peteg's blog 2006-12-12-ArrowsTheorem.autumn

Discover Magazine: May the Best Man Lose

An oldie but a goodie about the vagaries of all voting systems, and specifically the one used in US presidential elections.

Alan D. Taylor: Social Choice and the Mathematics of Manipulation

This book is linked from a lot of social-choice related pages on Wikipedia (a great way to advertise your book, for sure), covers an impressive selection of topics and got a good review. It's available as an eBook for , or I could pay more, wait a month and get a dead tree from Alibris. What the heck, I think, how good can PDF DRM lockdown be these days?

I paid my money, Adobe Reader's iceberg functionality kicked in to download the book in a very mysterious way, and then crashed. No worries, I download it again, and this time it works. Not confidence-inspiring, and apparently eBooks.com has had plenty of experience on that score.

After futzing around with it for a while, the cons of the Digital Editions DRM become apparent. It seems to be locked to this machine. No other PDF viewer groks the file. The DRM-removing tools tend to not work for eBooks. Ultimately this means I cannot print the whole thing (which is as the publisher intends) or share it as one would by loaning the dead tree to someone. The pros are that one saves about a third of the cost and doesn't have to wait on the post, neither of which justifies such a loss of utility to me.

Sen's Liberal Paradox.

This proof is pretty straightfoward, but its implications are tricky to fathom. I tend to think that his notion of liberalism is pretty weird, and there's plenty of discussion on it. Again, while even Isabelle is convinced that the theorem is an analytic truth, there is plenty of doubt that Sen's mathematization is right.

For the keen there's an entire volume of Analyse & Kritik (1996 (18) Heft 1) dedicated to this topic. Eventually I hope to get around to looking at fellow Nobel laureate James M. Buchanan's criticism.

I've updated the document and also the darcs repo. Some linting, much more to do. Onwards! — to the Gibbard-Satterthwaite theorem.

The beginnings of a proof of Arrow's Theorem in Isabelle.

On the side I've been chugging through Amartya Sen's classic Collective Choice and Social Welfare (1970), trying to convince Isabelle of some of the classic results in what is ultimately a theory of voting. As Richard Routley observes in his Repairing Proofs of Arrow's General Impossibility Theorem and Enlarging the Scope of the Theorem (Notre Dame Journal of Formal Logic, Volume XX, Number 4, October 1979):

The importance of a logically adequate proof is in no way diminished because, as it fortunately turns out, the theorem is correct under the intended (if often inadequately formulated) conditions. But that makes it easy to say that it is trivial to fuss over quantificational details of the standard proofs (proof failure comes ultimately in every case from quantificational errors, omission of necessary quantifiers or mistaken orderings of quantifiers, both major sources of invalidity in logic and mathematics) for every economist knows what is meant by the theorem, that it is essentially correct and its proof intuitively clear, and that a rigorous proof can be produced. The claim is false, as will emerge, even of the economic textbook writers. The textbooks have failed to produce what it is essential to have, especially in the case of a theorem with such far-reaching consequences (even if it is after all only an exercise in second-order quantificational logic), namely a correct and rigorous proof. The history of mathematics is replete with cases where what everyone was thought to know proved false, or where what was intuitively clear turned out to be mistaken or correct only under restrictive conditions.

In other words, perfect for formalisation in Isabelle.

At this point I have formalised the definitions given by Sen (and others) and shown Arrow's General Possibility Theorem (commonly known as Arrow's Impossibility Theorem) and the positive result about Social Decision Functions (SDFs). It's a huge space and there's plenty more to do.

arrows_theorem.pdf: A rough Isabelle-generated document of the development.
darcs get http://peteg.org/isabelle/arrows_theorem/: A darcs repository for the development.