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Decision Analysis
Chapter 16: Hillier and Lieberman
Chapter 11: Decision Tools for Agribusiness
Dr. Hurley’s AGB 328 Course
Terms to Know

Alternative, State of Nature, Payoff, Payoff
Table, Prior Distribution, Prior
Probabilities, Maximin Payoff Criterion,
Maximum Likelihood Criterion, Bayes’
Decision Rule, Crossover Point, Joint
Probabilities, Posterior Probabilities,
Probability Tree Diagram, Expected Value
of Perfect Information, Expected Value of
Experimentation, Nodes, Branches,
Decision Node, Event Node
Terms to Know Cont.

Backward Induction Procedure, Spider
Chart, Tornado Chart, Utility Function for
Money, Decreasing Marginal Utility for
Money, Risk Averse, Increasing Marginal
Utility of Money, Risk Neutral, Risk
Seekers, Exponential Utility Function
Goferbroke Company Example






Trying to maximize payoff from land that
may have oil on it
The company has two options: drill or sell
the land
If the company drills for oil and oil exists,
they expect a payoff of $700K
If the company drills for oil and oil does not
exist, they expect a payoff of -$100K
If the company sells the land it receives
$90K whether the oil exists or not
There is a 1 in 4 chance that oil exists
Payoff Table for Goferbroke
Nature
Alternatives
Oil Exists
Oil Does Not Exist
Drill
$700K
-$100K
Sell
$90K
$90K
Prior Probability
25%
75%
Maximin Payoff Criterion

This criterion identifies the worst payoff
for each decision that you could make
and maximizes the highest of these
amounts
◦ For Goferberoke this would be to sell the
land

This criterion is for the very cautious
Maximum Likelihood Criterion
This criterion requires you to select the
best payoff from the highest likelihood
state of nature
 For Goferbroke, the best decision based
on this criterion is to sell the land

Bayes’ Decision Rule
This criterion calculates the expected value of
each decision and then chooses the maximum of
these expected values
 For Goferbroke, the expected payoff for drilling is
100K while for selling it is 90K
 A nice attribute about Bayes decision rule is that
you can conduct a sensitivity analysis to find what
probability would cause you to change your
decision from the given prior probabilities

◦ You can do this by finding the probability that will
cause one decisions expected payoff to equal another
decisions expected payoff
Quick Statistics Lesson
Suppose that one of three events can
occur: A, B, or C.
 Suppose that each of these events has a
probability of occurring but that the
events do not overlap.
 Let P(•) represent the probability
operator that maps the event into a
probability from 0 to 1.
 Since one of the three events has to
occur, we know that P(A)+P(B)+P(C)=1

Quick Statistics Lesson Cont.
C
A
B
Quick Statistics Lesson Cont.
Now suppose that one of three events
can occur: A, B, or C.
 Suppose that each of these events has a
probability of occurring and that events A
and B can overlap.
 Because we have overlapping probabilities,
we have four possibilities that we care
about: P(A), P(B), P(C), and P(A and B).

Quick Statistics Lesson
C
A
A and B
B
Quick Statistics Lesson Cont.

Intuitively we should know that:
◦ P(A) + P(B) + P(C) - P(A and B) =1.

Notation:
◦ P(AB)= P(A and B)
  is known as the intersection operator
◦ P(AB)= P(A) + P(B) - P(AB)
  is known as the union operator
Quick Statistics Lesson Cont.
The conditional probability function
P(A|B) shows what is the probability that
event A occurs given that you see event
B.
 Notation:

◦ P(A|B)=P(AB)/P(B)
◦ From this we can infer:
 P(B|A)=P(AB)/P(A)
Bayes Theorem
Let Ai represent the true state is i where i =
1,2,…,n
 Let Bj represent the finding/event j occurring
where j = 1,2,…,m
 Let P(•) represent the probability operator
and P(•|•) represent the conditional
probability operator
 Then Bayes Theorem states:
𝑃 𝐵𝑗 𝐴𝑖 𝑃(𝐴𝑖 )
𝑃 𝐴𝑖 𝐵𝑗 = 𝑛
𝑘=1 𝑃 𝐵𝑗 𝐴𝑘 𝑃(𝐴𝑘 )

Joint and Posterior Probabilities

In the previous slide, the joint
probabilities are represented as:
◦ 𝑃 𝐵𝑗 𝐴𝑖 𝑃(𝐴𝑖 )

The posterior probabilities are
represented as:
◦ 𝑃 𝐴𝑖 𝐵𝑗
Bayes Theorem Using a Tree
Diagram
P(B1 |A1)
P(A1)
P(A2)
P(B2 |A1)
P(B1 |A2)
P(B2 |A2)
𝑃 𝐴1 𝐵1 =
P(B1|A1)P(A1)
P(B2|A1)P(A1)
P(B1|A2)P(A2)
P(B2|A2)P(A2)
𝑃 𝐵1 𝐴1 𝑃(𝐴1 )
𝑃 𝐵1 𝐴1 𝑃 𝐴1 + 𝑃 𝐵1 𝐴2 𝑃(𝐴2 )
Using Bayes Theorem Using a Tree
Diagram for Goferrbroke
Let the probability of finding oil be 25%
and 75% for not finding oil given no prior
information
 Let the probability of finding oil be 60%
given information that is favorable to
finding oil
 Let the probability of finding oil be 40%
given information that is not favorable to
finding oil

Using Bayes Theorem Using a Tree
Diagram for Goferrbroke Cont.
Let the probability of not finding oil be
20% given information that is favorable to
finding oil
 Let the probability of not finding oil be
80% given information that is not
favorable to finding oil

Using Bayes Theorem Using a Tree
Diagram for Goferrbroke
P(Favorable |Oil)=0.6
P(Favorable|Oil)P(Oil)
= 0.25*0.6 = 0.15
P(Unfavorable |Oil)=0.4
P(Unfavorable|Oil)P(Oil)
= 0.25*0.4 = 0.1
P(Oil)=0.25
P(Favorable |No Oil)= 0.2
P(No Oil)=0.75
P(Unfavorable |No Oil)=0.8
P(Favorable|No Oil)P(No Oil)
= 0.75*0.2 = 0.15
P(Unfavorable |No Oil)P(No Oil)
= 0.75*0.8 = 0.6
𝑃 𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑂𝑖𝑙 𝑃(𝑂𝑖𝑙)
𝑃 𝑂𝑖𝑙 𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 =
𝑃 𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑂𝑖𝑙 𝑃(𝑂𝑖𝑙) + 𝑃 𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑁𝑜 𝑂𝑖𝑙 𝑃(𝑁𝑜 𝑂𝑖𝑙)
0.15
1
=
=
0.15 + 0.15 2
In-Class Activity (Not Graded)

What are the other posterior
probabilities:
◦ P(Oil|Unfavorable)
◦ P(No Oil|Favorable)
◦ P(No Oil|Unfavorable)
Calculating Expected Payoffs of the Alternatives
Given Information from Seismic Study

Expected payoff if you drill given that the
findings were unfavorable:
◦ =P(Oil|Unfavorable)*(Payoff of drilling when
oil exists)+P(No Oil|Unfavorable)*(Payoff of
drilling when oil does not exist)-Cost of
survey
◦=
1
7
700
6
+
7
−100 − 30 = −15.7143
Calculating Expected Payoffs of the Alternatives
Given Information from Seismic Study Cont.

Expected payoff if you sell given that the
findings were unfavorable:
◦ =P(Oil|Unfavorable)*(Payoff of selling when oil
exists)+P(No Oil|Unfavorable)*(Payoff of
selling when oil does not exist)-Cost of
survey
◦=
1
7
90 +
6
7
90 − 30 = 60
Calculating Expected Payoffs of the Alternatives
Given Information from Seismic Study Cont.

Expected payoff if you drill given that the
findings were favorable:
◦ =P(Oil|Favorable)*(Payoff of drilling when oil
exists)+P(No Oil|Favorable)*(Payoff of drilling
when oil does not exist)-Cost of survey
◦=
1
2
700
1
+
2
−100 − 30 = 270
Calculating Expected Payoffs of the Alternatives
Given Information from Seismic Study Cont.

Expected payoff if you sell given that the
findings were favorable:
◦ =P(Oil|Favorable)*(Payoff of selling when oil
exists)+P(No Oil|Favorable)*(Payoff of selling
when oil does not exist)-Cost of survey
◦=
1
2
90 +
1
2
90 − 30 = 60
Expected Payoff with Perfect
Information (EPPI)
◦ EPPI calculates the expected value of the
decisions made given perfect information
 This measures assumes that you will have chosen
the best alternative given the state of nature that
occurs
 Hence you will multiply the probability of the state
of nature by the best payoff achievable in that state
 For the Goferbroke example, if oil exists you would choose
to drill receiving 700 and if oil does not exists you would
choose to sell receiving 90
◦ Goferbroke’s EPPI = 0.25*700+0.75*90 =
242.5
Expected Value of Perfect
Information (EVPI)

EVPI = Expected Payoff with Perfect
Information – Expected Payoff without
Perfect Information
◦ Expected Payoff without Perfect Information is
just the value you get by using Bayes Decision
Rule of maximizing expected payoff
Goferbroke’s EVPI = 242.5-100=142.5
 If the seismic survey was a perfect indicator,
you would choose to do it because the EVPI
is greater than the cost of the survey

Expected Payoff with
Experimentation (EPE)

EPE =
𝑚
𝑗=1 𝑃
𝐵𝑗 ∗ 𝐸(𝑝𝑎𝑦𝑜𝑓𝑓|𝐵𝑗 )
◦ Where:
 P(Bj) is the probability that finding j occurs
 E(payoff|Bj) represents the expected payoff that you
get if finding j occurs
 Note that this payoff does not factor in the cost of
collecting the needed information

Goferbroke’s EPE =
P(Favorable)*E(payoff|Favorable) +
P(Unfavorable)*E(payoff|Unfavorable) =
0.3*300 + 0.7*90 = 153
Expected Value of Experimentation
(EVE)
EVE = expected payoff with
experimentation – expected payoff
without experimentation
 Goferbroke’s EVE = 153 – 100 = 53

◦ Since this exceeds the cost of the information,
Goferbroke would proceed with undergoing
the survey
Decision Trees

Decision trees can be a useful tool when
examining how to make the optimal
decisions when there is multiple alternatives
to choose from
◦ In the trees, you have decision nodes which are
represented as squares and event/chance nodes
that are represented by circles
◦ You also have the payoffs that occur due to a
sequence of decision and event nodes occurring

To solve these decision trees you work your
way from the end of the tree to the
beginning of the tree
Goferbroke’s Decision Tree Example

Discussed in class
In-Class Activity (Not Graded)
Do problem 15.2-7
 Do Problem 15.4-3
