A Question About Arrow's Impossibility Theorem

Arrow's Impossibility Theorem is a theorem about voting methods that says that any voting method devised cannot have all of the following properties

• unanimity, if everyone prefers A over B, then A will be ranked higher than B.
• independence of irrelevant alternatives, if everyone prefers A over B, if asked to choose between A, B, and C, A should still rank higher than B regardless of how C is ranked.
• transitivity, if everyone prefers A over B and B over C, then A should be preferred over C.
• non-dictatorship, there should be no individual whose choices determine the overall results for society.

Now, I certainly agree that the first three are essential for a good voting method, put I'm uncertain against the fourth. Obviously, having a voting system that just follows the choices of one voter is a bad idea, but this is not the way the criteria is generally considered. The proofs I've read through has shown that if one starts changing voters preferences in a systematic but arbitrary way, there will be some voter who when they change their vote, the final result will change as well, and this voter in fact acts as a dictator.

The existence of such a dictator does not seem to be such a big problem. This is not a dictator who gets to decide the outcome. This is a dictator who is the straw that breaks the camel's back. Also, it is impossible to determine beforehand which voter will be the dictator. Finally, in a real election, there will generally be many people whose vote matches the final outcome, but that does not mean that only their vote determined the result.

Given that I don't really see why the existence of such an arbitrary dictator is a big problem, it does detract from the significance of Arrow's Impossibility Theorem. It's interesting, but not really relevant to anything (admittedly true of much of maths on first sight). If I were to design a voting method, I'd be more concerned about the transitivity property, which is much more troublesome just on its own.

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Ahead of his time

It would appear that Leonhard Euler has a number of papers up on arXiv.org. This is doing pretty good for a guy who died in 1783, a good two hundred and some years before arXiv.org started.

The papers are actually in the history section, but it's still pretty cool that they are there. I haven't actually read any yet, but I may. We'll see if any of the titles sound particularly interesting.

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Euler or Broke

Two weekends ago via XKCD I found out about Project Euler. It's a series of maths/computer programming problems to be solved. It's proved an interesting project to while away my hours. I don't think I've played Civilization IV since I started on the problems.

I think though I've now reached the point where I've got most of the low lying fruit. The first problems were easy enough, but now they're taking a lot longer to solve. It also seems to be a test of the strengths of your language of choice. So far I've been using a Matlab clone. On one problem it wasn't giving me the right answer despite everything I did to refine my code. When I put the same algorithm into Mathematica, it gave me the correct answer right away. That really annoyed me. I'm not sure if I make a full switch, since that'll take a fair bit of learning to get up to the same speed on Mathematica as I'm at with Matlab, although it seems to make a few things a lot easier. One problem can be done in one line of Mathematica that took me about 60 lines in Matlab. Admittedly it was a rather crude algorithm but still, that's a big change, and it ran a lot quicker too.

There is no doubt though that I will not complete all the problems any time soon. Problem #202 seems well beyond me (take a look for yourself if you think I'm wussing out), and there are more, tougher problems on the way.

I have learnt a few things along the way, and it's been good to flex the old maths muscles again.

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A Maths Quiz

If I were a Springer-Verlag Graduate Text in Mathematics, I would be Frank Warner's Foundations of Differentiable Manifolds and Lie Groups. I give a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. I include differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provide a proof of the de Rham theorem via sheaf cohomology theory, and develop the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find me extremely useful. Which Springer GTM would you be? The Springer GTM Test

This is a little surprising because for all the maths I've studied, I've never even looked at Lie groups, which looking back seems a little odd given nearly everyone else I knew doing physics knew something about them.

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