Arbitrage and Rational Decisions: (Chapman and Hall/CRC Financial Mathematics Series)

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.

Table of Contents:
1 Introduction 1.1 Social physics 1.2 The importance of having money 1.3 The impossibility of measuring beliefs 1.4 Risk-neutral probabilities 1.5 No-arbitrage as common knowledge of rationality 1.6 A road map of the book 2 Preference axioms, fixed points, and separating hyperplanes 2.1 The axiomatization of probability and utility 2.2 The independence axiom 2.3 The difficulty of measuring utility 2.4 The fixed point theorem 2.5 The separating hyperplane theorem 2.6 Primal/dual linear programs to search for arbitrage opportunities 2.7 No-arbitrage and the fundamental theorems of rational choice 3 Subjective probability 3.1 Elicitation of beliefs 3.2 A 3-state example of probability assessment 3.3 The fundamental theorem of subjective probability 3.4 Bayes’ theorem and (not) learning over time 3.5 Incomplete preferences and imprecise probabilities 3.6 Continuous probability distributions 3.7 Prelude to game theory: no-ex-post-arbitrage and zero probabilities 4 Expected utility 4.1 Elicitation of tastes 4.2 The fundamental theorem of expected utility 4.3 Continuous payoff distributions and measurement of risk aversion 4.4 The fundamental theorem of utilitarianism (social aggregation) 5 Subjective expected utility 5.1 Joint elicitation of beliefs and tastes 5.2 The fundamental theorem of subjective expected utility 5.3 (In)separability of beliefs and tastes (state-dependent utility) 5.4 Incomplete preferences with state-dependent utilities 5.5 Representation by sets of probability/utility pairs 6 State-preference theory, risk aversion, and risk-neutral probabilities 6.1 The state-preference framework for choice under uncertainty 6.2 Examples of utility functions for risk-averse agents 6.3 The fundamental theorem of state-preference theory 6.4 Risk-neutral probabilities and their matrix of derivatives 6.5 The risk aversion matrix 6.6 A generalized risk premium measure 6.7 Risk-neutral probabilities and the Slutsky matrix 7 Ambiguity and source-dependent risk aversion 7.1 Introduction 7.2 Ellsberg’s paradox and smooth non-expected-utility preferences 7.3 Source-dependent utility revealed by risk-neutral probabilities 7.4 A 3x3 example of a two-source model 7.5 The second-order-uncertainty smooth model 7.6 Discussion 7.7 Some history of non-expected-utility 8 Noncooperative games 8.1 Introduction 8.2 Solution of a 1-player game by no-arbitrage 8.3 Solution of a 2-player game by no-arbitrage 8.4 Games of coordination: chicken, battle of the sexes, and stag hunt 8.5 An overview of correlated equilibrium and its properties 8.6 The fundamental theorem of noncooperative games 8.7 Examples of Nash and correlated equilibria 8.8 Correlated equilibrium vsNash equilibrium and rationalizability 8.9 Risk aversion and risk-neutral equilibria 8.10 Playing a new game 8.11 Games of incomplete information 8.12 Discussion 9 Asset pricing 9.1 Introduction 9.2 Risk-neutral probabilities and the fundamental theorem 9.3 The multivariate normal/exponential/quadratic model 9.4 Market aggregation of means and covariances 9.5 The subjective capital asset pricing model (CAPM) 10 Summary of the fundamental theorems and models 10.1 Perspectives on the foundations of rational choice theory 10.2 Axioms for preferences and acceptable bets 10.3 Subjective probability theory 10.4 Expected utility theory 10.5 Subjective expected utility theory 10.6 State-preference theory and risk-neutral probabilities 10.7 Source-dependent utility and ambiguity aversion 10.8 Noncooperative game theory 10.9 Asset pricing theory 11 Linear programming models for seeking arbitrage opportunities 11.1 LP models for arbitrage in subjective probability theory 11.2 LP model for for arbitrage in expected utility theory 11.3 LP model for for arbitrage in subjective expected utility theory 11.4 LP model for ex-post-arbitrage and correlated equilibria in games 11.5 LP model for arbitrage in asset pricing theory 12 Selected proofs Bibliography Index

About the Author :
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.