The Development of Decision Analysis Jason R. W. Merrick Based on - - PowerPoint PPT Presentation

The Development of Decision Analysis

Jason R. W. Merrick

Based on Smith and von Winterfeldt (2004). Decision Analysis in Management Science. Management Science 50(5) 561-574.

Why making decisions can be hard?

There are trade-offs between the alternatives

Consider buying a car, a computer or a phone

There is uncertainty about the outcomes

Consider playing the lottery, investing in the stock market,

• r choosing health insurance

Th i f d i i t k

There is a sequence of decisions to make

Consider choosing a major and then a career

There are disagreements between stakeholders

Consider making any decision with your spouse or

significant other

There is a large range of alternatives available

confined by constraints

Go see Drs. Brooks, Hardin, and McLay!

Elements of a Decision

Values and Objectives

What you are trying to achieve?

Decisions and Alternatives

What you are choosing between to get what you What you are choosing between to get what you

want?

Uncertainties and Probabilities

The uncertain events that affect you getting what

you want?

The Decision Context

Keeney (1992) uses the concept of a decision frame

to explain the decisions that people make.

A decision frame consists of a decision maker’s set of

alternatives and the objectives that the decision maker is attempting to achieve when choosing.

Suppose you are looking for a car.

What objectives might you have if you wanted a car to get

to work, go shopping, and get around town?

Suppose you are looking transportation for the same

purpose

How does this change your objectives for just the car

choice?

Development of Decision Analysis

Bernoulli 1738 Bayes 1763 Ramsey 1931 DeFinetti 1937 von Neumann Morgenstern 1944 Savage 1954

• Concerned with the fact that people generally do not follow the expected value

model when choosing amongst gambles (e.g. buying insurance).

• Proposed the expected utility model with a logarithmic utility function to

explain the deviations from the expected value model.

Development of Decision Analysis

Bernoulli 1738 Bayes 1763 Ramsey 1931 DeFinetti 1937 von Neumann Morgenstern 1944 Savage 1954

• Interested in the revision of probability based on observations and proposed

the updating procedure that is now known as Bayes Theorem

( | ) ( ) ( | ) ( | ) ( ) ( | ) ( ) P B A P A P A B P B A P A P B A P A = +

Development of Decision Analysis

Bernoulli 1738 Bayes 1763 Ramsey 1931 DeFinetti 1937 von Neumann Morgenstern 1944 Savage 1954

• Recognized the notion of probability and utility as intrinsically intertwined and

showed that subjective probabilities and utilities can be inferred from preferences among gambles.

Development of Decision Analysis

Bernoulli 1738 Bayes 1763 Ramsey 1931 DeFinetti 1937 von Neumann Morgenstern 1944 Savage 1954

• Followed a similar path as Ramsey by developing a system of assumptions

about preferences among gambles that allowed him to derive subjective probabilities for events.

• DeFinetti’s interest was primarily in the representation of beliefs as subjective

probabilities, not in the derivation of utilities.

Development of Decision Analysis

Bernoulli 1738 Bayes 1763 Ramsey 1931 DeFinetti 1937 von Neumann Morgenstern 1947 Savage 1954

• “Theory of Games and Economic Behavior”: Primary purpose was to lay the

foundation for the study of games, but also established foundations for decision analysis.

• Provided an axiomization of the expected utility model showing that the

cardinal utility function could be created from preferences among gambles.

• Analysis took the probabilities as a given and their axioms led to the

conclusion that decision makers should make decisions to maximize their expected utility.

• This is now referred to as the expected utility model.

Development of Decision Analysis

Bernoulli 1738 Bayes 1763 Ramsey 1931 DeFinetti 1937 von Neumann Morgenstern 1944 Savage 1954

• Extended the work of von Neumann and Morgenstern to consider cases in

which the probabilities are not given.

• Savage’s goal was to provide a foundation for a “theory of probability based
• n the personal view of probability derived mainly from the work of DeFinetti.”
• Savage proposed a set of axioms about preferences among gambles that

enabled him to simultaneously derive the existence of subjective probabilities for events and utilities for outcomes

• Combined the ideas of utility theory from economics and subjective

probability from statistics in to the subjective expected utility model.

Lotteries

Let’s see what your answers would be

1 ? 1-?

• $10,000

$30,000 $0

≈

What would your answer be? What would your answer be? Etc…

1 1-?

• $10,000

$30,000 $500 ≈

How should we decide?

Complete Ordering Axiom

2 1 2 1 2 1

• r
• r

r r r r r r ≈ p f

3 1 3 2 2 1

and r r r r r r f f f ⇒

These are the minimal mathematical conditions

for a complete ordering

What does this mean?

How should we decide?

Continuity Axiom

s.t. and

3 2 2 1

> ∃ ⇒ c r r r r f f

1 c

1

r

This is rather like the mean value theorem in

calculus

What does this mean?

1 1-c

3

r

2

r ≈

How should we decide?

Independence Axiom

c r r r and then if

3 2 1

∀ ≈

c

r

c

r

What does this mean?

1-c

2

r

3

r

1-c

1

r

3

r ≈

How should we decide?

Unequal Probability Axiom

then and if

2 1

q p r r > f

q

r

p

r

What does this mean?

1-q

1

r

2

r

1-p

1

r

2

r f

How should we decide?

Compound Lottery Axiom

1 p 1-p

2

r

3

r

1

r ≈

What does this mean?

q 1-q

1

r

4

r ≈

1-q

4

r

q p 1-p

2

r

3

r ⇒

Expected Utility Wins

Criteria that don’t satisfy these axioms

Maximin Maximax Minimax regret Minimax regret They fail the continuity, unequal probability and

the compound lottery axioms

Criteria that do satisfy these axioms

Expected value Expected utility

Three Viewpoints

There are three major angles of study about gambles and

decisions

• Normative: the study of rational choice.
• Normative models are built on basic assumptions (axioms) that

people consider as providing logical guidance for their decisions.

• Examples include the expected utility model and the subjective

t d tilit d l expected utility model.

• Descriptive: the study of how people actually think and behave.
• Descriptive studies may develop mathematical models of behavior,

but such models are judged by the extent to which their predictions correspond to the actual choices people make.

• Major example is prospect theory.
• Prescriptive: focused on helping people make better decisions.
• Uses normative models, but with awareness of the limitations and

descriptive realities of human judgment.

Decision Analysis

Focused on the prescriptive power of the subjective

expected utility model and Bayesian statistics.

Robert Schlaifer at Harvard wrote “Probability and Statistics

for Business Decisions” in 1959.

Howard Raiffa and Schlaifer wrote “Applied Statistical Howard Raiffa and Schlaifer wrote Applied Statistical

Decision Theory” in 1961.

Ron Howard at Stanford first used the term decision

analysis.

• Howard (1966) “Decision Analysis: Applied Decision Theory”.
• Howard (1968) “The Foundations of Decision Analysis”.