Arbitrage and Rational Decisions

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. Professor Nau is an internationally known authority on mathematical models of decision making under uncertainty. His research has been supported by the National Science Foundation, 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. He was a co-recipient of the Decision Analysis Society Best Publication Award. One of the themes in Professor Nau’s research is that models of rational decision making in various fields are linked by a single unifying principle, namely the principle of no-arbitrage, i.e., avoiding sure loss at the hands of a competitor. This principle is central to modern finance theory, but it can also be shown to be the fundamental rationality concept that underlies Bayesian statistics, decision analysis, and game theory. Professor Nau has taught the core MBA courses on Decision Models and Statistics in several programs, and he developed an MBA elective course on Forecasting which he has taught throughout his career. He also teaches a course on Rational Choice Theory in the Ph.D. program that draws students from other departments and schools at Duke University.

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