The economics profession has been for a long time divided into two camps — ADs and ACs.

On the AD camp, there is me, claiming that a bit of exciting bizarreness brought to game theory by the axiom of choice is not worth the headache of checking measurability of strategies and the embarrassment caused by undetermined gale-stewart games; The Axiom of Determinacy, on the other hand, offers a paradise in which every function is measurable and every zero-sum game with perfect monitoring is determined. When pressed, we — that is to say, I — are willing to accept the axiom of countable choice (without which you say goodbye to pretty much all of analysis) and the axiom of dependent choice (which essentially says that if at every day you have an action that let you stay alive one day longer, then you have a strategy that makes you immortal). So, the AD camp should perhaps more accurately be called the ZF+DC+AD camp: Zermelo Frenkel + Dependent Choice + Axiom of Determinacy.

The AC guys insist of sticking to the axiom of choice, mainly, I suppose, because of finitely additive probabilities.

Well, here is a nice compromise, called ZFC+PD, suggested to me by Alexander Kechris (In fact, I was partially aware of it already through the work of Maitra and Sudderth — more on that in a moment,) which I hereby commit to accept if the AC-s will collaborate. PD stands for the axiom of projective determinacy. Projective sets are essentially everything you encounter in real life. Not only Borel sets are projective, but also the images of Borel sets under Borel maps (AKA analytic sets) and their complements, and their Borel images, etc. This is a nice sigma-algebra of sets, which is closed under more operation then Borel sets, and also admits nice uniformization theorems. Also, every projective map is measurable.

Projective determinacy is undecidable in ZFC, but from what I understand it is by now a rather acceptable addition to ZFC among sets theorists and logicians. Hence ZFC+PD.

AC-s get to keep their finitely additive probabilities; AD-s get to keep all the nice regularity and closure properties we like in our sets, every gale-stewart game which can be even vaguely described is determined. And, in ZFC+PD the optimal reward function in a Borel gambling house is a measurable function of your initial capital !

(jstor, pdf)

Nice, huh ?

## 3 comments

August 7, 2009 at 8:48 am

noam neer“The economics profession has been for a long time divided into two camps — ADs and ACs.”

what’s this got to do with economics? even (most) mathematicians don’t like this kind of stuff. my cousin told me that in the US the subject of set theory is quite dead, and mathematical logic is taught in the departments of PHILOSOPHY, not mathematics. anyway, the real question is whether you accept Martin’s axiom and if you do, do you believe that the communist regime was erected by mathematicians who wanted to solve the problem of maximum welfare using forcing.

BTW, check out this one-

http://israblog.nana10.co.il/blogread.asp?blog=635903&blogcode=11022833

August 7, 2009 at 10:57 am

EranDude, now I am confused. I think it is likely that you realize that `economic profession’ is a hyperbole — what I actually meant to say is `Me, myself, and I have been arguing for some time about the best axioms for doing game theory’. But I think it is not very likely that you think that I think that you realize this.

I suppose that’s the problem with irony — you never really approximate common knowledge. But better writers than I am can at least climb a bit higher in the belief hierarchy. Anyway, nice to see you around. I like the ears.

August 7, 2009 at 11:22 am

noam neerwell, I didn’t realize its only you, I thought there’s a community of (infinite) game theorists arguing about this point whilst calling themselves economists cause that’s a good way to get funding. would “real” mathematicians be interested in this field? specifically, what would M.T. say?

P.S. I’m not a “real” mathematician. while its true I’m irrational, I’m way to complex for that.