DECISION
THEORY
Decision Tree Analysis
– A decision tree is a physical representation of a decision
situation.
– It provides an overview of the total process, thereby
helping the decision maker examine possible outcomes.
– In a decision tree, let a rectangle represent a decision
point or a place where a choice must be made; while a
circle represents a chance events or expected value.
Example 1
Consider the problem of a student who has to decide whether to stop
his studies and work for a job paying ₱ 1,500 monthly or continue his
studies, after which a job awaits him paying ₱ 4,000 a month,
provided he passes his remaining subjects. He feels that the
probability that he will pass his remaining subjects is 40%. Which
choice is better for the student?
₱ 4,000
₱ 1600
₱0
₱1,600
₱ 1,500
Example 2
A manager has to decide whether to prepare a bid or
not. It costs ₱ 5000 to prepare the bid. If the bid is
submitted, the probability that the contract will be
awarded is 50%. If the company is awarded the contract,
it may earn an income of ₱ 100,000 if it succeeds, or pay
a fine of ₱ 8,000 if it fails. The probability of success is
estimated to be 80%. Should the manager prepare the
bid?
Example 3
Assume P:
(no outbreak)=P1=95%
(small outbreak)=P2=4.5%
(pandemic)=P3=0.5%
Example 4
Mr. Reyes, the president of the CPA Corporation is faced with deciding whether to
purchase a patent to develop a new product or not. If the company purchases the
patent, it should develop the product. The selling price of the patent is ₱50,000.
There are two ways of developing the product: Modern Method and Traditional
Method.
It costs ₱20,000 to use the Modern Method and ₱15,000 for the Traditional
Method. The probability of success in the Modern Method is 60% while it is 70% for
the Traditional Method.
If the product is successfully developed, it will give an income of ₱500,000.
Should the company purchase the patent? If so, what method of development
should be used?
Expected Monetary Value
– Is how much money you can expect to make from
a certain decision.
– In statistics, it is the sum of all possible values of a
random variable, or any given function of it,
multiplied by the respective probabilities of the
values of the variable.
– EMV = probability * impact
Example 1
Katniss identifies an opportunity with a 40% chance of
happening. However, if this positive risk occurs it may help her gain ₱
2,000. Calculate the EMV for this risk event.
Given:
Probability of risk = 40%
Impact of Risk = ₱ 2,000
EMV = probability * impact
= 0.4 * ₱ 2,000
= ₱ 800
Example 2
Ellise had identified two risks with a 20% and 15% chance of occurring. If
both of these risks occur it will cost her ₱ 1000 and ₱ 2000 respectively. What is the
expected monetary value of these risks events?
– EMV of two risk events = EMV of the first event + EMV of the second event
– EMV of the first event = 0.20 * (-1000)
= -200
– EMV of the second event = 0.15 * (-2000)
– = -300
The EMV of these two risks events = (-200) + (-300)
= -500
Example 3
– Suppose you are leading a construction project. Weather, cost of construction
material, and labor turmoil are key project risks found in most construction
projects:
– Project Risks 1 - Weather: There is a 25 percent chance of excessive snow fall
that’ll delay the construction for two weeks which will, in turn, cost the project
$80,000.
– Project Risks 2 - Cost of Construction Material: There is a 10 percent probability
of the price of construction material dropping, which will save the project
$100,000.
– Project Risks 3 - Labor Turmoil: There is a 5 percent probability of construction
coming to a halt if the workers go on strike. The impact would lead to a loss of
$150,000. Consider your industry and geographic area to determine whether this
risk would have a higher probability.
The Expected Monetary Value for the project risks:
– Weather: 25/100 * (-$80,000) = - $ 20,000
– Cost of Construction Material: 10/100 * ($100,000) = $ 10,000
– Labor Turmoil: 5/100 * (-$150,000) = - $7,500
The project’s Expected Monetary Value based on these project risks is:
-($20,000) + ($10,000) – ($7,500) = - $17,500
Example 4
CONDITIONAL PROFIT
Aling Lumen owns a jeepney which she uses to
transport lanzones from Laguna which she buys on a
wholesale basis at ₱4.50 per kilo. She buys and sells
the fruit in the multiple of five number of kilos, and
distributes them to sidewalk vendors at ₱12.00 per
kilo. The maximum number of kilos she can sell in one
day is 120 kilos and the maximum is 100. It has been
observed that in the past 100 days, 100 kilos were
demanded on 20 days, 105 on 30 days, 110 on 20 days,
115 in 10 days and 120 in 20 days. Her son studying
business course advises her to use decision analysis to
determine the right number of kilos to purchase daily.
The Conditional Profit Table of Aling Lumen
STOCK
Demand
Probability
100
105
110
115
120
100
20
100
₱750.00
727.50
705.00
682.50
660.00
105
30
100
750.00
787.50
765.00
742.50
720.00
110
20
100
750.00
787.50
825.00
802.50
780.00
115
10
100
750.00
787.50
825.00
862.50
840.00
120
20
100
750.00
787.50
825.00
862.50
900.00
Expected Monetary Value of Aling Lumen
(probability times the conditional profit)
Possible Stock
Demand
Probability
100
105
110
115
120
100
.2
₱150.00
145.50
141.00
136.50
132.00
105
.3
225.00
236.25
229.50
222.75
216.00
110
.2
150.00
157.50
165.00
160.50
156.00
115
.1
75.00
78.75
82.50
86.25
84.00
120
.2
150.00
157.50
165.00
172.50
180.00
Expected Value
₱750.00
775.50
783.00
778.50
768.00
Above the table shows ₱783 as the highest expected value corresponding to
stock of 110 kilos.
Decision for Aling Lumen: Buy 110 kilos daily
Conditional Profit of Aling Lumen with Salvage Value
Suppose that the extra number of kilos of lanzones
left cannot be sold by Aling Lumen the next day, since the
fruit will not be fit for human consumption. Suppose also
that a factory manufacturing insecticide and fertilizer buys
overripe lanzones at ₱2.00 per kilo. If Aling Lumen can sell
her left-over lanzones at the end of the day at a salvage
value of ₱2.00 per kilo, construct the conditional profit table
of Aling Lumen with salvage value.
Computation of Profit with Salvage Value
If Aling Lumen has a stock of 120 kilos, but the demand for the day is
100 kilos only, then 20 kilos will be sold at a salvage value of ₱2.00 per
kilo which should be added to her profit without the salvage value.
Possible Stock
Demand
Probability
100
105
110
115
120
100
.2
₱ 750.00
737.50
725.00
712.50
700.00
105
.3
750.00
787.50
775.00
762.50
750.00
110
.2
750.00
787.50
825.00
812.50
800.00
115
.1
750.00
787.50
825.00
862.50
850.00
120
.2
750.00
787.50
825.00
862.50
900.00
EV Table for Aling Lumen with Salvage
(probability times conditional profit with salvage)
Possible Stock
Demand
Probability
100
105
110
115
120
100
.2
₱ 150.00
147.50
145.00
142.50
140.00
105
.3
225.00
236.25
232.50
228.75
225.00
110
.2
150.00
157.50
165.00
162.50
160.00
115
.1
75.00
78.75
82.50
86.25
85.00
120
.2
150.00
157.50
165.00
175.50
180.00
₱ 750.00
777.50
790.00
792.50
790.00
– The preceding table shows that the highest expected value with
salvage is ₱ 792.50, that is if Aling Lumen has stock of 115 kilos.
Therefore, the decision of Aling Lumen if there is a salvage value is
to have a stock of 115 kilos.
– Note that her highest expected profit with salvage is higher by
₱ 9.50 than that without salvage.
(₱ 792.50 - ₱ 783.00 = ₱ 9.50)
Bayes’ Theorem
– Named after 18th Century British mathematician Thomas
Bayes, is a mathematical formula for determining
conditional probability.
– It is a theorem about conditional probabilities: the
probability that an event A occurs given that another
event B occurred is equal to the probability that event B
occurs given that A has already occurred multiplied by the
probability of occurrence of event A and divided by the
probability of event B.
If, P(A ∩ B) = P(A) x P(B) = P(B) x P(A|B)
then, P(A|B) = [P(A) x P(B|A)] / P(B)
Where P(A) and P(B) are the probabilities of A
and B without regard to each other.
P(B|A) is the probability that B will occur given
A is true.
P(A|B) is the conditional probability of A
occurring given B is true.
Example 1
– Ella is interested in finding out a patient’s probability of having liver
disease if they are an alcoholic. “Being an alcoholic” is the test (kind
of like a litmus test) for liver disease.
– A could mean the event “Patient has liver disease.” Past data tells that
10% of patients entering the clinic have liver disease. P(A) = 0.10.
– B could mean the litmus test that “Patient is an alcoholic.” Five
percent of the clinic’s patients are alcoholics. P(B) = 0.05.
– Among those patients diagnosed with liver disease, 7% are alcoholics.
This is our B|A: the probability that a patient is alcoholic, given that
they have liver disease, is 7%.
Given:
P(A) = 0.10
P(B) = 0.05
P(B|A) = 0.07
Bayes’ theorem tells that:
P(A|B) = [P(A) x P(B|A)] / P(B)
P(A|B) = (0.10 x 0.07) / 0.05 = 0.14
Example 2
In a particular pain clinic, 10% of patients are
prescribed narcotic pain killers. Overall, five
percent of the clinic’s patients are addicted to
narcotics (including pain killers and illegal
substances). Out of all the people prescribed
pain pills, 8% are addicts. If a patient is an addict,
what is the probability that they will be
prescribed pain pills?
– Given:
– P(A) = 0.1
– P(B) = 0.05
– P(B|A) = 0.08
P(A|B) = P(B|A) * P(A) / P(B) = (0.08 * 0.1)/0.05 = 0.16
The probability of a person being prescribed by pain pills given
he is an addict is 0.16 (16%).
𝑃(𝐸|𝐻)
P(H|E) =
(PH)
𝑃(𝐸)
– P(H|E) is called the posterior probability
– P(E|H)/P(E) is called the likelihood ratio
– P(H) is called the prior probability
Bayes’ theorem can be rephrased as “posterior probability
= likelihood ratio * prior probability”
If a single card is drawn from a standard deck of playing cards.
Using the Bayes’ theorem what is the probability that the card
is a king given it is a face card.
P(King|Face) =
𝑃(𝐹𝑎𝑐𝑒|𝐾𝑖𝑛𝑔)
𝑃(𝐹𝑎𝑐𝑒)
P(Face|King) = 1
P(Face) =
12
52
P(King) =
4
52
𝑜𝑟
or
P(King|Face) =
=
1
3
3
13
1
13
1
3/13
x
1
13
P(King)
Expected Value of Perfect
Information
– An amount equal or less than the expected cost of
uncertainty the decision maker is willing to pay.
– It is defined as the expected profit with perfect
information minus the expected profit under
uncertainty.
Payoff table for Business Sale
(Figures in thousands of dollars)
Events
E1 Contract Awarded
E2 Contract not Awarded
PROBABILITIES
A1
SELL NOW
A2
HOLD A YEAR
.20
.80
80
80
100
70
The calculation of the expected monetary values of the
actions confronted by the business owner who wants to
sell her business are performed in the same way.
The EMV’s of the actions are:
EMV1 = $80(.20) + $80(.80) = $16 + $64 = $80
EMV2 = $100(.20) + $70(.80) = $20 + $56 = $76
It appears that the best decision based on EMV is to sell
the business now.
– In the previous example, the EVPI for the business owner facing the decision
about selling her business was calculated to be $4000.
Best Payoffs under Certainty for Business Sale
EVENT KNOW TO BE IMMINENT
E1 Contract Awarded
E2 Contract not Awarded
BEST ACTION
PAYOFF FOR BEST ACTION
A2 Hold
A1 Sell
100
80
EPUC = 100(.20) + 80(.80) = 20 + 64 = 84
The EMV was found to be $80,000 so the expected value of perfect information is
found to be
EVPI = |84 – 80| = 4
Mr. Reyes, the president of the CPA Corporation is faced with deciding whether to
purchase a patent to develop a new product or not. If the company purchases the
patent, it should develop the product. The selling price of the patent is ₱50,000.
There are two ways of developing the product: Modern Method and Traditional
Method.
It costs ₱20,000 to use the Modern Method and ₱15,000 for the Traditional
Method. The probability of success in the Modern Method is 60% while it is 70% for
the Traditional Method.
If the product is successfully developed, it will give an income of ₱500,000.
Should the company purchase the patent? If so, what method of development
should be used?
Profit with Perfect Information
Suppose that there is a man who volunteers to give Aling Lumen information on
the actual orders of the sidewalk vendors, what is the right amount Aling Lumen
should give him daily?
If Aling Lumen had perfect information about the demand of lanzones for the day,
she would always purchase exactly the right amount. Thus if the demand for the
day is 110 kilos, Aling Lumen would purchase only 110 kilos.
Demand
Probability
Maximum Profit
Expected Value
100
.2
₱ 750.00
₱ 150.00
105
.3
787.50
236.25
110
.2
825.00
165.00
115
.1
862.50
86.25
120
.2
900.00
180.00
Expected Value with Perfect Information
To find the value of perfect information:
₱ 817.50 - ₱ 792.50 = ₱ 25.00
Aling Lumen should give the man a maximum of ₱ 25.00
₱ 817.50
STOCK
Demand
Probability
100
105
110
115
120
100
20
100
₱ 750.00
727.50
705.00
682.50
660.00
105
30
100
750.00
787.50
765.00
742.50
720.00
110
20
100
750.00
787.50
825.00
802.50
780.00
115
10
100
750.00
787.50
825.00
862.50
840.00
120
20
100
750.00
787.50
825.00
862.50
900.00
EV Table for Aling Lumen with Salvage
(probability times conditional profit with salvage)
Possible Stock
Demand
Probability
100
105
110
115
120
100
.2
₱ 150.00
147.50
145.00
142.50
140.00
105
.3
225.00
236.25
232.50
228.75
225.00
110
.2
150.00
157.50
165.00
162.50
160.00
115
.1
75.00
78.75
82.50
86.25
85.00
120
.2
150.00
157.50
165.00
175.50
180.00
₱ 750.00
777.50
790.00
792.50
790.00
Sensitivity Analysis
– Is a technique used to determine how different values of
an independent variable impact a particular dependent
variable under a given set of assumptions.
– Also referred to as what-if or simulation analysis, is a way
to predict the outcome of a decision given a certain range
of variables.
– Determining the range of probability for which an
alternative has the best expected payoff.
Expected Utility Theory
– a branch of decision theory that deals with the issue of
measuring the “value” of payoffs to decision makers
Utility is a measure of relative satisfaction. We can plot a graph of amount of money spent vs.
“utility” on a 0 to 100 scale. Typical shapes for different types of risk takers generally follow
the patterns shown below.
U(100)
U(100)
U(0) 0
100
U(0) 0
Risk seeker
100
U(0) 0
Risk averse
U(100)
U(0) 0
U(100)
Risk neutral
U(100)
100
U(0) 0
100
U(100)
100
U(0) 0
100
Graphs above show that to achieve 50% utility, risk seekers will pay
maximum, risk averse will pay minimum and risk neutral will pay an
average amount.
Imagine for a moment that a gambler offered two alternatives. He would pay you $10,000 for
sure or he would allow you to engage in this game:
You toss a coin; if it comes up heads you win $200,000, and if it comes up tails you must pay
$160,000.
Payoff Table for Gamble
Events
E1 Heads
E2 Tails
Probabilities
A1
Sure Thing
A2
Toss Coin
.5
.5
10,000
10,000
200,000
-160,000
EMV1 = 10,000(.5) + 10,000(.5)
= 5,000 + 5,000 = 10,000
EMV2 = 200,000(.5) +(-160, 000(.5))
= 100,000 + (-80,000) = 20,000
The decision to toss the coin has the EMV of $20,000 and thus
appears to be better than the decision simply to accept the $10,000.
Yet you might justifiably feel that you would rather have the $10,000
straight out than toss the coin and run the risk of having to pay the
gambler $160,000. That is you prefer the $10,000 to a gamble with
an EMV of 20,000.
this very natural feeling is based on fact that people make decisions
in terms of utility.
One person’s utility for an amount of money may be very different
from another person’s utility for the same amount.
A millionaire might easily choose to engage in the gamble of tossing a
coin for a $200,000 gain or a loss of $160,000. This is because
millionaire could afford the loss of $160,000 without much difficulty,
and the possible gain of $10,000 is peanuts.
Suppose you are offered the chance to take $.10 for sure or toss a coin and
receive $2.00 for a head and pay $1.60 for a tail.
EMV1 = $.10(.5) + $.10(.5)
= 0.05 + 0.05 = 0.10
EMV2 = $2.00(.5) +(-1.60(.5))
= 1 + (-0.8) = 0.20
Many people who would not have tossed the coin when the payoffs were $200,000 and $160,000 would not hesitate to do so with the smaller profits.
This is because, in general, for small amounts of money the monetary value is good measure
of utility.
The expected monetary value seems to be good decision-making
criterion when the payoffs involved are small relative to the overall
wealth, assets, or budget of the decision maker, but when the payoffs
are large, the utility criterion must be considered