I have been enjoying reading Ken Binmore’s new book “Rational Decisions”. I am currently only halfway through it but already it has proved to be a stimulating survey of decision theory.

According to its blurb, the aim of the book is to show why Bayesian decision theory was rightly restricted by its strongest proponent, Leonard Savage, to what he called “small worlds” and why its application to “large worlds” is misguided.

A small world is one in which Savage believes you can “look before you leap”. That is, you can take into account all possible actions, outcomes and future pieces of information when forming subjective probabilities. This is perfectly reasonable for suitably rarefied models of the world but is certainly not the case in more complicated “large world” situations. According to Savage, these large worlds are those where you can only “cross certain bridges when they are reached” and where the possibility of surprises means it is foolish to obstinately stick to a plan.

Any serious discussion of decision theory must grapple with probability theory in its many formulations and philosophical slants. Binmore does this in a very readable manner through expositions of key definitions and paradoxes that their highlight differences. During the discussion of the measure theoretic basis of probability Binmore wheels out the usual suspects—the Banach-Tarski paradox and Hausdorff paradox—to demonstrate the counter-intuitive implications of the axiom of choice.

Rather than arguing why these assaults on common-sense should have the axiom of choice sent off the field, Binmore unasks the question:

I think it is fruitless to ask such metaphysical questions. The real issue is whether a model in which we assume the axiom of choice better fits the issues we are seeking to address than a model in which the axiom of choice is denied.

His argument for leaving open the possibility of using the axiom of choice is that it is equivalent to “saying that nature has ways of making choices that we can’t describe using whatever language is available to us”. Hamlet puts it more poetically: “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy”, which leads Binmore to coin

The Horatio Principle:

Some events in a large world are necessarily nonmeasurable.

I am also wary of metaphysics but my response to the paradoxes arising from assuming the axiom of choice was to choose to not believe it and embrace my inner intuitionist. Binmore’s response is arguably more subtle and sophisticated. Rather than outright denying the axiom of choice he reserves the right to assume it when its useful and deny it when it is not. Hopefully, the remainder of the book expands on this position.

My only concern with following his lead in this type of hedge is that I doubt my ability to know ahead of time when assuming the axiom of choice will not result in paradox.

Mathematics, after all, is a very large world.

Mark Reid March 3, 2009 Canberra, Australia