the development of decision analysis

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, or choosing health insurance

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

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.


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 AP A B

P B A P A P B A P A


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.


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.


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.


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 on 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

What would your answer be?

What would your answer be? Etc…

1

?

1-?-$10,000

$30,000

$0

11-?

-$10,000

$30,000

$500

How should we decide?

Complete Ordering Axiom

These are the minimal mathematical conditions for a complete ordering

What does this mean?

212121 or or rrrrrr

313221 and rrrrrr


Continuity Axiom

This is rather like the mean value theorem in calculus


s.t. 0 and 3221 crrrr

1

c

1-c

1r

3r2r


Independence Axiom


crrr and then if 321

c

1-c

2r

3r

c

1-c

1r

3r


Unequal Probability Axiom


then and if 21 qprr

q

1-q

1r

2r

p

1-p

1r

2r


Compound Lottery Axiom


q

1-q

1r

4r

1

p

1-p

2r

3r1r

1-q4r

q

p

1-p

2r

3r

Expected Utility Wins

Criteria that don’t satisfy these axioms Maximin Maximax 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 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

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”.

the development of decision analysis

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