Decision Theory Lecture Notes.pdf

Decision theory as the name would imply is concerned with the process of making decisions. The extension to statistical decision theory includes decision making in the presence of statistical knowledge which provides some information where there is uncertainty. The elements of decision theory are quite logical and even perhaps intuitive. The classical approach to decision theory facilitates the use of sample information in making inferences about the unknown quantities. Other relevant information includes that of the possible consequences which is quantified by loss and the prior information which arises from statistical investigation. The use of Bayesian analysis in statistical decision theory is natural. Their unification provides a foundational framework for building and solving decision problems. The basic ideas of decision theory and of decision theoretic methods lend themselves to a variety of applications and computational and analytic advances.

1. Decision Theory
Lecture Notes for B.Com/M.Com Students

Dr. Tushar Bhatt
5/21/2022

2. Decision Theory


2
Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

5.1: Introduction
Decision theory as the name would imply is concerned with the
process of making decisions. The extension to statistical decision
theory includes decision making in the presence of statistical
knowledge which provides some information where there is
uncertainty. The elements of decision theory are quite logical and
even perhaps intuitive. The classical approach to decision theory
facilitates the use of sample information in making inferences about
the unknown quantities. Other relevant information includes that of
the possible consequences which is quantified by loss and the prior
information which arises from statistical investigation. The use of
Bayesian analysis in statistical decision theory is natural. Their
unification provides a foundational framework for building and
solving decision problems. The basic ideas of decision theory and of
decision theoretic methods lend themselves to a var iety of
applications and computational and analytic advances.
A principle of decision making can be useful in deciding the best act
from the given alternatives. In decision processes making generally
the following steps are involved:
1.
Different alternatives or acts available for the problem.
2.
Proper understanding of the problem about which decision is
to be taken.
3.
The estimates of gain or loss of different alternatives.
4.
The choice of proper alternative from available alternatives.
5.
Implementation of the selected act.
5.2. Terms which are useful in the process of decision making
- The following terms are used throughout the chapter as defined
below:

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

(a) Act
- If a manufacture has to decide about the production of any one of
the o items x, y, z. then these items are different acts or strategies.
Similarly monsoon, before rain starts a farmer has to decide about
the crop to be en the different crops are different alternatives or acts
for him. He has decided any one of the crops from rice, groundnut,
cotton, bajari etc. Thus a farmer these are different acts and he has
to select the best act which has him the maximum gain.
(b) States of nature or Events
- The selection of any act depends upon the external conditions
which are not in the control of decision maker. These states of
nature depend upon as mal factors. In monsoon there may be heavy
rain or moderate rain or rain. These are different states of nature.
These states of nature are not in the control of the farmer. From
past experience or by some guess he on at the most estimate
probabilities of different states of nature.
- Similarly in deciding about the production of anyone of the three
items be manufacturer has to estimate the demand condition of the
market. The demand may be heavy or moderate or very less.
- From market research he has to estimate the probabilities of these
conditions. The different conditions of demand are different states
of nature. He does not have perfect information about this state’s of
nature. However by his experience or by market survey by guess
work he estimates the probabilities of different states of nature.
(c) Pay off matrix
- The monetary gain or loss from the combination of state of nature
and act is known as its Pay off, and the arrangement of different
pay offs from different states of nature and acts is known as a pay

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Off matrix. A pay off matrix can be understood from the following
illustration.
The estimates of the profit per acre which can be obtained from the
crops rice, cotton and bajari under different conditions are given in
the following table:
Crop(Act)
Event or State of nature Rice Cotton Bajari
Heavy rain 1500 1200 600
Moderate rain 1000 1500 1000
Less rain 300 500 1200
This matrix is known as a pay off matrix.
(d) Estimates of probabilities of different states of nature
- The pay offs of different crops are dependent on different states of
nature as seen in the above table. If it rains more, the crop of rice
may be advantageous. But before monsoon it is not p ossible to
decide whether will rain heavily or not. From past experience or by
any other suitable method we can at the most make p redictions
about rain and probabilities for different states.
- The decision maker makes estimates of the probabi lities of
different states of nature from his past experience or from the
experience of others or by some scientific method or intuitionally.
Thus the probabilities of different states of nature are only the
estimates.
5.3. Types of Decision Making Environment
- In all decision making problems the decision maker has to decide
the course of action out of number of available alternatives. There
are two type of situations in which a decision is required to be
taken:

5. Decision Theory


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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

• Decision making under certainty
- When the result of each alternative can be easily known, the
decision maker has to decide the best action that gives maximum
pay-off. To understand this let us take a simple example. Suppose a
manufacturer of buckets can produce and sell a cheaper variety of
buckets at Rs. 20 per bucket and earns Rs. 2 per bucket. He can
sell 1000 buckets per day. On the other hand if he produces a
superior quality of buckets and sells at Rs. 40 each, he can earn Rs.
4 per bucket. But he can sell only 600 buckets of superior quality.
We have to decide about the type of the buckets the manufacturer
should produce.
For cheaper quality of buckets he can earn 1000 × 2 = 2000 rupees
per day.
For superior quality of buckets he can earn 600 × 4 = 2400 rupees
per day. As the profit on the superior quality is more he should
produce buckets of superior quality.
• Decision making under uncertainty
-
When the results of different courses of actions are not known
with certainty the process of taking a decision is relatively difficult.
The problems decision making under uncertainty can be studied
under two heads.

(1)
Decision making under uncertainty when probabiliti es of
different states of nature are not known.
- There are many methods of decision making when the
probabilities of differences states of nature are not known. The
commonly used criteria are:
(i) Maxi-min principle (ii) Maxi-max principle (iii) Horwich principle
(iv) Laplace principle (v) Mini-max regrets.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

(i) Maxi-min principle:
According to this principle the minimum pay off of each act is
obtained from the pay off matrix, and the maximum o f these
minimum values is decided. The act corresponding to this
maximum value is regarded as the best act according to maxi-min
principle. This is a pessimistic approach towards the selection of
the act. The following example will illustrate the Maxi-min principle.
Events

Act





i
10 2 -8 8
i
-6 14 16 -4
i
18 8 0 10
i
15 8 -6 10
Minimum
Pay off of
the act
-6 2 -8 -4

In the last row minimum pay offs of different acts are shown. The
maximum pay off among these minimum pay-offs is 2 wh ich
corresponds to act
. Hence according to maxi-min principle act A₂
should be selected.
(ii) Maxi-max principle:
- According to this principle the maximum pay-offs of different acts
are obtained. From these values the maximum value is decided. The
act against this maximum value is the best act according to Maxi-
max principle. This approach is an optimistic approach towards the
selection of the act.

7. Decision Theory


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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Events

Act





i
10 2 -8 8
i
-6 14 16 -4
i
18 8 0 10
i
15 8 -6 10
Maximum
Pay off of
the act
18 14 16 10

In the above illustration the maximum pay-offs the highest pay-off
is 18, which corresponds to the act
. Hence act
should be
selected according to Maxi-max principle.
(iii) Horwich principle
Horwich principle can be interpreted as a compromise between the
optimistic approach of Maxi-max principle and pessi mistic
approach of Maxi-min principle.
According to this principle co-efficient of optimism is selected and it
is multiplied by the maximum value of an act.
The minimum value of that act is multiplied by. T e.
For all the acts the sum of products are similarly found out. The act
giving the maximum sum of products is selected as t he best act
according to Horwich principle. i.e. For any act
.e.mswhaflf.pwk T g(()+m T e)mfafaflf.pwk T g(().
The coefficient e is in between 0 and 1.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

E.g. Consider the following data:
Events

Act





i
10 2 -8 8
i
-6 14 16 -4
i
18 8 0 10
i
15 8 -6 10

Here we are taking e I 5bS
For act
: m0.7)× 18 +m1 − 0.7)×m−6)= 12.6 − 1.8 = 10.8
For act
: m0.7)× 14 +m1 − 0.7)× 2 = 9.8 + 0.6 = 10.4
For act
: m0.7)× 16 +m1 − 0.7)×m−8)= 11.2 − 2.4 = 8.8
For act
: m0.7)× 10 +m1 − 0.7)×m−4)= 7 − 1.2 = 5.8
The highest value is for act
hence according to Horwich principle
act
should be selected.
(iv) Laplace - principle:
- In this method equal probabilities are assigned to different states
of nature. In other words the average pay-offs of different acts are
found out. The act for which the average pay-off is maximum,
selected as the best act according to this principle.
Let us consider the following data, for better understanding of
Laplace – principle.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Events

Act





i
10 2 -8 8
i
-6 14 16 -4
i
18 8 0 10
i
15 8 -6 10
Total 37 32 2 24

For act
the average pay-off =
$%m&')%(%)

=
*

= 9.25
For act
the average pay-off =
%%(%(

=


= 8
For act
the average pay-off =
m3M)xoDxyxm3D)

=


= 0.5
For act
the average pay-off =
(%m&)%$%$

=


= 6
The maximum average pay-off is for act
and hence according to
Laplace principle act
should be selected.
(v) Mini-max Regret:
- In this criterion a regret table or opportunity loss table is
prepared. For constructing an opportunity loss table from the given
pay off table, the pay off of each act for a state of nature is
subtracted from the highest pay off of that state of nature. Similarly
values are obtained for each state of nature for different acts. It is
obvious that no value in the regret table will be negative. After
preparing regret table, the maximum regret or opportunity loss of
each act is found out. The act with minimum of all m aximum

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

opportunity losses is the best act according to Mini-max regret
criterion.
E.g. Consider the following data for the creation of Mini-max regret
table.

Events

Act





t
10 2 -8 8
t
-6 14 16 -4
t
18 8 0 10
t
15 8 -6 10



Events

Act





t
10-10=0 2-10=-8 -8-10=-18 8-10=-2
t
-6-16=-22 14-16=-2 16-16=0 -4-16=-20
t
18-18=0 8-18=-10 0-18=-18 10-18=-8
t
15-15=0 8-15=-7 -6-15=-21 10-15=-5

Remember no values in the regret table are negative . Therefore
consider all as a positive. Hence the above table can be shows as
follows:

11. Decision Theory


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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Events

Act





i
0 8 18 2
i
22 2 0 20
i
0 10 18 8
i
0 7 21 5
Maximum
regret
22 10 21 20
Among all maximum regrets the minimum is 10 for act
. Hence
according to Mini-max regret criterion
is the best act.
Ex – 1: Construct the opportunity loss table and obtain the best act
using Mini-max regret criterion, from the following pay- off table.
State
of
Nature
Act




i
18 15 13
i
12 14 16
i
14 11 17
i
9 11 16
Solu.: Now first we construct the opportunity loss table.
State of Nature
Act




i
|18−18|=0 |15−18|=3 |13−18|=5
i
|12−16|=4 |14−16|=2 |16−16|=0
i
|14−17|=3 |11−17|=6 |17−17|=0
i
|9−16|=7 |11−16|=5 |16−16|=0
Maximum Opportunity Loss 7 6 5

12. Decision Theory


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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Minimum of maximum opportunity losses is 5 for act
. Hence
according to Mini-max regret criterion
is the best act.
Ex – 2: For the following pay – off matrix find the best act using (i)
Maxi-min principle (ii) Maxi – max principle (iii) Laplace principle
(iv) Horwich principle.m–VC.e I 5bS).
Event

Act





)
i
10 25 10 15 20
i
-5 10 -5 10 -5
i
15 5 10 10 10

Solu.: Now first we construct the pay-off table.
Event
Act





)
i
10 25 10 15 20
i
−5 10 −5 10 −5
i
15 5 10 10 10
Minimum pay-off of act −5 5 −5 10 −5
Maximum pay-off of act 15 25 10 15 20

(I)
Maxi – min principle: From all the minimum pay-offs
has
maximum pay-off 10. So
is the best act according to Maxi-min
principle.

(II)
Maxi-max principle: Among all the maximum pay-offs, the pay
off for act
is the highest i.e. 25. So according to Maxi-max
principle
is the best act.

13. Decision Theory


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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

(III) Laplace principle: We shall find average pay offs for all the acts.

For
, the average pay-off =
$&)%)

=
$

= 6.67
For
, the average pay-off =
)%$%)

=
$

= 13.33
For
, the average pay-off =
$&)%$

=
)

= 5
For
, the average pay-off =
)%$%$

=
)

= 11.67
For
), the average pay-off =
$&)%$

=
)

= 8.33

(IV)
Horwich Principle: For each act we shall calculate

f cB.fkg.T.cxfv u qbbOB dD u lO iB.gCg.T.cxfv u qbbO
For
, the average pay-off= 0.7 × 15 +B1 − 0.7O×B−5O= 9
For
, the average pay-off= 0.7 × 25 +B1 − 0.7O×B5O= 19
For
, the average pay-off= 0.7 × 10 +B1 − 0.7O×B−5O= 5.5
For
, the average pay-off= 0.7 × 15 +B1 − 0.7O×B10O= 13.5
For
), the average pay-off= 0.7 × 20 +B1 − 0.7O×B−5O= 12.5
From above all values the value of
is maximum. Hence
is the
best act according to Horwich Principle.

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

(2) Decision making under uncertainty when probabilitie s of
different states of nature are known.
Sometimes it is possible to estimate the probabilities of different
states of nature. Using the probabilities, the best act can be
obtained by any one of the following criterion.

(I)
Expected Monetary Value (EMV) Criterion
This is a very commonly used method of decision mak ing. In this
method probabilities are assigned for different states of nature. This
can be done by using past experience or by intuition.

A pay-off matrix is obtained for different acts and different states of
nature. The expected monetary value for each act is then found out.

Expected monetary value of any act is the sum of products of pay-
offs and the corresponding probabilities of the states of nature.

Hence 4s2 I 34
56∙ 8
6
where 8
6I 89g:w:a;aCk.g(.<
=>
.?.0/,∑8
6I .wOA.
4
56I 8wk T g((.g(.a
=>
.wBC.wOA.<
=>
.?.0/.
Thus EMV for each act is calculated and that act is selected for
which the EMV is maximum.

Ex – 1: Using the following pay-off matrix, decide the best act
according to EMV criterion.

State of Nature
Probability
Act




i
0.2 125 -20 -100
i
0.5 300 500 400
i
0.3 600 700 800

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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Solu.: For each act we shall multiply pay-off with the corresponding
probability of state of nature and shall find their sum. This gives
EMV for that act. The EMV for different act are found and the act
giving the maximum EMV regarded as the best act.


State of Nature Probability
Act

h8
h8
h8
i
0.2 125 25 -20 -4 -100 -20
i
0.5 300 150 500 250 400 200
i
0.3 600 180 700 210 800 240
EMV 355 456 420

Here EMV for act
is maximum and hence
is the best act
according to this principle.

(II)
Expected Opportunity Loss (EOL) Criterion.
For a state of nature the opportunity loss of different acts can be
found out by subtracting the pay-offs of the acts from the maximum
pay-off of that state of nature.
The expected opportunity loss is defined as
14- = ∑-
56∙ 8
6
Where 8
6I 89g:w:a;aCk.g(.<
=>
.?.0/,∑8
6I .wOA
-
56I 4ppg9ClOaCk.;gCC.g(.a
=>
.wBCagO.(g9.CDV.<
=>
.CCwCV.g(.OwCl9V.
The act with minimum expected opportunity loss is the best act.

16. Decision Theory


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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

Ex – 1: Find the best act using EOL criterion from the following
data.
Event
Act Event
Probability




i
4 1 -2 -3 0.2
i
11 4 19 29 0.5
i
19 23 39 34 0.3
Solu.: Now we shall construct opportunity loss table first.
Step-1: Find the maximum value from each row (Act) and subtract
the same into corresponding row and consider only th e positive
difference and denote it into ;
,…,;
respectively.
Event

Act Event
Probability




i
4 1 -2 -3 0.2
i
11 4 19 29 0.5
i
19 23 39 34 0.3

Event
Act Event
Probability
;

;

;

;

i
4 0 5 3 -2 6 -3 7 0.2
i
11 18 4 25 19 10 29 0 0.5
i
19 20 23 16 39 0 34 5 0.3

Step – 2: Multiply ;
56 with 8
6 and obtain ;
568
6 for each event.
Event

Act Event
Probability
;
;
6 8
6
;
;
6 8
6
;
;
6 8
6
;
;
6 8
6
i
4 0 0 5 3 0.6 -2 6 1.2 -3 7 1.4 0.2
i
11 18 9 4 25 12.5 19 10 5 29 0 0 0.5
i
19 20 6 23 16 4.8 39 0 0 34 5 1.5 0.3
EOL 15 17.9 6.2 2.9
The expected opportunity loss for act
is minimum. Hence act

is the best according to EOL.

17. Decision Theory


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Dr. Tushar J. Bhatt, Assistant Professor in Mathematics, Atmiya University, Rajkot.

5.4 Decision Tree
It is a diagram through which the problem of decision making can
be represented. The decision tree is constructed starting from left to
right. The square nodes □ denote the points at which strategies are
considered and the decision is made. The circle ○ denote the chance
nodes and the various states of nature emerge from these nodes.
The pay-off of each branches are shown at the termin al of the
branch.
Ex – 1: Represent the following problem by decision tree and
decide the best act from minimum cost.
State of Nature
Probability of
fire
To take
insurance
Not to take
insurance
Fire during a
year
0.01 -100 - 8000
No fire during a
year
0.99 -100 0









As EMV -80 is more for the act of not taking assurance, that act
should be selected.
=-100
4s2 = 0.99∗m0)=0
4s2 = 0.01∗m−8000)=−80
4s2 = 0.01∗m−100)=−1
4s2 = 0.99∗m−100)=−99
No Fire
Fire
No Fire
Fire
To take insurance
Not to take insurance
1
2 =-80