Arbitrage
and Rational Decisions by Robert Nau, Fuqua
School of Business, Duke University
Published by Chapman
and Hall, 2025
Abstract:
“This unique book offers a unified
approach to the modeling of rational decision-making under conditions of
uncertainty and strategic and competitive interactions among agents. Its most
elementary axiom of rationality is the principle of no-arbitrage, namely that
neither an individual decision maker nor a small group of strategic competitors
nor a large group of market participants should behave in such a way as to
provide a riskless profit opportunity to an outside observer.
Both those who work in the finance
area and those who work in decision theory more broadly will be interested to
find that basic tools from finance (arbitrage pricing and risk-neutral
probabilities) have broader applications, including the modeling of uncertainty
aversion, inseparable beliefs and tastes, nonexpected utility, ambiguity, and
noncooperative games.
The book emphasizes the use of
money (rather than varieties of utility) in the quantification of rational
economic thought. It provides not only a medium of exchange and an objective to
maximize but also a language for cognition, interpersonal expression of
preferences, aggregation of beliefs, and construction of common knowledge in
terms of precise numbers. At the same time it provides an obvious standard of
economic rationality that applies equally to individuals and groups: don’t
throw it away or allow your pocket to be picked. The modeling issues that arise
here provide some perspective on issues that arise in quantitative modeling of
decisions in which objects of choice are less concrete or higher-dimensional or
more personal in nature.
One of the book’s key
contributions is to show how noncooperative game theory can be directly unified
with Bayesian decision theory and financial market theory without introducing
separate assumptions about strategic rationality. The no-arbitrage standard of
rationality leads straight to the conclusion that correlated equilibrium rather
than Nash equilibrium is the fundamental solution concept, and risk-neutral
probabilities come into play when agents are uncertainty-averse.
The book also provides some
history of developments in the field over the last century, emphasizing
universal themes as well as controversies and paradigm shifts. It is written to
be accessible to advanced undergraduates, graduate students, researchers in the
field, and professionals.”
What does the figure on the cover 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 red saddle is the set of independently randomized strategies. The
blue hexahedron is the set of correlated equilibria. Their three points of
intersection (black 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. See section 8.4 of the book for details.