Arbitrage and Rational Decisions

This book presents a unified theory of rational decision-making under conditions of uncertainty and strategic and competitive interactions among agents. Its most fundamental axiom of rationality is the principle of no-arbitrage, namely that neither an individual decision maker nor a small group of game players nor a large group of market participants should behave in such a way as to provide a riskless profit opportunity to an outside observer. This principle provides a foundation for classic models of subjective probability and expected utility theory, state-preference theory, asset pricing theory, and cardinal welfare theory, and it provides tractable pathways for their generalization. Basic tools from finance (arbitrage pricing and risk-neutral probabilities) are shown to have wider applications, including models of non-expected-utility preferences and solution concepts for noncooperative games. A central theme is that, notwithstanding the fields expansion over the last several decades toward new and more sophisticated methods for quantifying the psychology of economic decisions (particularly with regard to ambiguity), the language of money remains fundamental for numerically precise cognition, interpersonal communication, common knowledge of preferences, and aggregation of beliefs. It also provides an obvious standard of economic rationality that applies equally to individuals and groups: dont throw it away or allow your pocket to be picked. The monetary environment provides an elementary setting for studying foundational issues such as the finiteness and imprecision of behavioral measurements, intrinsic incompleteness of preferences, interpersonal observability of preferences, perturbation of beliefs by attempts to measure them, separability of beliefs and tastes, effects of unobserved prior stakes in events, state-dependent utility for outcomes, and source-dependence of aversion to uncertainty. What is or is not possible here provides some perspective on what may happen in settings where 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 notions of strategic rationality. The no-arbitrage requirement leads straight to the conclusion that correlated equilibrium rather than Nash equilibrium is the fundamental solution concept, and risk-neutral probabilities have a role to play in modeling uncertainty aversion in this setting. 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. Robert Nau is a Professor Emeritus of Business Administration in the Fuqua School of Business, Duke University. He received his Ph.D. in Operations Research from the University of California at Berkeley. His research deals with mathematical models of decision-making under uncertainty, and his papers have been published in journals such as Operations Research, Management Science, Annals of Statistics, Journal of Economic Theory, and the International Journal of Game Theory. Throughout his career he taught a Ph.D. course on rational choice theory that drew students from other departments and schools at Duke University, as well as graduate courses in decision modeling and statistical forecasting.