Notes on Decision Tree Analysis

University of ChicagoGraduate School of Business

Introduction to Computer Based Models Bus-36102-81Mr. Schrage Spring 2003

Decision Trees and Decision Analysis

Environment: - A strategic decision must be made,

- Number of alternatives is small.

- Uncertainty plays a key role.

Examples:

Go/No-go decisions on a

-New product, process, or plant.

Why Do It Carefully?

- We are not good at computing probabilities,

e.g., combinations of events.

- We are careless in proposing subjective probabilities.

- We under-enumerate possible scenarios.

What Are Decision Trees?

It is a methodology for

1) Displaying the alternatives in a sequential decision problem.

2) Calculating expected values of the various alternatives.

a) Simple case:

All probabilities about future events are given.

b) More complex:

New information about future probabilities can be bought,

e.g., by market research, testing, clinical trials, etc.

Example

(Based on an example of Gary Eppen)

We are about to introduce a new line of garden tractors.

How much money should we spend on marketing?

A very reasonable simplification is that there are three levels of marketing effort we could use:

Aggressive

Basic

Cautious

Similarly, the market/economy has two possible outcomes:

Strong or Weak.

Example:

So take the BASIC strategy and hope for a Strong market.

Note: We have assumed:

Linear utility, e.g., we are indifferent between:

.95 probability of a $1000 loss, and

.01 probability of a $95,000 loss.

That’s OK if this is one of many projects in a large firm…

So,

Prob{ market is strong, given study is encouraging}

= .270/.435 = .621

Prob{ market is weak | study is encouraging}

= .165/.435 = .379 ( = 1 - .621)

Prob { strong | discouraging} = .180/ .565 = .318

Prob{ weak | discouraging} = .385/.565 = .682

Sensitivity of Posterior Probabilities to Prior Probabilities

Prior Posterior

P(S) P(S|E) P(S|D) 0 0 0 .1 .18 .06 .2 .33 .13 .3 .46 .20 .4 .57 .28

.45 .621 .318 .5 .67 .36 .6 .75 .46

.7 .82 .57 .8 .89 .70 .9 .95 .84 1 1 1

Policy:

Make the study;

If it is encouraging, then take the Aggressive strategy.

If it is discouraging, then be Cautious.

The value of the study is $12.96M - $12.85M = $110,000

in excess of its cost.

Other Examples: Testing for HIV

ELISA test if very “sensitive”,

Prob{test negative | have HIV} .0001

Prob{test positive | not have HIV} .01;

Western Blot test is very “specific”,

Prob{test negative | have HIV} .001

Prob{test positive | not have HIV} .0001;

Western blot is expensive. About 1 in 300 people in U.S. are HIV positive.When should you use which test?

Example: Checking for Breast Cancer

Mammogram: Cost > $100;

60 out of 1000 women over 40 test positive on mammogram.

In fact 2.7 out of 1000 over 40 having mammogram have cancer.

Biopsy: Cost > $2000;

Essentially an exact test.

Should insurance be required to cover mammograms for women over age X?

Example: Spouse Battery and Homicide

O. J. Simpson was known to have battered his wife. She was murdered. What is the probability that O.J. did it?

Alan Dershowitz: FBI statistics show that out of 4M incidents of partner battery in 1992, there followed 1432 homicides by the partner. A 36 in 100,000 chance of subsequent murder provides essentially no support for the contention that O.J. did it.

Is Alan correct?

Other Federal statistics: In 1993, 5 out of 100,000 women were murdered by someone.

What is the relevant question?