peteg's blog - choice - social choice

May's Theorem in Isabelle

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Tobias Nipkow has written an article about mechanising some recent proofs of Arrow's Theorem and Gibbard-Satterthwaite in Isabelle, and has kindly cited my clumsy attempts at similar things. In an attempt at differentiation I have polished up my proofs from Sen's magnum opus and added one of May's Theorem. As one might expect from a positive result in social choice theory there are endless reworkings of the original conditions in the literature since 1952, and I am not entirely sure what the state of the art is.

The proof itself is fairly straightfoward, although Sen makes an uncharacteristic slip by unnecessarily arguing by contradiction in his proof of my anonymous_neutral_indifference. When I'm next bored and idle I'll see if that makes any difference in the bigger scheme of things.

The hope is to get all of this into the Archive of Formal Proofs and obtain some feedback on the nastier parts of the mechanisation.

Once more:

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

An interested soul has written a great overview of May's life work.

The Cowles Foundation Monographs in Economics

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Amazing stuff, one can now get a whole pile of old economics monographs for free off the internet. The drawcard is Kenneth Arrow's Social Choice and Individual Values. Pulls the spare life out of the back pocket and is not seen for the next twenty years.

Gibbard-Satterthwaite in Isabelle.

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Finally completed a long-winded proof of the Gibbard-Satterthwaite theorem in Isabelle. This is a bit more complex than the proof of Arrow's Theorem because of the requisite ballot manipulations, though most of the ideas are the same. I followed Taylor quite closely. Of perhaps more general interest, in Section 3.3: The Equivalence of Arrow's Theorem and the Gibbard-Satterthwaite Theorem he says:

In geometry, we can say that two versions of the parallel postulate are equivalent if each becomes a theorem when the other is added as an axiom to Euclid's original four. Similarly, we say that two versions of the axiom of choice are equivalent if each becomes a theorem when the other is added as an axiom to the Zermello-Frankel axioms for set theory.

The reasons these assertions have formal content is that the results whose equivalence is being claimed are independent of the remaining axioms (assuming the consistency of the remaining axioms). Absent this condition of independence, the theorem asserting that 2 + 2 = 4 would qualify as being equivalent to Andrew Wiles' elliptic curve result that settled Fermat's last theorem (each being provable from the standard axioms of set theory with the other added — or not added, as it turns out).

Equivalence, however, is also used in an informal sense inspired by the formal notion above. We say that two theorems are equivalent if each is "easily derivable" from the other, where the ease of the derivation is measured (intuitively) relative to the difficulty of the stand-alone proofs of the theorems whose equivalence is being asserted. It is in this informal sense that we want to ask about the equivalence of Arrow's theorem and the Gibbard-Satterthwaite theorem.

You'll have to read his book for his conclusions on this very important topic.

As it stands I have a grotty proof for the linear-ballots case and hope to extend it to the non-linear case in the near future.

What would evolution do?

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When the problem gets hard it pays to see what nature does; this, I guess, is a slightly more abstract reformulation of the human-based computation gambit.

So, when faced with the apparently impossible task of cooking up decent voting procedures, what did evolution do? Apparently honeybees engage in a kind of range voting.

Discover Magazine: May the Best Man Lose

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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

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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 $US20, 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?

Adobe sicle 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.

/choice/social-choice | Link

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.

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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.