Decision Theory
Varsha Varde
Introduction
• Decision theory provides a rational
methodology for making management
decisions.
• It does not generate alternative courses of
action
• It merely provides a rational way of
choosing among several alternative
strategies.
Examples
• Natural Resource Development
-Should an oil or gas well be drilled
-What set of seismic experiment be run
-What is the expected payoff of the investment in
exploration
• Agricultural applications
-What crops should be planted
-Should excess acreage be planted
-What actions should be taken to fight pests
Examples
• Financial Applications
-What is the proper investment portfolio
-What capital investments should be made this
year
-Whether to grant or not to grant credit to a
customer
• Marketing Applications
-Which new product should be introduced
-What is the best distribution channel to use
-What is the best inventory strategy
Examples
• Production Applications
-Which of several different types of
machines should be purchased
-What maintenance schedule should be
used
-What mix of products should be produced
Assumptions
• The decision maker can define all decision alternatives
or strategies or acts which are being considered. The
decision maker has a control over choice of these
• He can define various states of nature or events for the
decision setting which are not under his control
-various economic conditions
-various decisions of competitors
-various weather conditions
• He can estimate quantitatively benefits or costs of any
decision alternative with various states of nature. These
are called payoffs.
• The problem is to choose the best of the alternatives to
optimise the pay-offs
Conditions Under which Decisions are made
• Decision making under conditions of
certainty- Decision maker is certain as to
which state of nature is going to occur
• Decision making under conditions of
uncertainty-No knowledge of the likelihood
of the occurrence of various states of nature
• Decision making under conditions of riskshas sufficient knowledge of the states of
nature to assign probabilities to their
occurrence
Conditions of Certainty
• Conditions of certainty are rare.
• Decision is easy under conditions of
certainty
Illustration
• A mineral water company has to make
selection from amongst three strategies
A , B and C
• The three states of nature for decision
setting are
S1 ,S2 and S3
• Benefits of each option are known
Illustration
Company has three strategy options:
A: Revolutionize product & high price
(Oxygen enriched, vitamin fortified mineral water)
B: Modify packaging & small price increase
(300 ml easy-to-slip-into-purse-or-pocket bottle)
C: Change design & marginal price hike
(Four colour attractive picture on the lable).
Illustration
Three possible states of nature are
S1: Huge increase in sales
S2: No change in sales
S3: Decline in sales
Estimated Yearly Net Profit
PAY_OFF MATRIX
Rs.
S1
S2
S3
A
7,00,000
3,00,000
1,50,000
B
5,00,000
4,50,000
0
C
3,00,000
3,00,000
3,00,000
Certainty
• Under conditions of certainty we have to
choose an alternative which gives us
maximum profit
• Solution
• Under S1 select option A (Rs 7,00,000)
• Under S2 select option B (Rs 4,50,000)
• Under S3 select option C (Rs 3,00,000)
Uncertainty
• States of nature known but probability
of their occurrence not known
• Selection depends on whether decision
maker is pessimistic or optimistic
MAXIMIN and MAXIMAX CRITERION
• Pessimistic decision maker first identifies lowest
profit with each decision alternative and chooses
that alternative which gives maximum of the
minimum profits. This criterion is called
MAXMIN criterion
• Optimistic decision maker first identifies highest
profit with each decision alternative and chooses
that alternative which gives maximum of the
maximum profits. This criterion is called
MAXIMAX criterion
Illustration of Maximin
Minimum Benefit from A : Rs. 1,50,000
Minimum Benefit from B : Rs. 0
Minimum Benefit from C : Rs. 3,00,000
The Maximum of the three minimum
benefits is Rs. 3,00,000 for C
Hence, Maximin criterion directs you to
select option # C: Change design &
marginal price hike.
Illustration of Maximax
Maxmum Benefit from A : Rs. 7,00,000
Maxmum Benefit from B : Rs. 5,00,000
Maxmum Benefit from C : Rs. 3,00,000
The Maximum of the three maximum
benefits is Rs. 7,00,000 for A
Hence, Maximax criterion directs you to
select option # A: Revolutionize product
& high price.
Hurwicz Criterion
Inventor: L. Hurwitz
Decision maker is neither optimistic nor
pessimistic
He specifies Index of optimism  which
lies between 0 and 1.
Weighted profits are calculated as
(maximum profit for alternative)+
(1-  )(minimum profit for alternative
Alternative which gives maximum of
weighted profits is the decision chosen
Hurwitz Criterion
Let  = 0.6
Weighted profits are calculated as
0.6(maximum profit for alternative)+
0.4(minimum profit for alternative)
Alternative which gives maximum of
weighted profits is the decision chosen
Illustration of Hurwitz
Weighted Benefit from A :
0.6( 7,00,000)+.4(1,50,000)= 4,80,000
Weighted Benefit from B :
0.6( 5,00,000)+.4(0)= 3,00,000
Weighted Benefit from C :
0.6( 3,00,000)+.4(3,00,000)= 3,00,000
The Maximum of the three weighted
benefits is Rs. 4,80,000 for A
Hence, Hurwitz criterion directs you to
select option # A: Revolutionize product
& high price.
Minimax Regret Criterion
Inventor: L. J. Savage
Assumption: You may regret your decision
afterwards (after-thought)
Hence, it is designed to select the option
that MINIMIZES the MAXIMUM regrets
Determine ‘maximum regrets’ that can
accrue from implementation of each option
Select the one for which it is lowest.
Minimax Regret Criterion
Regret is the opportunity loss or opportunity
cost
Loss incurred by not selecting the best
alternative
It is measured by the difference between the
maximum profit we would have realised in
case of known state of nature and the profit
we realize
Estimated Yearly Net Profit
Rs. (‘000)
S1
S2
S3
A
700
300
150
B
500
450
0
C
300
300
300
Regret Matrix
Rs. (‘000)
S1
S2
S3
A
700 – 700 =
0
450 – 300
=150
300 – 150 =
150
B
700 – 500 = 450 – 450 = 300 – 0
200
0
300
C
700 – 300 = 450 – 300 = 300 – 300 =
400
150
0
=
Regret Matrix ( * = Maximum)
Rs. (‘000)
S1
S2
S3
A
700 – 700 =
0
300 – 450
=150*
150 – 300 =
150*
B
500 – 700 = 450 – 450 = 0 – 300 =
200
0
300*
C
300 – 700 = 300 – 450 = 300 – 300 =
400*
150
0
Illustration of Minimax Criterion
Maximum Regret from A : Rs. 1,50,000
Maximum Regret from B : Rs. 3,00,000
Maximum Regret from C : Rs. 4,00,000
The Minimum of the three maximum
regrets is Rs. 1,50,000 for A
Hence, Savage’s Minimax Regret
criterion directs you to select option # A:
Revolutionize product & fix high price.
Laplace Criterion
First three criteria are based on the best or
worst outcome. They ignore the others.
Laplace Principle: ‘Don’t ignore any info.’
Assign equal probability to all possible
outcomes of each strategic option
Compute Expected Value of each option
Select the one for which EV is highest.
Estimated Yearly Net Profit
Rs. (‘000)
S1
S2
S3
A
700
300
150
B
500
450
0
C
300
300
300
Illustration of Laplace Criterion
Rs.
(‘000)
S1
S2
S3
EV
A
700
300
150
383.33
B
500
450
0
316.67
C
300
300
300
300.00
Decision Making Under Risk
• All possible states of nature are known
• Probabilities can be assigned to their
likelihood of occurrence
• Probabilities could be subjective based upon
decision maker’s feelings and experience or
• Probabilities could be objective based upon
collection and analysis of numerous data
related to states of nature
• Expected values are used to evaluate
decisions under uncertainty
• Alternative with Maximum EMV is selected
Illustration Under Risk
• Let P(S1)=0.5,
P(S2)=0.3 and P(S3)=0.2
• EMV(A)=.5x700+.3x300
+.2X150= 470
• EMV(B)=.5x500+.3x450+
.2x0=400
• EMV(C)=.5x300+.3x300+
.2x300=300
• Alternative A is selected
Rs.
(‘00
0)
A
S1
S2
S3
700
300
150
B
500
450
0
C
300
300
300
Expected Value Of Perfect Information
• Accurate and complete information about future is
known as perfect information.
• When perfect information is available (at
additional cost) the decision maker would select
that alternative which has maximum profit under
the known state of nature .This is known as
conditional profit
• The maximum possible expected profit is worked
out as weighted average of conditional profits with
weights as probabilities of various states of nature.
This is called Expected profit under certainty
• The expected value of perfect information is the
difference between the expected profit under
certainty and the best expected profit without
perfect information
Conditional Profit Table Under Certainty & EPVI
• Let P(S1)=0.5,
P(S2)=0.3 and P(S3)=0.2
UNDER PERFECT
INFORMATION
• P(S1)EMV(Decision/S1)=
.5x700= 350
• P(S2)EMV(Decision/S2)=
.3x450 =135
• P(S3)EMV(Decision/S3)=
.2x300 = 60
• EMV under certainty
=545
• EMV under risk=470
• EVPI=545-470=75
Rs.
(‘000
)
S1
A
700
B
C
S2
S3
450
0
300
Decision Tree Analysis
• Decision tree is a mathematical model of decision
situations
• It guides a manager to arrive at a decision in an
orderly fashion
• It contains decision nodes
from which one of
several alternatives may be chosen
• It contains state of nature nodes
out of which
one state of nature would occur
• The tree is constructed starting from left &
moving towards right
• Problem represented by a decision tree is solved
from right to left
Decision Tree
• Identify all decision alternatives & their order
• Identify chance events or states of nature that can occur
after each decision
• Develop a tree diagram showing the sequence of decisions
& states of nature.
• Obtain probability estimates of each state of nature
• Obtain esimates of the consequences of all possible
decisions & states of nature
• Calculate expected value of all possible decisions
• Select decision offering most attractive expected value
Illustration
• A company has to take a decision to either expand
by opening a new outlet or to maintain the current
status. In case the company decides to expand it will
earn an additional profit of Rs 30 lakh provided the
economy grows. However if the economy declines
the company will lose Rs 50 lakh. In case company
maintains status-quo it will neither gain or lose.
Draw a decision tree &state the best action under the
assumption of 70% chance of economic growth.
Also work out the action under 50% chance of
economic growth.
Process
Economic growth rises
0.7
Expand by opening new outlet
Economic growth declines
0.3
Expected outcome
Rs 30,00,000
Expected outcome
– Rs 50,00,000
Maintain current status
Rs 0
The circle denotes the point where different outcomes could occur. The estimates of the
probability and the knowledge of the expected outcome allow the firm to make a calculation of
A square
the pointit where
a decision is made, Here, they are contemplating
the likely
return. denotes
In this example
is:
opening
new
outlet.
Thenothing
uncertainty
is the state
the economy.
If the
economy
There
is also athe
option
to do
and maintain
theof
current
status quo!
This
would have
Economic
growth
rises:
0.7
x
Rs
30,00,000
=
Rs
21,00,000
continues
to
grow
healthily
the
option
is
estimated
to
yield
profits
of
Rs
30,00,000.
an outcome of Rs 0.
However, if it fails to grow as expected, the potential loss is estimated at Rs 50,00,000.
Economic growth declines:
0.3 x – Rs 50,00,000 = – Rs 15,00,000
Calculation suggests it is wise to go ahead with the decision: net ‘benefit’ of +Rs 6,00,000
Process
Economic growth rises
0.5
Expected outcome
Rs 30,00,000
Expand by opening new outlet
Economic growth declines
0.5
Expected outcome
– Rs 50,00,000
Maintain current status
Rs 0
Look what happens however if the probabilities change. If the firm is unsure of the potential for
growth, it might estimate it at 50:50. In this case the outcomes will be:
Economic growth rises:
0.5 x Rs 30,00,000 =
Rs 15,00,000
Economic growth declines:
0.5 x – Rs 50,00,000 = – Rs 25,00,000
In this instance, the net benefit is –Rs 10,00,000. The decision looks less favourable!
Marginal Analysis
•
•
•
•
•
•
•
•
At a particular activity level
Marginal Profit (MP): Additional profit generated by
increasing activity level by one unit
Marginal loss (ML) :Loss incurred by increasing activity
level by one unit and not profiting by it
Probability (P) of generating additional profit by increasing
activity level by one unit
Probability(1-P) of incurring loss by increasing activity
level by one unit
Expected (MP)= P x MP
Expected (ML)=(1-P) x ML
Optimum level of activity occurs when
Expected (MP)=Expected (ML)
P x MP=(1-P)xML thus optimum level of activity P* is
P*=ML/(ML+MP)
P* represents the minimum required probability to justify
increase in activity level by one unit
Marginal Analysis
• Let initial stock be x units. Increase it to (x+1) units
• There would either be profit MP with probability P
or loss ML with probability 1-P
• Then P=P(D>X) and 1-P=P(D≤ X)
• Optimum level of stock occurs when
Expected (MP)=Expected (ML)
P x MP=(1-P)xML
P*=ML/(ML+MP)
• P(D>X)= ML/(ML+MP)
Illustration
• Classic Burger Shoppe sells chicken
burgers. The cost of preparation comes to
Rs11 and selling price is Rs 18.Demand for
burger is normally distributed with mean
190 and SD 40.How many burgers should
shop prepare so as to reduce losses from
spoilage ?
• We work out minimum required probability
P* to justify preparation of an additional
burger
• MP=7 ,ML=11
P*=11/11+7=11/18= 0.61
• Let X* be the value of stock to be kept
• Then P(X>X*)=0.61 where X is N(190,40)
• From normal tables we find that X*=178.8
• Since MP is a decreasing function we round
it downwards to 178
• Therefore the shop should prepare 178
burgers to avoid losses due to spoilage