**Arbitrage, Rationality, and Equilibrium**

by Robert F. Nau and Kevin F. McCardle* *(*Theory and Decision*
**31** 199-240, 1991)

**Abstract:** No-arbitrage is the fundamental principle of
economic rationality which unifies normative decision theory,
game theory, and market theory. In economic environments where
money is available as a medium of measurement and exchange, no-arbitrage
is synonymous with subjective expected utility maximization in
personal decisions, competitive equilibria in capital markets
and exchange economies, and correlated equilibria in noncooperative
games. The arbitrage principle directly characterizes rationality
at the market level; the appearance of deliberate optimization
by individual agents is a consequence of their adaptation to the
market. Concepts of equilibrium behavior in games and markets
can thus be reconciled with the ideas that individual rationality
is bounded, that agents use evolutionarily-shaped decision rules
rather than numerical optimization algorithms, and that personal
probabilities and utilities are inseparable and to some extent
indeterminate. Risk-neutral probability distributions, interpretable
as products of probabilities and marginal utilities, play a central
role as observable quantities in economic systems.

**Key words: **Bayesian decision theory, game theory, arbitrage,
coherence, incompleteness, competitive equilibrium, correlated
equilibrium, capital asset pricing model, risk-neutral probabilities.

**Comments:** This paper discusses applications of the no-arbitrage
characterization of individual rationality and mutually expected
rationality in a variety of economic contexts: asset pricing,
decision analysis, welfare economics, and game theory. As such,
it integrates and/or lays the groundwork for the results in a
number of my other papers. The game theory results were originally
developed in the papers "Coherent Behavior in Noncooperative Games"
(also with McCardle) and "Joint Coherence in Games of Incomplete Information."
The decision analysis results are extended in the paper (with
Jim Smith) on "Valuing Risky Projects: Option Pricing Theory and Decision Analysis."
The game theory and decision analysis results are extended in
two more recent papers, "Coherent Decision Analysis with Inseparable Probabilities and Utilities"
and "Arbitrage-Free Correlated Equilibria."
The role of risk neutral probabilities in the characterization
of common knowledge is discussed in more depth in "The Incoherence of Agreeing to Disagree."
The phenomenon of incomplete (indeterminate) personal probabilities,
which is discussed here in the context of an investor in a securities
market, is modeled in more detail in "Indeterminate Probabilities on Finite Sets"
and "Decision Analysis with Indeterminate or Incoherent Probabilities."
(See also the description of this paper in my research summary.)