Probability and Decision
Analysis
Introduction
1
Course content
•
Introduction to Decision Analysis – Chapters 1 and 2
•
Structuring Decisions and Choices (Decision Trees and Influence Diagrams) –
Chapters 3 and 4
•
Sensitivity Analysis – Chapter 5
•
Organizational Use of Decision Analysis – Chapter 6
•
Basic Probability Concepts and Models – Chapters 7 and 9
•
Subjective Probability Models – Chapter 8
•
Decision Making Using Data and Simulation – Chapters 10 and 11
•
Value of Information – Chapter 12
•
Introduction to Utility Theory – Chapters 14 and 15
2
What is decision analysis?
• Decision analysis is a process that provides a
structured method along with analytical tools
designed to improve one’s decision-making
skills.
– It enables you to base your decisions on more
than intuition or hunches.
– It teaches you how to rigorously analyze decisions
to gain helpful insights.
3
Decision analysis helps us address
decisions.
• Decision analysis adds clarity and insight.
– Uses a conceptual framework for thinking systematically
– Breaks the problem down into smaller, more easily
understood units for individual analysis
• Decision analysis supplies analytical tools that
improve insight and understanding of the problem.
• Decision analysis instills a sense of confidence in the
solution we choose to enact.
4
Yet, it doesn’t make decision making
“easy”.
• Decision analysis cannot provide us with “the”
answer:
– Even after analysis, hard decisions are still difficult
to decide.
– We must think through all the various aspects of
the decision.
– We cannot simply feed our inputs into a computer
to get the one and only answer.
– Decision analysis provides us with alternatives to
choose between.
5
But why are decisions hard?
•
•
•
•
Complexity
Uncertainty
Multiple objectives
Competing view points
6
But why are decisions hard?
• Complexity
– Can quickly overload the decision maker
• Decision analysis reduces complexity:
– It helps identifying and isolating the individual
components of the bigger decision.
– Each component can then be fully understood
separately.
– Then interlinkages are addressed.
7
But why are decisions hard?
• Uncertainty
– If the future is perfectly known, then decisions are trivial.
– In practice, there is no way to know precisely how all of the factors not
under your control will play out.
• Decision analysis addresses uncertainty:
– It breaks each uncertainty down by listing the different outcomes that
could occur.
– It determines both the consequence and the likelihood of that
outcome occurring.
• Such analysis does not lead to immediate clarity, but it can provide
a richer understanding of the problem and its alternative solutions.
8
But why are decisions hard?
• Multiple objectives
– In most cases, decision makers are interested in
several goals with different tradeoffs involved:
• Financial goals vs. non-financial goals
• Short-term goals vs. long-term goals
• Decision analysis provides a framework and
specific tools for dealing with multiple objectives:
– It allows the explicit identification of goals.
– It facilitates ranking them in order of importance.
9
But why are decisions hard?
• Competing view points
– Beyond strictly personal decisions, several decision
makers are generally involved in important decisions.
– This leads to:
• Different goals and views of their relative importance
• Different inputs and biases
• Different approaches to adopt
– Even when a single decision maker is involved, a
different perspective could affect the whole
perception of the decision context.
• Decision analysis provides a systematic process to
approach consensus.
10
Why study and use decision analysis?
“The basic presumption of decision analysis is not at all
to replace the decision maker’s intuition, to relieve him
or her of the obligations in facing the problem, or to
be, worst of all, a competitor to the decision maker’s
personal style of analysis, but to complement,
augment, and generally work alongside the decision
maker in exemplifying the nature of the problem.
Ultimately, it is of most value if the decision maker has
actually learned something about the problem and his
or her own decision-making attitude through the
exercise.” (Bunn 1984, p. 8)
11
Discussion example 1
12
Was it a bad decision for Goldman
Sachs to invest?
• There is a difference between a good decision and a
good outcome!
• Rubin states: “While the result may have been bad,
the investment decision wasn’t necessarily wrong.
After a deal broke up, we’d always reexamine it,
looking for clues we might have missed. But even a
large and painful loss didn’t mean that we misjudged
anything. As with any actuarial business, the essence of
[risk-]arbitrage is that if you calculate the odds
correctly, you will make money on the majority of the
deals and on the sum total of all your deals.”
13
What is the relation between decision
analysis and subjective judgements?
• Subjective judgments and personal beliefs are important
inputs to decision analysis.
• Decision analysis does not attempt to remove them from
the decision making process.
• They are necessary because ultimately a person, not a
machine, has to make the decision.
• However, decision analysis helps us to correct limitations
inherent to human thinking by making us aware of
them and providing us with tools for addressing them.
14
What is the relation between decision
analysis and subjective judgements?
Decision analysis not only provides a structured
way to think about decisions, but also more
fundamentally provides a structure within which
a decision maker can develop beliefs and
feelings, those subjective judgments that are
critical for a good solution.
15
A high-level decision analysis process
1
Identify the
decision situation
and understand the
objectives
•
•
•
•
Sometimes the surface problem hides the real problem.
Define the problem as exactly as possible.
Identify financial and non-financial objectives.
Understanding objectives is critical to subsequent steps.
16
A high-level decision analysis process
1
Identify the
decision situation
and understand the
objectives
2 Identify the
alternatives
• Analysis of objectives may reveal alternative solutions not initially
obvious.
• This step involves both discovery and creativity.
17
A high-level decision analysis process
3
1
Identify the
decision situation
and understand the
objectives
2 Identify the
alternatives
Decompose and model
the problem:
1. Model of problem
structure
2. Model of uncertainty
3. Model of preferences
• Decomposition entails structuring the problem into smaller and more
manageable pieces.
• Consider elements of uncertainty in the different pieces
• Modeling involves reassembling the pieces of the problem into a
simplified representation.
• Keep the relevant pieces and discard the irrelevant ones.
• Models can be mathematical or graphical.
• Mathematical models allow for formal analysis that can provide
insights not otherwise obvious.
18
A high-level decision analysis process
3
1
Identify the
decision situation
and understand the
objectives
2 Identify the
alternatives
Decompose and model
the problem:
1. Model of problem
structure
2. Model of uncertainty
3. Model of preferences
4
Choose the
best alternative
• Which alternative is “preferred”?
• Which alternative best serves the objectives?
• Remember that decision making requires human judgment and is not
a process of “solving” a problem for the right answer.
19
A high-level decision analysis process
3
1
Identify the
decision situation
and understand the
objectives
2 Identify the
alternatives
Decompose and model
the problem:
1. Model of problem
structure
2. Model of uncertainty
3. Model of preferences
4
Choose the
best alternative
5
Sensitivity
analysis
• “What if” you change one of more aspects of the preferred
alternative or the input parameters?
• What are the inputs that, if modified, largely affect the outputs?
20
A high-level decision analysis process
3
1
Identify the
decision situation
and understand the
objectives
2 Identify the
Modified frame
alternatives
Decompose and model
the problem:
1. Model of problem
structure
2. Model of uncertainty
3. Model of preferences
New alternatives
4
Choose the
best alternative
Probabilistic analysis
Yes
• Sensitivity analysis often leads to new insights that may require
repeating the previous steps.
• Decision-making cycle: think of the overall decision making process as
iterative.
• Usually, it is necessary to cycle through the different steps several times
• Continue iterations until you arrive at the requisite decision model, i.e.,
the model in which no new insights or intuitions are gained by another
cycle of analysis, or which contains every essential for solving the
problem.
5
Sensitivity
analysis
6
Further
analysis?
21
A high-level decision analysis process
3
1
Identify the
decision situation
and understand the
objectives
2 Identify the
Modified frame
alternatives
Decompose and model
the problem:
1. Model of problem
structure
2. Model of uncertainty
3. Model of preferences
New alternatives
4
Choose the
best alternative
Probabilistic analysis
5
Sensitivity
analysis
6
Yes
Further
analysis?
No
• Without implementation, the whole decision analysis process is
worthless.
• It is essential to follow through the results of the analysis and proceed
to implementation.
7
Implement
the chosen
alternative
22
We will use PrecisionTree and @RISK
software tools.
23
DECISION ELEMENTS
24
Four elements characterize decisions
1.
2.
3.
4.
Values and objectives (the decision context)
Decision to make
Uncertain events
Consequences
25
1- Values and objectives
• It is critical to align actions with goals.
– Values: things that matter to us
– Objectives: what we specifically want to achieve
• Most people do not make the effort or take
the time to define values and objectives
explicitly, systematically and consciously.
• But it is critical to do so!
26
Discussion example 2
• As a new graduate, you will have to choose from among a
number of job opportunities. Your values are the things you
care about in choosing the job.
• Typical values of graduates include compensation, prestige,
healthy workplace environment, housing, and cost-of-living
concerns, along with time off to spend with family and
friends and social concerns.
• As you can imagine, you want to know which values are
important to you and how they rank with each other before
you are able to decide on which job to choose.
27
Making money: a special objective!
• Money is a special and unique objective because
it is usually tied in with achieving your other
objectives:
– This is called a proxy objective.
• For example, if you value traveling, you will need
money.
• Pricing out: By placing a monetary value on your
different objectives, you can compare them and
decide about trade-offs more easily.
28
Discussion example 3
• You are considering buying an alarm system or a large
LED TV. Assume you cannot afford both.
• how much more would you be willing to pay for each?
• By attaching monetary values to the two items (pricing
out), you can compare them. The TV costs $400 while
the alarm system costs $800.
• Is the alarm system twice as valuable as the TV? Are
there other values involved besides money?
29
Using money as an objective has
advantages and shortcomings.
• Advantages:
– Universally understood
– Measurable
– Simplifies trade-offs because it allows different
objectives to be put on the same scale via pricing out
• Shortcomings:
– Monetary values cannot be assigned to all values and
objectives.
– Safety, family, lives, environment, …
30
The decision context englobes the
circumstances in which decisions occur.
• Every decision situation involves a specific
context, and that context determines which
objectives need to be considered.
31
2- Decisions to make
• For every decision, two or more alternatives are
possible.
– It is important to identify the alternatives under
consideration.
• Decisions can be urgent (have to be made
immediately) or not so urgent.
– It is crucial to assign a decision deadline for every decision
under consideration.
• There is always a tradeoff between timeliness and
information.
– The more you delay the decision the more information you
have.
32
Sequential decisions
• It is common to encounter decisions that have to be made in
sequence.
– A decision triggers another one or depends on decisions made earlier.
• It is important to understand and properly model this sequence.
– You will need to consider future decisions when making immediate
decisions.
33
3- Uncertain events
• Decisions generally have to be made without
knowing exactly what will happen in the future or
exactly what the ultimate outcome will be.
• Every uncertain event has more than one
possible outcome.
• Not all uncertain events are relevant. Determine
which ones are; ignore the rest.
– Relevant events are those whose outcomes impact
one or more of your objectives.
• Information availability should not determine
which uncertain events you focus on.
34
Sequencing uncertain events and
decisions - Dovetailing
• For every uncertain event, determine when its
uncertainty will be resolved with respect to
the considered decisions. This is called
Dovetailing.
– Dovetailing is important as it identifies what
uncertainties are resolved and what others are
not at the time when every decision has to be
made.
35
Dovetailing uncertain events and
decisions
36
4- Consequences
• Each objective in the decision context will have a
final consequence.
– Multiple objectives mean multiple consequences.
• Determine measures for each consequence so
you can determine the extent to which each
objective was met.
– Monetary or non-monetary consequences?
• Can you price out any of the non-monetary consequences?
– What are the trade-offs between various objectives?
37
Dovetailing + consequences
38
Planning horizon
• The timeline of your consideration represents
the planning horizon
– How long will you continue tracking the
consequences of your decisions?
• A compromise should be made to decide on
the time period to consider. Choose it such
that:
– Events and decisions happening beyond it are not
essential parts of the current decision context.
39
Talking about trade-offs – Time value
of money, a special trade-off
• “A dollar today is worth more than a dollar
tomorrow.”
– Trade-offs exist between dollars today and dollars
tomorrow
• Why?
– Due to the opportunity that the dollar today can
create until tomorrow!
• This is manifested in the interest rate.
40
The interest rate is an equivalence metric
representing the time value of money.
• A dollar today is equivalent to: 1 × (1 + 𝑟) dollars one year
from now where 𝑟 is the yearly interest rate of the investor.
• More generally:
𝐹𝑉 = 𝑃𝑉 × 1 + 𝑟
𝐹𝑉
𝑃𝑉 =
1+𝑟
𝑛
𝑛
Where:
𝐹𝑉 is the future value at period 𝑛
𝑃𝑉 is the present value (now)
𝑛 is the number of periods between the present and the future
𝑟 is the interest rate per period
41
Example 4
• What is the future value of $100 placed in a
savings account now at 5% annual interest
rate for two years?
FV = PV (1+ r)n
FV = $100 (1+0.05)2
FV = $110.25
42
The NPV is the net present value of a
series of cashflows.
• Let 𝑥𝑖 be the cashflow in period 𝑖.
• The NPV is the net present value (equivalent
value) of all cashflows at time 0 (now).
43
Example 5
• What is the present value of the following
series of cashflows?
r= 4%/year
$600
$400
$350
0
$400
3
1
2
years
4
5
$400
$800
44
Example 6
• Fouad, Leila and Tarek have each $100,000 that they plan to
invest in buying a small chalet. The chalet will then be offered
for rent for 1 year for $9,000 then will be sold again. Assume
that the sale price of the chalet one year from now is
$103,000. The table below provides the interest rates of the
three potential investors.
– Is this investment good for Fouad? Leila? Tarek?
– Do the previous answers change if they instead offer the chalet
for rent for two years then sell it at the same price of 103,000?
Investor
Yearly interest rate
Fouad
4.5%
Leila
12%
Tarek
13%
45
Readings
• Chapter 2:
– Larkin Oil case (Page 36)
– Case 1: The Value of Patience
– Case 2: Early Bird Inc.
46
Discussion case – Larkin Oil
• Read the Larkin Oil example in the text first (Page 36).
• Discussion questions:
–
–
–
–
–
–
–
–
–
–
–
What is the decision context?
What are Larkin’s objectives?
What decisions need to be made?
Is this a sequential decision problem?
Are there any immediate decisions needed?
What uncertain events need to be considered?
What is the appropriate planning horizon?
What are the different consequences of the decisions?
How can these consequences be valued? Monetarily?
What are the trade-offs?
How should Larkin allocate its limited resources?
47
Larkin Oil – putting everything
together
48