**Robert Nau**

**Professor Emeritus**

**Fuqua School of Business**

**Duke University**

**Biographical information and web pages**:

- Duke web page
- Choice theory course web site
- Forecasting course web
site
- RegressIt Excel add-in web site

**What does this figure represent?**

If you guessed "battle
of the sexes," you are correct. The figure illustrates a theorem
concerning the geometry of the set of solutions of a noncooperative game, as it
applies to the 2x2 game known as battle-of-the-sexes. (He prefers the boxing
match, she prefers the ballet, but they would like to go somewhere together
rather than separately. What should they do?) The gray saddle is the set of
independently randomized strategies. The green hexahedron is the set of
correlated equilibria. Their three points of intersection (red dots) are Nash
equilibria. The obvious fair solution (flipping a coin) is the midpoint of the
long edge, which is *not* a Nash equilibrium. This picture is generic in
the sense that Nash equilibria always lie on supporting hyperplanes of the set
of correlated equilibria and as such they cannot exist in its interior when it
has full dimension as it does here. For more details see the following paper.

**Which one of them was
right?**

**Research highlights:**

My research area lies within
the broad field of rational choice theory, the theory of the expected-utility
maximizing, game-playing, equilbrium-seeking rational economic person. My own
work has primarily focused on foundational issues: primitive assumptions
(axioms) that govern the preferences of rational agents and mathematical
representations of mental processes that might explain such preferences.
Classical work in the field, from the 1920's to 1960's, emphasized the
subjective expected utility model for decisions made under conditions of
uncertainty, in which agents act as if they assign numerical probabilities to
events and numerical utilities to consequences and they make choices so as to
maximize the expected value of their utility. The** expected utility** model was formalized by von Neumann and
Morgenstern (1944/1947) with a sweeping scope: "We shall assume that the
aim of all participants in the economic system, consumers as well as
entrepreneurs, is money, or equivalently a single monetary commodity. This is
supposed to be unrestrictedly divisible and substitutable, freely transferable,
even in the quantitative sense, with whatever 'satisfaction' or 'utility' is
desired by each participant." (p. 8) In a situation involving two or more
agents, such as a game or market, they seek an** equilibrium **in which each one optimizes his or her expected
utility against the actions of others, ceteris paribus, independently randomizing
their choices for strategic purposes if necessary. The existence of such a
solution was proved by Nash (1951) via a fixed point theorem, and the elegance
and power of this result made it the focal point of game theory as the field
developed. A similarly sweeping formalization of **subjective probability **was introduced by Savage (1954), in which
agents are assumed to be able to assign equally precise probabilities to any
kinds of events, and it is the foundation on which "Bayesian"
statistics and decision theory are built.

Beginning in the 1970's this
paradigm was challenged on many fronts as contrary findings emerged from
behavioral experiments and as the preference axioms that support the subjective
expected utility model were reappraised on normative as well as descriptive
grounds. More general models of "non-expected utility preferences"
began to be explored, along with weakenings and strengthenings of solution
concepts for games. However, in nearly all of this newer work, great importance
is still attached to preserving a clean separation between beliefs about events
(represented by probabilities or generalizations thereof) and preferences among
consequences (represented by utilities or generalizations thereof), and the
parameters of utility are still assumed to be measurable with unlimited
precision, regardless of whether the conquences are material goods to be
received or personal experiences to be enjoyed.

Much of my own work has taken
a different approach and has been motivated by analogies with modern physics,
in which variables that were formerly assumed to be measurable on absolute
scales (e.g., space and time) are instead measured relative to the viewpoints
of observers with their own frames of reference, which makes those variables
inseparable to some extent. The economic analog of this phenomenon is that
agents do not in general have observable absolute positions in the space of
material goods and investments and personal well-being that they may possess,
nor are utility functions their natural tools of thought with respect to such
things, let alone functions having values that are interpersonally measurable.
All that can be observed in public are contracts that agents are willing to
sign and relative rates and prices at which they are willing to make exchanges
with each other, which makes it impossible to separate the effects of beliefs
and preferences and endowments when interpreting their behavior. Money plays
(literally) a cardinal role in quantifying the rates of exchange in precise
terms, which is one of its functions. The more we abstract from it, the fuzzier
the numbers become. It may seem that giving a distinguished role to money
rather than utility in the modeling of rational economic behavior would create
problems for decision analysis, which often deals with multiattribute outcomes,
as well as for noncooperative game theory, in which equilibrium probabilities
in randomized strategies are determined from payoff matrices whose units are
utilities. In various papers I have shown that this is not true: there are
natural extensions of these models that do not require the strict separation of
probability from utility. Central to this program is the idea that the
principle of **no-arbitrage** (avoiding
a sure loss in units of money) is the key axiom that unifies the domains of
rational choice theory that deal with choice under uncertainty: personal
decisions (the 1-body problem), noncooperative games (the problem of 2, 3, 4 .
. . bodies), and investments in markets (the n-body problem for large n).

This approach to modeling
rational choice under uncertainty does not require that the agent only cares
about money or that the only situations that can be analyzed are those whose
primary consequences are monetary. Rather, side bets in units of money which
are conditioned on the agent's own actions can be used as a precise yardstick
for indirectly quantifying her attitude toward risk and her preferences for
other attributes of consequences. The optimal actions for an individual or
equilibrium actions for players in a game are those which do not give rise to
arbitrage in the side bets. To the extent that this story of hypothetical
monetary side bets stretches the imagination, it is an even greater stretch to
assume that agents can know each others' probabilities and utilities or
equilibrate on their beliefs in games and markets with any sort of numerical
precision, particularly when they all may have private information,
idiosyncratic interests, and unobservable background risk.

The fundamental representation
theorems in the three domains (decisions, games, markets) are all applications
of the duality theorem of linear programming, a special case of the separating
hyperplane theorem for convex sets, which has many other applications in
economics. In each domain there is a primal problem whose variables are actions
of the body (moves to be made, offers to bet or trade) and the objective is for
an observer to extract an arbitrage profit, and there is a corresponding dual
problem whose variables are properties of mind (probabilities, utilities, or
combinations thereof) and the objective is to find values of them that
rationalize the actions of the body in the sense of individual or joint
maximization of expected value or expected utility. This correspondence is well
known in subjective probability theory (in de Finetti's version of it) and in
asset pricing theory, but noncooperative game theory, as it is conventionally
presented, takes a very different approach to characterizing rational behavior.
It assumes the expected utility model, performs some hand-waving about common
knowledge of utilities, and then directly imposes an equilibrium condition
(usually Nash equilibrium). What I have contributed is to show that game theory
fits into a continuum with subjective probability theory and asset pricing
theory. Rational behavior in games can be given a primal definition in terms of
avoidance of arbitrage by the players as a group ("joint coherence"),
and application of the duality theorem of linear programming then leads to **correlated equilibrium**, first proposed
by Robert Aumann (1974), rather than Nash equilibrium as the fundamental
solution concept. The very act of revealing the rules of the game via
conditional side bets on its outcome (thereby solving the common knowledge
problem up front) is equivalent to asserting that its solution lies in the set
of correlated equilibria, and outcomes that do not have positive probability in
some correlated equilibrium are those which expose the set of players to
arbitrage. Moreover, the existence proof for correlated equilibria does not
require the use of a fixed point theorem. (The obsession with fixed points in
economics is unfortunate: they usually lack a good story about selection and
convergence. By comparison, the no-arbitrage criterion for joint rationality is
self-enforcing in the presence of an alert observer.)

In the most general case, the
units of preference modeling are **risk
neutral probabilities**, which are the rates at which an agent is publicly
willing to make small side bets on events. This term originated in the finance
literature, where it refers to the probability distribution of a "risk
neutral representative agent" who determines arbitrage-free asset prices.
However, these risk neutral probabilities are not the true subjective
probabilities of a typical real agent, who is risk averse. Rather, they are
interpretable as the product of her subjective probabilities and her
state-dependent marginal utilities for money, with those two psychic dimensions
not being separately measurable, the economic equivalent of space-time. In my
work on this topic I've shown that risk aversion can be modeled without
separating probabilities from utilities by using a generalization of the
Arrow-Pratt risk aversion measure that refers to first and second derivatives
of the risk neutral probabilities rather than first and second derivatives of
utility functions.

The traditional subjective
expected utility model also imposes subtle (and arguably unreasonable)
restrictions on the shapes of indifference curves in payoff space, requiring
the agent to be equally risk averse to all sources of risk. My more general
approach based on derivatives of risk neutral probabilities allows risk
aversion to be source-dependent, which accomodates the phenomenon of **ambiguity aversion** (also called
"uncertainty aversion") that is illustrated by Daniel Ellsberg's
(1961) famous paradox: most persons would prefer to bet on events whose
probabilities are known (say, the color of a ball drawn from an urn that is
known to hold equal numbers of red and black balls) rather than events whose
probabilities are undetermined (the color of a ball drawn from an urn in which
the numbers are unknown). Ellsberg's subversive idea** **has inspired the development of many new models in decision theory
over the last few decades.

Some of my other work in the
area of game theory addresses various issues concerning the geometry of the
sets of Nash and correlated equilibria, one of which is illustrated by the
figure above. In the most general case, where risk-averse agents must
reciprocally measure each other's payoffs to establish the rules of the game,
the parameters of correlated equilibrium distributions are risk neutral
probabilities rather than measures of pure belief. Another important analogy
with physics is that the act of measurement tends to perturb whatever is being
measured, resulting in fundamental indeterminacies. This same concept is
relevant to attempts to measure personal probabilities in terms of the rates at
which agents are willing to bet money on events. The very fact that a second
agent may eagerly take the other side of a bet that the first agent has offered
could perturb that agent's beliefs. My work in the area of **indeterminate probabilities** (the "confidence weighted probability"
model) addresses this phenomenon.

Yet another connection with
physics arises in the modeling of differences in beliefs. **Cross-entropy** serves as a measure of information gain in a physical
experiment, and this same principle (even the same mathematics) describes the
situation in which a risk averse agent with fixed personal probabilities and no
prior investments comes into contact with a market in which asset prices are
determined from some other probability distribution. In the special case where the
agent has logarithmic utilities, the cross-entropy (KL divergence) between the
two probability distributions is the gain in expected utility that the agent
achieves in transacting with the market until an indifference point is reached,
where her own risk neutral probabilities agree with those that price the
assets. More general utility functions lead to well-known generalizations of
cross-entropy.

**Teaching
and software**:

Throughout my career I have
taught a continuously-evolving elective course on statistical forecasting to
MBA students and graduate students from other departments around the
university. Some of my older, unplugged notes for it are posted on a public web
site, statforecasting.com, which
receives over 1 million daily visitors per year. In connection with the course,
I have designed a couple of software tools. One is the
"user-specified-model" forecasting procedure in a commercial
statistics package, Statgraphics. (The
user manual for the procedure is here.) It
allows the user to construct forecasting models from combinations of data transformations
(log, power, deflation, seasonal adjustment) and model types (regression,
random walk, moving average, exponential smoothing, ARIMA) and to do rigorous
side-by-side testing and comparison of up to 5 models at once. The other is an
Excel add-in, RegressIt, which performs
linear and logistic regression analysis. It offers high-quality, interactive
table and chart output and includes a 2-way interface with the R programming
language that allows Excel to be used as a front end and/or back end for linear
and logistic regression analysis in R. RegressIt provides an array of novel
tools for model space exploration and model comparison and testing, with many
interactive features that are useful for classroom demonstrations and
evaluation of student work as well as for supporting good analytical practices
in general, and it includes extensive built-in teaching notes for regression.

I have also regularly taught
a Ph.D. course on rational choice theory (most recently in 2015) which is taken
by students from a number of departments around the university. Some older
notes for this course can be found here.

**Published papers:**

- Risk
Neutral Equilibria of Noncooperative Games (
*Theory and Decision**, v. 78, no. 2, 2015*) - Imprecise
Probabilities in Noncooperative Games (Electronic
proceedings of the Seventh International Symposium on Imprecise
Probability: Theories and Applications, 2011) Powerpoint
slides
- A
Theorem for Bayesian Group Decisions (with Ralph Keeney;
*Journal of Risk and Uncertainty*, v. 43, no. 1, 2011) - Risk,
Ambiguity, and State-Preference Theory (
*Economic Theory*, v. 49, no. 1, 2011) - Duality Between
Maximization of Expected Utility and Minimization of Relative Entropy When
Probabilities are Imprecise ( Electronic proceedings of the Sixth
International Symposium on Imprecise Probabilities and Their Applications,
2009)
- Sensitivity
to Distance and Baseline Distributions in Forecast Evaluation (with Victor
Richmond Jose and Bob Winkler, Management Science, v. 55, no. 4, 2009
- Scoring Rules,
Generalized Entropy, and Utility Maximization (with Victor Richmond Jose
and Bob Winkler,
*Operations Research*, v. 56, no. 4, 2008) Spreadsheet example Contour plots - Extensions of the Subjective Expected
Utility Model, in
*Advances in Decision Analysis,*Cambridge Univ. Press, 2007, edited by Edwards, Miles, and von Winterfeldt) - The Shape of Incomplete
Preferences (
*The Annals of Statistics*, v. 34, no. 5, 2006) Supplement: mathematical programs for example - Uncertainty
Aversion With Second-Order Utilities and Probabilities (
*Management Science,*v. 52, no.1, 2006) Proofs of theorems Spreadsheet example - On
the Geometry of Nash Equilibria and Correlated Equilibria (
*International Journal of Game Theory*, v. 32, no. 4, 2004) - A
Generalization Of Pratt-Arrow Measure To
Non-Expected-Utility Preferences And Inseparable Probability And Utility (
*Management Science*v.49 n.8, 2003) - The Aggregation
of Imprecise Probabilities (
*J. Stat. PIanning & Inference*v. 105 n.1, 2002) - De
Finetti Was Right: Probability Does Not Exist (
*Theory and Decision*v. 51 n. 2-4*,*2001) - Uncertainty
Aversion with Second-Order Probabilities and Utilities (Electronic
proceedings of the Second International Symposium on Imprecise
Probabilities and Their Applications, 2001)
- Arbitrage,
Incomplete Models, and Other People's Brains (in
*Beliefs, Interactions, and Preferences in Decision Making*, Machina & Munier, eds., Kluwer, 1999) - Valuing
Risky Projects: Options Pricing Theory and Decision Analysis
(with Jim Smith,
*Management Science*v. 41 n.5, 1995) - Coherent
Decision Analysis with Inseparable Probabilities and Utilities (
*J. Risk and Uncertainty*v. 10, 1995) - The
Incoherence of Agreeing to Disagree (
*Theory and Decision*v. 39, 1995) - Indeterminate Probabilities on
Finite Sets (
*The Annals of Statistics*v. 20 n. 4, 1992) - Joint
Coherence in Games of Incomplete Information (
*Management Science*v. 38 n. 3, 1992) - Arbitrage,
Rationality, and Equilibrium (with Kevin McCardle,
*Theory and Decision*v. 31, 1991) - Coherent
Behavior in Noncooperative Games (with Kevin McCardle,
*J. Econ. Theory*v. 50 n. 2, 1990) - Decision
Analysis with Indeterminate or Incoherent Probabilities (
*Annals of Operations Research*v. 19, 1989) - Blau's
Dilemma Revisited (
*Management Science*v. 33, 1987) - Should
Scoring Rules Be 'Effective'? (
*Management Science*v. 31, 1985) - Adaptive
Filtering Revisited (
*J. Operational Research Society*v. 30 n. 9, 1979)

**Edited volume:**

**Older working papers**:

- Bayesianism Without Priors, Acts
Without Consequences (September 2005)
- Coherent Assessment of Subjective
Probability (1981)

**Other web pages:**