Decision theory provides a rational methodology for decision-making under uncertainty. It involves identifying decision alternatives, possible future states of nature, and assigning payoffs for each alternative-state combination. Payoff and loss tables are used to evaluate the alternatives. Decision trees graphically display the decision process over time. Non-probabilistic decision rules like maximin (conservative) and maximax (risky) are used when probabilities are unknown, while the Bayes decision rule maximizes expected payoff when probabilities are known. In the example, the firm assessed probabilities for sales levels and the standard truck was chosen as it had the highest expected profit of $18.35.

1. Decision Theory
Decision Analysis – provides a rational methodology for decision making in the face of
uncertainty.
Components of a Decision Problem:
1. Decision Alternatives / Set of actions- the alternatives form which the decision maker is
to choose.
2. Events/ State of Nature – a list of possible events that might occur after the decision is
made.
Payoff Tables- a table which shows the reward obtained if a particular decision is made and the
event occurs.
Payoff Table
Decision States of Nature
Alternatives S1 S2 S3
a1 r11 r12 r13
a2 r21 r22 r23
a3 r31 r32 r33
a4 r41 r42 r43
Example 1:
Blockwood Inc. is a newly organized manufacturer of furniture products. The firm must
decide what type of trick to purchase for use in the company’s operations. Use truck is needed to
pick up raw material supplies, to make deliveries and to transport product samples to commercial
exhibits during the coming year. Three alternatives were identified by the firm:
(1) a small commercial import truck
(2) a standard size pickup
(3) a large flatbed truck
It is expected that sales in the 1st year will fall in one of four categories:
(1) 0-200,000 (low)
(2) 200,000 – 400,000 (moderately low)
(3) 400,000 – 600,000 (moderately high)
(4) Above 600,000 (high)
The payoff table for the firm would be:
Payoffs in ‘000 profits
Actions States of Nature
(truck type) (1) L (2)ML (3) MH (4) H
a1= Import 20 10 15 25
a2= Standard 15 25 12 20
a3= Flatbed -20 -5 30 40

2. Loss Tables – a table of opportunity cost corresponding to the losses incurred for not
choosing the action corresponding to the highest payoff.
Procedure:
For each of the states of nature identify the highest payoff. Then subtract each entry
in the column from the highest payoff.
Loss Table
Actions States of Nature
(truck type) L ML MH H
Import 0 15 15 15
Standard 5 0 18 20
Flatbed 40 30 0 0
Meaning of Losses:
If we purchase Import and Even 1 occurred, we have no opportunity cost or regret
since we have chosen the best out. If we had purchased standard, payoff is5 which is 5
less that the best, our opportunity cost would then be 5.
Decision Trees- a graphical method of expressing in chronological order the alternative actions
available to the decision maker and the possible states of nature.
Types of nodes in a Decision Tree:
1. Decision Node – a point in time in which the decision maker selects and alternatives
(represented by a rectangle).
2. Event Node- makes the occurrence of one of the possible states of nature after a decision
is made. (represented by a circle)
Represents an
event that can
occur at the
Decision event node
Node Branch
(represents a
course of Event
action taken)
Node

3. Decision Tree for Example 1:
0.2 20
ML
0.35 10
import
2 MH
0.3 15
H
0.15 25
L
0.2 15
ML
0.35 25
1 standard 3 MH
0.3 12
H
0.15 20
L
0.2 -20
ML
0.35 -5
flatbed 4
MH
0.3 30
H
0.15 40
Decision Making Under Uncertainty
I. Non Probabilistic Decision Rules- management does not have reasonable estimates
of the likelihood of the occurrence of various events.
A. Maximin Rule
For each decision alternative, identify the minimum payoff. Select the decision
alternative having the largest of the minimum payoffs.
Import 10 Therefore, Choose
Standard 12 Standard
Flatbed -20
Note: Maximin is a conservative or pessimistic decision rule. For each of the
alternative we assume most event and we maximize their pessimistic outcome.
B. Maximax Rule
Identify the maximin payoff for each decision alternative. Select the decision
alternative having the largest of the payoffs.
Import 25 Therefore, Choose
Standard 25 Flatbed
Flatbed 40
Note: This is a risky or optimistic decision rule. We assume that for each decision
alternative, the best event will occur. We maximize these optimistic outcomes.

4. C. Minimax Rule
Here, we work with the loss table. For each decision alternative, identify the
maximum possible loss. Select the decision alternative having the smallest of the
losses.
Import 15 Therefore, Choose
Standard 20 Import
Flatbed 40
Note: The rule is considered to be neither pessimistic nor optimistic.
II. Probabilistic Decision Rule- the decision maker is able to assign probabilities to the
various events that may occur.
Sources of Probabilities:
1. Sample Information- a study or research analysis of the environment is used to
assess the probability or occurrence of the event.
2. Historical Records- available from to files.
3. Subjective Probabilistic- probability may be subjectively assessed based on
judgment, sample information and historical records.
Example:
Suppose that the firm in Example 1 has assessed the probabilities for the 4 sales levels as:
P(1) = 0.20 P(3)= 0.30
P(2)= 0.35 P(4)= 0.15
What decision would be reached?
Tool:
Bayes Decision Rule – Select the decision alternative having the maximum expected
payoff (minimum expected loss)
Solution:
0.2 20
ML
0.35 10
import
2 MH
0.3 15
H
0.15 25
Therefore, purchase
the standard truck.
18.35 18.35 L
0.2 15
ML
0.35 25
1 standard 3 MH
0.3 12
H
0.15 20
9.25
L
0.2 -20
ML
0.35 -5
flatbed 4
MH
0.3 30
H
0.15 40

5. Expected Value of Perfect Information – The worth to the decision maker to have access to an
information source that would indicate for certain which of the events will occur.
Consider previous example…
If a perfect information source will reveal that eh following events will occur:
Event Choice Payoff Probability
1 a1 20 0.2
2 a2 25 0.35
3 a3 30 0.3
4 a4 40 0.15
Expected Profit using Perfect Information Source (EPPI) would be:
EPPI = 0.20(20) + 0.35(25) + 0.30(30) + 0.15(40)
= 27.75 or P27,750
Without such perfect information, the expected profit based on Bayes Rule is P18.35
Therefore the expected value of perfect information would be:
EVPI = 27.75 − 18.35 = 9.4 (or P 9,400)
The primary use of EVPI is to determine the maximum amount a decision maker should be
willing to pay for additional information (imperfect information) that could be employed to
refine further the probability estimates.
Sequential Decision Making- one in which the decision maker must initially select a decision
alternative; once the outcome following that decision is observed, an opportunity again
exists to select another decision alternative.
Example:
The city of Metropolis is planning to construct a street that will run through the city
perpendicular to the main East-West Street. The city planner have to make a choice between a
modern, wide (4-lane) street that would cost P2M or a lesser-quality, narrower street that would
cost P1M. We shall denote these 2 alternatives as W1 and N1. After 4 years, depending on
whether the traffic on the street turns out to be light or heavy( L1 or H1), the city will have the
option of widening the street., The probability of these traffic condition are estimated by city
planner and economists as P(L1)=0.25 And P(H1)=0.75. If W1 is selected, maintenance
expenses during the 1st 4 years will be P5,000 or P75,000 depending whether the traffic is light
or heavy. If N1 is selected, there costs are light or heavy. If N1 is selected, there costs are
expected to be P 30,000 and P150, 000 respectively. Suppose street W1 is built, then at the end
of 4 years, no further work is required. If heavy, either a minor or major repair must be made at
costs of P150, 000 and P200, 000 respectively. If street N1 is built, then at the end of 4 years, if

6. traffic has been light, either a minor or major repair must be made at costs of P50,000 or
P100,000 respectively. If traffic has been heavy, a major repair must be made at a cost of
P900,000.traffic during the next 6 years will be classified as light or heavy (L2 or H2). The
probabilities of these 2 events in years 1-4, are given as follows.
P(L2/L1) = 0.75 P(L2/H1) = 0.10
P(H2/L1) = 0.25 P(H2/H1) = 0.90
Maintenance Costs over year 5-10 will depend on which street was built in year 1, what type of
repair was made at the end of year 4, and the amount of traffic during years 5-10.
Street Repair Traffic Maintenance
Year 1 Year 5-10 Year 5-10
W1 None L2 200,000
H2 250,000
Minor L2 150,000
H2 175,000
Major L2 125,000
H2 100,000
N1 Minor L2 200,000
H2 250,000
Major L2 175,000
H2 150,000
a.) Construct a decision tree for this problem.
b.) Determine the optimal sequential strategy for the city of Metropolis.

7. Solution:
0.2125 0.75 0.20
L2
.005 6
H2
0.3375 L1 75 0.25 0.25
0. 25 1
0. 2
2
0.10 0.125
0
0.1025
.7
5
L2
0.2
H1 7 5
7
.3
H2
7
0.3025
30 r
0. ajo
0.
0.10
25
0.90
0
75
M
4
0.10 0.15
M 32 2
0.1725
0.
P2 1
in 5
M
W
L2
or
5
0.15
37
8
H2
2.3
0.90 0.175
1.975
1 Expected Value= Green
Cost In M= Blue
Probability = Red
1 .9
75
0.16875 0.75 0.175
L2
N1
1M
0.1 9
H2
0.15
26 r
0.25
0. ajo
5
87
0.2625
M
0.975 0.0 3
5
L1 5 .2
M
0.
25 92 0.2125 0.75
0.2
ino 2 5
3 0 .2 L2
6
r
0.05 10
H2
0.7
5
0.25 .25
1.2
H1 5
02
0.175
0 .1
0.1525 0.10
5
Major 0.90 L2
11
1.0525 H2
0.90 0.15
Year 1 – Build narrower street
Year 4- Minor Repair if traffic is light
Posterior Analysis – New information is obtained in order to refine the probability estimates of
the events, thus, it is hoped, leading to a better decision. The information obtainable may be
sample data, marketing research, data collected by electronic testing or surveillance or it may
involve purchasing the advice of an expert.
Prior Probabilities – Original probabilities of the various events. They exist prior to the use of
sample information.
Posterior Probabilities- revised probabilities calculated after the sample information is
obtained.

8. Example:
Suppose the fiorm in the truck eample acquires the services of a consulting firm, ABC Inc.
ABC will conduct a market study that will result in one of the 2 outcomes:
(1) O1 will be a favorable indicator of the market for the firm’s products.
(2) O2 will be an unfavorable indicator O1 and O2 are referred to as sample outcome.
The following conditional probabilities were arrived at from considerable ABC experience,
using historical market-research record in ABC’s files and the statistician’s judgment
P(Oj/Si) where:j=1 ,2 ; i=1,2,3,4
Sales
S1 S2 S3 S4
O1 0.05 0.30 0.70 0.90
O2 0.95 0.70 0.30 0.10
Solution:
Probability Tree
P (K1∩K2)
0.01
O1
5
2 0.0
O
0. 9 2
5
0.19
S1 .2
0 0.105
O1
3 0.3
S2 O
0.35 0. 7 2
0
1 0.245
S3
0.3 0.21
0 O1
0 .7
4
O2
S4 5
0.3
0. 1
0
0.09
0.135
O1
0
0.7
5 O
0. 3 2
0
0.015
P(O1) = 0.01 + 0.105 + 0.21 + 0.135
= 0.46
P(O2) = 0.54

9. Revised Probabilities
S 1 17
02
0.
S2
3
0.228 Sample Computation:
2
S3
0.45 P ( S1 ∩ O1)
P ( S1O1) =
65
0.2 S4
93
5 P (O1)
0.01
0. 1
O
46
=
0.46
=0.0217
1
O
0. 2
54
S 1 19
5
0. 3
S2
0.4537
3
S3
0.166
7
0. S4
02
78
Final Decision Tree:
16.902 L
0.0217 20
ML
0.2283 10
Import 4 MH
0.4565 15
H
0.2935 25
23.35 17.381 L
0.0217 15
ML
0.2283 25
Z1=favorable 5
2 standard
0.46 MH
0.4565 12
H
0.2935 20
23.35 L
0.0217 -20
ML
0.2283 -5
flatbed 6
MH
0.4565 30
H
0.2935 40
1
14.785 L
0.3519 20
ML
0.4537 10
import 7 MH
0.1667 15
H
0.0278 25
19.194 19.194 L
0.3519 15
ML
Z2=unfavorable 0.4537 25
0.54 3 standard 8 MH
0.1667 12
H
0.0278 20
-3.114
L
0.3519 -20
ML
0.4537 -5
flatbed 9
MH
0.1667 30
H
0.0278 40
Summary of Decision
Sample Outcome Action Profit
O1 Flatbed(a3) P23,859.5

10. O2 Standard (a2) P19,177.40
Expected Value of Sample Information (Preposterior Analysis)
- indicates whether it would pay us to purchase the sample information.
EVSI= Expected Payoff with - Expected Payoff without
Sample Information Sample Information
For the truck Example:
Expected Payoff with Sample Information = 0.46(23.8575) + 0.54(19.1774)
= 21.330246
EVSI = 21.330246 − 18.35
= 2.980246 or (P2,980.25)
Thus, we can hire the services of ABC ( for additional information) for as much as or less than
P2,980.25. If ABC chooses P1000, the expected net gain of sampling would be:
ENGS = 2,980.25 -1,000
= 1,980.25
Generally speaking, the sample information should be purchased if ENGS >0.