Harvard University FAS
Spring 2001

Choice and Chance: The Mathematics of Decision Making



This course develops mathematical ideas that can help individuals make rational choices. We study both decisions whose results are predictable and those made under uncertainty, including cases designed for professional school classes. Topics range from optimization under constraints to Bayesian probability theory, and from iterated dynamical systems to empirical surprises concerning how people make estimates and bets in practice.


The only prerequisities are high school algebra and a willingness to think hard.


Daniel Goroff

Howard Raiffa

Professor of the Practice of Mathematics

Professor Emeritus of Managerial Economics

Science Ctr. 427 (M 12:30-2:30 & appt.)

Harvard Business and Kennedy Schools




Tuesdays and Thursdays from 10:00-11:30 in Science Center E.

Teaching Fellows/Sections


Daniel Rauch

David Savitt


Science Ctr. 321c

Science Ctr. 421e




Office Hours



Section Meets

Mondays 3:00

Science Ctr. 113

Mondays 8:00 pm

Science Ctr. 103b

Primary Resources

Text: Making Hard Decisions with Decision Tools by Robert T. Clemen and Terence Reilly. Duxbury Press 2001.

Text: Smart Choices by John S. Hammond, Ralph L. Keeney, and Howard Raiffa. Harvard Business School Press 1999.

Sourcebook: Available at the HPPS window in the Science Center Basement.

Software: Microsoft Excel available in Science Center Labs or in a student edition from TPC.

Excel Add-On: Decision Tools Suite by Palisade for Windows comes bundled with the Clemen and Reilly text. This software and interactive tutorials for using it will also be available on Windows machines in FAS computing labs.

Supplementary References

Judgement in Managerial Decision Making by Max Bazerman. Wiley 1998.

The Art of Decision-Making by Morton Davis. Springer 1986.

Rational Choice in an Uncertain World by Robyn M. Dawes. Harcourt 1988.

Decisions with Multiple Objectives by L. Keeney and H. Raiffa. Wiley 1976.

Making Decisions by D. V. Lindley. Wiley 1971.

Trust in Numbers by Theodore M. Porter. Princeton 1995.

Individual and Small Group Decisions by K. J. Radford. Springer 1989.

Decision Analysis by Howard Raiffa. Addison Wesley 1968.

Excel Add-On: TreePlan software available for course use for the Mac on the FAS server under courseware, or for Windows through the course webpage,www.fas.harvard.edu/~qr26.


The course progresses from learning techniques to looking deeper into underlying ideas about quantification and reasoning. While the course concentrates on prescriptive theory concerning how you can learn to make better choices, we will also distinguish between descriptive theory concerning real behavior and normative theory concerning ideal behavior. Trying in this way to understand what is consistent, rational, and quantifiable about decision-making can also help us understand better what is not like that at all.

Theme I: The Logic of Preferences

Formulating personal decision problems. Formalizing the mathematical properties, relations, and conditions for describing how individuals order alternatives in practice and in theory.

Theme II: Tradeoffs Under Certainty

How equal swapping and weighted scoring schemes can help sort through choices among items with multiple attributes, including the special case when payoffs occur in a stream over time.

Theme III: Optimization and Mathematical Programming

Geometrical and spreadsheet-based techniques for maximizing a quantity that depends on constrained variables, especially the useful and highly structured case of linear programming.

Theme IV: Judgement and Decisions Under Uncertainty

Probability as a way individuals can order uncertain alternatives. How to separate attitudes towards risk from attitudes towards reward. Simple decision trees. Behavioral anomalies.

Theme V: Inference, the Value of Information, and Sequential Choice

Deciding whether to act now or collect more information. Revising Bayesian probabilities. Solving compound and evolving decision trees. Common mistakes, biases, and anomalies.

Theme VI: Perspectives and Extensions

How ideas developed for personal decision problems relate to game theory, negotiation theory, portfolio theory, statistics, voting theory, group decision theory, social choice theory, etc.


CR=Clemen and Reilly, SC=Smart Choices, SB=Source Book


Primary Sources

Further Reading

I. Problems, Trades, and Time

Problems, Objectives, Alternatives, Consequences, Trade-Offs

CR Chap. 1

SC Chaps. 1, 2, 3

Values, Net Present Value, and Interest Rates

CR Chap. 2

SC Chaps. 4, 5, 6

Constructing Decision Trees and Diagams

CR Chap. 3

SB WF pp. 132-140

Solving Decision Trees and Diagrams

CR Chap. 4

SB WF pp. 153-157

Scales, Weights, and Quantification

Comap Handout

CR Chap. 6

II. Probability, Data, and Estimation

Expectation, Conditioning, and Bayes' Theorem

CR Chap. 7

SC Chap. 7

Judgement and Subjective Probability

CR Chap. 8

Bayesway Website

Inference and the Value of Information

CR Chap. 12

SB HR pp. 7-38,104-128

Normal Forms, Dominance, and Sensitivity

CR Chap. 5

SB RP pp. 67-85

Linearity, Programming, and Optimization

SB HH pp. 495-530

CR Chap. 10

III. Preferences, Risk, and Multiple Objectives

Utility Functions and Attitudes towards Risk

CR Chap. 13

SC Chaps. 8, 9

Axioms, Paradoxes, and Behavioral Anomalies

CR Chap. 14

SC Chap. 10

Additive Utilites and Preferential Independence

CR Chap. 15

SB RP pp. 11-46

Conflicting Objectives and Group Decisions

CR Chap. 16

SC Chap. 11


A study guide will be distributed at the beginning of each unit indicating goals, readings, vocabulary, and assignments such as regular problem sets and preparations for case discussions.

Grades will be based on the following calculation:

10% Participation in class, section, and case discussions.

35% Homework including problems sets most weeks.

20% Midterm Examination in class on Thursday, March 22, 2001.

35% Final Examination based on exercises and essays from the course.

Sample Questions

By the end of QR 26, you should feel confident answering questions like these:

If you know that a family has two children, one of whom is a son, what is the probability the other child is a daughter? (If you think the answer is one half, think some more.)

A friend has a streak of heads tossing a fair coin and wants to bet heavily that the next toss must finally turn up tails. What would you advise?

A friend choosing among three items says that he has good reasons to prefer alternative x to y, and y to z, but z to x. What rationalizations could account for this behavior and what would you advise? From what would you want to protect your friend?

What about the following data convinced the U.S. Congress not to pass a proposal for requiring couples applying for marriage licenses to take the HIV blood test: about 0.3% of the U.S. population carries the HIV virus; a person with the HIV virus has a 95% chance of testing positive; and an HIV virus free person has a 4% chance to test positive? [AC]

The Yankees are playing the Dodgers in a World Series. The Yankees win each game with probability 0.6. What is the probability that the Yankees win the series? (The series is won by the first team to win four games.) [GS]

How could you calculate a transportation plan for minimizing the total cost of moving goods from several warehouses with given inventories to several outlets with given orders?

Each of two people of comparable age and health has a revolver. The first gun has bullets in three of its six chambers; the second has a bullet in only one of its six chambers. Each person is going to spin the cylinder, put his gun to his head, and shoot. Knowing nothing else about the situation, you may remove one bullet from one gun beforehand. Which bullet do you take? Compare this to the problem of a doctor who has the resources to treat only one of two patients, both of whom are seriously ill but with different survival chances. [F]

There is lively debate in Massachusetts over whether to add a new runway at Logan or greatly expand the airport in Worcester. If you were advising a state official on non-political aspects of this decision, how would you begin formulating the problem? What data would you need?

What kinds of biases, inconsistencies, and irrational behaviors do experimenters routinely observe when people make decisions in practice? How can you avoid or use these traps?

How can you think better about multiple and conflicting objectives in decisions such as choosing a job, buying a car, renting an apartment, etc.?

How can you explore and summarize your own basic attitudes toward risk in order to help you handle complex risky choices?

In cases concerning medical, drug, or pollution testing, for example, when should you act under present imperfect information as opposed to gathering additional information at some cost that might change your mind about key uncertainties?

In the purse snatching case People v. Collins, a couple seen fleeing the scene in Los Angeles were described as a black man with a beard and a blond girl with a ponytail driving a partly yellow car. Malcom and Janet Collins were arrested because they and their car matched this description. A mathematician testified that the chances of a randomly chosen couple having these characteristics were smaller than one in twelve million, and they were convicted. How might you, as an expert witness, reach such a conclusion? What calculation did lawyers present on appeal that made the California Supreme Court overturn the lower court's initial guilty verdict nevertheless? [GS]

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