision Making Under Certainty
If we knew Large
this ===> Rise
Gold
-100
Bond
250
Stock
500
C/D
60
Choose => Stock
Small No Small Large
Rise Change Fall Fall
100 200 300
0
200 150 -100 -150
250 100 -200 -600
60
60
60
60
Stock Gold Gold C/D
• Decision Making Under Uncertainty
– The decision criteria are based on the decision maker’s
attitude toward life.
– These include an individual being pessimistic or optimistic,
conservative or aggressive.
– Criteria
• Maximin Criterion - pessimistic or conservative approach.
• Minimax Regret Criterion - pessimistic or conservative approach.
• Maximax criterion - optimistic or aggressive approach.
• Principle of Insufficient Reasoning.
Decision Making Under Uncertainty
• The decision criteria are based on the decision
maker’s attitude toward life.
• These include an individual being pessimistic or
optimistic, conservative or aggressive.
• Criteria
– Maximax (Minimin), Maximin (Minimax), Minimax Regret,
Principle of Insufficient Reasoning
• The Maximax Criterion
– This criterion is based on the best possible scenario.
– It fits both an optimistic and an aggressive decision maker.
• An optimistic decision maker believes that the best possible
outcome will always take place regardless of the decision made.
• An aggressive decision maker looks for the decision with the
highest payoff (when payoff is profit)
– To find an optimal decision.
• Find the maximum payoff for each decision alternative.
• Select the decision alternative that has the maximum of
the “maximum” payoff.
TOM BROWN - continued
The Maximax criterion
Maximum
Decision Large rise Small rise No changeSmall fall Large fall Payoff
Gold
-100
100
200
300
0
300
Bond
250
200
150
-100
-150
200
Stock
500
250
100
-200
-600
500
C/D
60
60
60
60
60
60
• The Maximin Criterion
– This criterion is based on the worst-case scenario.
– It fits both a pessimistic and a conservative decision
maker.
• A pessimistic decision maker believes that the worst possible
result will always occur.
• A conservative decision maker wishes to ensure a guaranteed
minimum possible payoff.
– To find an optimal decision
• Record the minimum payoff across all states of nature for
each decision.
• Identify the decision with the maximum “minimum payoff”.
TOM BROWN - Continued
Decisions
Gold
Bond
Stock
C/D account
The Maximin Criterion
LargeRrise Small Rise No change Small Fall
-100
250
500
60
100
200
250
60
200
150
100
60
300
-100
-200
60
Minimum
Large Fall Payoff
0
-150
-600
60
-100
-150
-600
60
• The Minimax Regret Criterion (“I shoulda”)
– This criterion fits both a pessimistic and a conservative
decision maker.
– The payoff table is based on “lost opportunity,” or “regret”.
– The decision maker incurs regret by failing to choose the
“best” decision.
– To find an optimal decision
• For each state of nature.
– Determine the best payoff over all decisions.
– Calculate the regret for each decision alternative as the difference
between its payoff value and this best payoff value.
• For each decision find the maximum regret over all states of
nature.
• Select the decision alternative that has the minimum of these
“maximum regrets”.
500 - (-100) = 600
500
-100
500TOM BROWN
- continued
500
-100
The Payoff Table
500 -100
Decision Large rise-100
Small rise No change Small fall Large fall
Gold
-100-100 100Investing200
300 a regret0
in
Gold
incurs
Bond
250 500 200
150
-100
-150
when
the
market
exhibits
Stock
500 500 250
100
-200
-600
a large rise
C/D
60
60
60
60
60
The Regret Table
Maximum
Decision Large rise Small rise No change Small fall Large fall Regret
Gold
600
150
0
0
60
600
build the Regret
Table
Bond
250 Let us50
50
400
210
400
Stock
0
0
100
500
660
660
C/D
440
190
140
240
0
440
• The Principle of Insufficient Reason
– This criterion might appeal to a decision maker who is
neither pessimistic nor optimistic.
– It assumes all the states of nature are equally likely to
occur.
– The procedure to find an optimal decision.
• For each decision add all the payoffs.
• Select the decision with the largest sum (for profits).
TOM BROWN - continued
– Sum of Payoffs
•
•
•
•
Gold $600
Bond $350
Stock $50
C./D $300
– Based on this criterion the optimal decision
alternative is to invest in gold.
Summary
•
•
•
•
Maximax Decision -- Stock
Maximin Decision -- C/D
Minimax Regret -- Junk Bond
Insufficient Reason -- Gold
• Isn’t there a better way?
• Decision Making Under Risk
– The Expected Value Criterion
• If a probability estimate for the occurrence of each state of
nature is available, payoff expected value can be calculated.
• For each decision calculate its expected payoff by
Expected Payoff = S(Probability)(Payoff)
Over States of Nature
Based on past market performance the analyst predicts:
P(large rise) = .2, P(small rise) = .3, P(No change) = .3,
P(small fall) = .1, P(large fall) = .1
TOM BROWN - continued
The Expected Value Criterion
Expected
Decision Large rise Small rise No changeSmall fall Large fall Value
Gold
-100
100
200
300
0
100
Bond
250
200
150
-100
-150
130
Stock
500
250
100
-200
-600
125
C/D
60
60
60
60
60
60
Prior Probability
0.2
0.3
0.3
0.1
0.1
(0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130
• 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.
Expected Value of Perfect Information
(EVPI)
• The Gain in Expected Return obtained from
knowing with certainty the future state of nature
is called:
Expected Value of Perfect Information
Therefore, the EVPI is the expected regret
corresponding to the decision selected
using the expected value criterion
(EVPI)
• It is also the Smallest Expect Regret of any
decision alternative.
TOM BROWN - continued
If it were known with certainty
that there will be a “Large Rise” in the market
The
- Expected Value of Perfect Information
Large
riseSmall Rise No change Small Fall Large Fall
Decision Large
Rise
100-100
Gold
100
200
300
0
Bond
200
150
-100
-150
250250
Stock
500
250
100
-200
-600
C/D
60
60
60
60
60
Probab.
0.2
0.3
0.3
0.1
0.1
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
Bayesian Analysis - Decision Making
with Imperfect Information
• Bayesian 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.
TOM BROWN - continued
• Tom can purchase econometric forecast
results for $50.
• The forecast predicts “negative” or “positive”
econometric growth.
• Should
StatisticsTom
regarding
the forecast.
purchase
the Forecast ?
The Forecast
When the stock market showed a...
Large Rise Small Rise No Change
predicted
Positive econ. growth
Negative econ. growth
80%
20%
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%
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.
• 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”)
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
P(B |A i)P(A i)
P(A i | B) =
[ P(B | A 1)P(A 1)+ P(B | A 2)P(A 2)+…+ P(B | A n)P(A n) ]
Posterior Probabilities for “positive” economic forecast
States of
Nature
Large Rise
Small Rise
No Change
Small Fall
Large Fall
Prior
Probab.
Conditnal
Probab.
Joint
Probab.
Posterior
Probab.
0.2
0.8 = 0.16
0.286
0.2 X
0.286
0.3
0.7
0.21
0.375
0.3
0.375
0.3
0.5
0.15
0.268
0.3probability
0.268
The
that
the
stock
market
Observe
the
revision
in
The
Probability
that
the
forecast
is
0.1
0.4
0.04
0.071
0.1
0.071
the
prior
probabilities
shows
“Largeand
that 0.000
thegiven
stock 0market
0.1
0Rise”
0.1 “positive”
0.000
Sum
= ”. “Positive”
0.56
theshows
forecast
predicted
“Large
Rise
0.16
0.56
Bayesian Prob. for “Negative” Forecast
s
Large Rise
Small Rise
No Change
Small Fall
Large Fall
P(s) P(Neg|s) P(Neg and s) P(s|Neg)
.20
.30
.30
.10
.10
.20
.30
.50
.60
1
.04
.09
.15
.06
.10
.44
.04/.44= .091
.09/.44= .205
.15/.44= .341
.06/.44= .136
.10/.44= .227
• Expected Value of Sample Information
– The expected 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.
Bond
• EV(Invest in…….
Gold |“Positive” forecast) =
-100
100)+.268( 150
200)+.071( -100
300)+0(
=.286( -100
)+.375( 100
250
200
200
300
-100
100
200
300
Gold100
-100 in Bond
200
• EV(Invest
…….
| “Negative”
forecast) =300
00 ) =$180
-150
$84
0
0
=.091(-100
250 )+.205( 100
200 )+.341( 200
150 )+.136( -100
300 )+.227(-150
0 ) = $120
$ 65
EREV
= Expected
Value
WithoutEV
Sampling
Information =in130
– The
rest of the
revised
s are calculated
a
similar manner.
Expected Value of Sample Information - Excel
Prior
Decisions/
large rise
small rise
no change
small fall
large fallEV
Gold
-100
100
200
300
0
100
Bond
250
200
150 -100 -150
130
Stock
500
250
100 -200 -600
125
C/D Account
60
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
So,
0.56
Should Tom purchase the Forecast ?
Invest=inExpected
Stock when
the with
Forecast
is “Positive”
ERSI
Return
sample
Information =
(0.56)(250)
+ (0.44)(120)
$193 is “Negative”
Invest in Gold
when the=forecast
0.44
• 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
Decision Trees
• The Payoff Table approach is useful for a
single decision situation.
• Many real-world decision problems consists
of a sequence of dependent decisions.
• Decision Trees are useful in analyzing multistage decision processes.
• Characteristics of the Decision Tree
– A Decision Tree is a chronological representation of the
decision process
– There are two types of nodes
• Decision nodes (represented by squares)
• State of nature nodes (representing by circles).
– The root of the tree corresponds to the present time.
– The tree is constructed outward into the future with
branches emanating from the nodes.
• A branch emanating from a decision node corresponds to a
decision alternative. It includes a cost or benefit value.
• A branch emanating from a state of nature node corresponds
to a particular state of nature, and includes the probability of
this state of nature.
BILL GALLEN DEVELOPMENT COMPANY
– B. G. D. plans to do a commercial development on a property.
– Relevant data
•
•
•
•
Asking price for the property is $300,000
Construction cost is $500,000
Selling price is approximated at $950,000
Variance application costs $30,000 in fees and expenses
– There is only 40% chance that the variance will be approved.
– If B. G. D. purchases the property and the variance is denied, the property can
be sold for a net return of $260,000
– A three month option on the property costs $20,000 which will allow B.G.D. to
apply for the variance.
• A consultant can be hired for $5000
– P (Consultant predicts approval | approval granted) = 0.70
– P (Consultant predicts denial | approval denied) = 0.90
What should BGD do?
• Construction of the Decision Tree
– Initially the company faces a decision about hiring the
consultant.
– After this decision is made more decisions follow regarding
• Application for the variance.
• Purchasing the option.
• Purchasing the property.
0
3
2
Buy land
-300,000
4
Apply for variance
-30,000
11
Apply for variance
-30,000
1
6
Build
-500,000
7
120,000
Sell
950,000
8
5
9
13
Buy land
-300,000
14
Sell
260,000
Build
-500,000
-70,000
10
15
Sell
950,000
16
100,000
12
17
-50,000
0
3
ing
oth
n
o
D
0
Do
t
no
2
t
an
ult
s
n
co
ih re
Buy land
-300,000
Pu
rch
-20 ase
op
,00
tio
0
n
4
Apply for variance
-30,000
5
ed
rov
p
p
A
0.4
Den
ied
0.6
ved
pro
Ap
1
11
Hi
re
c
-50 onsu
lta
00
n
Apply for variance
-30,000
12
6
Build
-500,000
7
9
13
Buy land
-300,000
14
120,000
Sell
8
950,000
Sell
260,000
Build
-500,000
0.4
Den
ied
0.6
t
18
Let us consider
This is where
the decision
we are at
to this
hire stage
a consultant
-70,000
10
15
Sell
950,000
16
100,000
17
-50,000
2
-5000
20
1
19
Buy land
21
-300,000
28
18
35
-300,000
-30,000
Apply for variance
-30,000
-5000
36
Buy land
Apply for variance
37
44
Apply for variance
-30,000
Apply for variance
-30,000
23
22
Build
-500,000
24
Sell
950,000
115,000
25
0.8235
?
0.1765
?
26
Sell
260,000
-75,000
27
The consultant serves as a source for additional information
about denial or approval of the variance.
Therefore, at this point we need to calculate the
posterior probabilities for the approval and denial
of the variance application
Posterior Probability of approval | consultant predicts approval) = 0.8235
Posterior Probability of denial | consultant predicts approval) = 0.1765
• Determining the Optimal Strategy
The rest of the Decision Tree is built in a similar manner.
Now,
• Work backward from the end of each branch.
• At a state of nature node, calculate the expected value of the
node.
• At a decision node, the branch that has the highest ending
node value is the optimal decision.
• The highest ending node value is the value for the decision
node.
Let us illustrate by evaluating one branch -Hire consultant and consultant predicts approval
115,000
•Predict
approval
22
•81483
23
0.8235
?
0.1765
?
75,000
26
115,000
Build
-500,000
-75,000
115,000
115,000
24
Sell
950,000
-75,000
-75,000
Sell
260,000
115,000
115,000
115,000
25
-75,000
-75,000
-75,000
27
•$81,483 is the expected value if consultant predicts approval and land is bought
•In a similar manner, the expected value if consultant predicts approval and the
option is purchased is $68,525
•The expected value if the consultant predicts approval and BGD does nothing is $5,000 (the consultant’s fee).
Thus if consultant recommends approval -- BUY LAND
Evaluation of Other Branches
• DO NOT HIRE CONSULTANT
– Do nothing
– Buy land
– Purchase option
EV = $0
EV = $6,000
EV = $10,000 <==BEST
• HIRE CONSULTANT -- PREDICTS DENIAL
– Do nothing
– Buy land
– Purchase option
EV = -$5,000 <===BEST
EV = -$40,458
EV = -$27,730
BEST COURSE OF ACTION
• EV(DO NOT HIRE CONSULTANT) = $10,000
• EV(HIRE CONSULTANT) =
.4(81,483) + .6(-5000) = $29,593.20
OPTIMAL STRATEGY -• HIRE CONSULTANT
– If consultant predicts approval -- buy the land
– If consultant predicts denial -- do nothing
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.
• 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)
– Sequence of Moves
• Sequential - Each player gets a play in a given sequence.
• Simultaneous - All players play simultaneously.
IGA SUPERMARKET
• The town of Gold Beach is served by two
supermarkets: IGA and Sentry.
• Market share can be influenced by their
advertising policies.
• The manager of each supermarket must decide
weekly which area of operations to discount and
emphasize in the store’s newspaper flyer.
• Data
– The weekly percentage gain in market share for IGA, as
a function of advertising emphasis.
IGA's
Meat
Emphasis Produce
Grocery
Meat
2
-2
2
Sentry's Emphasis
Produce Grocery Bakery
2
-8
6
0
6
-4
-7
1
-3
– A gain in market share to IGA results in equivalent loss
for Sentry, and vice versa (i.e. a zero sum game)
IGA needs to determine an advertising emphasis that
will maximize its expected change in market share
regardless of Sentry’s action.
SOLUTION
• IGA’s Perspective - A Linear Programming model
– Decision variables
• X1 = the probability IGA’s advertising focus is on meat.
• X 2 = the probability IGA’s advertising focus is on produce.
• X 3 = the probability IGA’s advertising focus is on groceries.
– Objective Function For IGA
• Maximize expected market change (in its own favor) regardless
of Sentry’s advertising policy.
• Define the actual change in market share as V.
– Constraints
• IGA’s market share increase for any given advertising
focus selected by Sentry, must be at least V.
– The Model
Maximize V
ST
Sentry's Meat
Advertising Produce
Emphasis Groceries
Bakery
IGA expected change in
market share
2X1
2X1
(-8X1)
6X1
X1
- 2X2
+6X2
- 4X2
+ X2
+ 2X3
- 7X3
+ X3
- 3X3
+ X3




The variables are probabilities
=
X1, X2, X3, are non negative: V is unrestricted
V
V
V
V
1
• Sentry’s Perspective - A Linear Programming
model
– Decision variables
•
•
•
•
Y1 = the probability that Sentry’s advertising focus is on meat.
Y2 = the probability that Sentry’s advertising focus is on produce.
Y3 = the probability that Sentry’s advertising focus is on groceries.
Y4 = the probability that Sentry’s advertising focus is on bakery.
– Objective function
• Minimize changes in market share in favor of IGA
– Constraints
• Sentry’s market share decrease for any given advertising focus
– The Model
Minimize V
ST
IGA's Meat
2Y1
Advertising Produce -2Y1
Emphasis Groceries 2Y1
Y1
Sentry's expected change in
market share
+ 2Y2
- 7Y2
+ Y2
+
+
+
8Y3
6Y3
Y3
Y3
+ 6Y4
- 4Y4
- 3Y4
+ Y4



=
Y1, Y2, Y3, Y4 are non negative; V is unrestricted
V
V
V
1
• Optimal solutions
– For IGA
• X1 = 0.3889; X2 = 0.5; X3 = 0.111
– For Sentry
• Y1 = 0.6; Y2 = 0.2; Y3 = 0.2; Y4 = 0
– For both players V =0 (a fair game).