Dear SIPTA colleagues,

I’d like to spread a few words about the recent publication of my book “Arbitrage and Rational Decisions”.  Publisher links, supporting materials, and a free draft copy of the entire book can be found at this web site:  https://people.duke.edu/~rnau/book/Arbitrage_and_Rational_Decisions.htm

It was in progress for many years, dating back to the dawn of SIPTA, but the issues that it considers are foundational and still timely.  It presents a unified view of the foundations of rational choice theory:  subjective probability and expected utility theory, state-preference theory, non-expected utility theory, noncooperative game theory, asset pricing theory, and social aggregation. The unifying theme is the principle of no-arbitrage, i.e., avoidance of sure loss at the hands of other actors in your scene.  In the context of subjective probability this is de Finetti’s concept of coherence, but it applies across the whole spectrum and leads to representation theorems that have a common structure in which a hyperplane (or many thereof) separates the convex set of arbitrage opportunities from the convex set of acceptable bets or trades, and the coordinates of the normal vector(s) are the observable parameters of the decision maker’s mind.  Imprecision in probabilities, utilities, state prices, and noncooperative equilibria is a natural state of affairs within this framework.  A running theme is that it makes a great deal of difference whether money is available as a medium of exchange and betting and (less well appreciated) as a language for constructing common knowledge of  beliefs and tastes and reciprocal stakes in events in numerical terms.  De Finetti had the right idea in using monetary bets as a measuring device. Nonlinear and/or state-dependent utility for money and/or non-expected-utility preferences are not problematic.  They merely change the visible units of analysis from personal probabilities to risk-neutral probabilities, as in asset pricing theory.  Probabilities do not need to be uniquely separated from utilities. When the setting is a noncooperative game, this method of analysis leads (via a slight twist on de Finetti’s theorem) to the result that Aumann’s correlated equilibrium, not Nash equilibrium, is the fundamental solution concept, which is illustrated on the book’s cover.  The generic solution of a game is imprecise:  a convex set of correlated equilibrium distributions.  I hope that you may find some of these perspectives to be of interest.

    Cheers and best wishes,

        --Bob

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Robert Nau

Fuqua School of Business

Duke University

https://www.fuqua.duke.edu/faculty/robert-nau

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