h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

These assumptions seem to rule out all the preference functions I can think of. First past the post with two policies will only satisfy the 'must always get a result' condition if there are an odd number of citizens. Proportional representation fails on the 'removing irrelevant policy' count.

So what the theorem seems to say is that, although these conditions seem sensible, they're not. They lead to a silly result.

Also, the only part of the proof that does not follow immediately from the definitions is the bit about reducing any system to one of only two voters, so I think it would have been more useful to present that bit and not the bit that was presented.

That's a fair criticism of the usefulness of the theorem. All it does is rule out a perfect voting system, it tells you nothing at all about whether or not you can get very good voting systems, or about which voting systems are better than others. It would be interesting to know if there is a mathematical (rather than intuitive) way of comparing different voting systems. Anyone know anything about this?

One thing that Arrow's theorem definitely does do is undermine the claim of economists that their subject is logically coherent, since the entire subject is predicated upon a falsehood (the existence of a social welfare function satisfying the conditions of Arrow's theorem). It is a standard result that any statement can be proved starting from a falsehood, and so anything can be proved on the assumptions of mainstream economics, making it a bit dodgy.

First Past The Post can be combined with, for example, a random tie-breakers {as in the UK} to satisfy the 'must always get a result' option. I can't figure out what you mean by PR failing the "removing irrelevant policy" count - Arrow really only applies to methods which result in a single candidate/policy winning - that's the 'must get a result' requirement...

Dogster asked for mathematical ways to compare electoral systems - umm - I could write a book on this - but basically, there are *lots* of different criteria, and some methods pass some of them, and some methods fails some of them. The question is, what criteria are really important - and sadly there is no agreement here.

Here's one criterion which works reasonably well. Each voter has a number between zero and one to each candidate, according to their normalised social utility for that voter. The voters vote. The candidate with the highest average social utility should be elected. The tricky bit is figuring out how voters should vote, especially if they are given partial information on other voters (IE, polls). There are also those who doubt the existance of social utilities, or the meaningfullness of normalised social utilities.

Dang - I need to go write entry...

Isn't what you've described with normalised social utilities just an explicit voting system rather than a criterion to judge between voting systems?

What I was thinking about was perhaps something that said, for example, "For voting system X with n voters and m policies to choose from, Arrow's conditions will be met in all but n^2 of the n^(m!) possible sets of preference relations". Or maybe, "For voting system X, assuming a relatively homogeneous population of more than 100,000 the probability of a failure to satisfy Arrow's conditions is less than 0.0001". That sort of thing.

No - and there's a very good reason why not... If you try and use normalised social utilities a a voting system then you run into *horrendous* strategy problems - and end up electing a candidate with potentially very low utility... So it is one of those criterion which you can get close to, but cannot ever achieve...

Rates of failure of Arrow - Arrow *can* be passed in precisely those cases where there is no Condorcet paradox. If there is a Condorcet paradox, then whoever you elect, you will fail IIAC, so you *must* fail.

A few facts about that:

A) As the number of policies increases, the chances of a paradox increases. (and vica versa)
B) The higher the correlation between voters, the lower the chances of a paradox. (and vica versa)