Log in

No account? Create an account

Previous Entry | Next Entry

Why Arrow's Theorem is wrong

I have never been all that convinced by Arrow's Impossibility Theorem, which "proves" that there is no perfect voting system and made its author the youngest ever winner of the Nobel Prize for Economics.

Arrow's original theorem stipulated five reasonable requirements of a fair voting method, two of which I don't have a problem with (that all voters' views should count, and that all outcomes are theoretically possible). The other three, however, are I think all questionable.

Arrow's stipulation of unrestricted domain or universality includes the requirement that a voting system should unique and complete ranking of societal choices. This is incorrect; most real elections require you only to sort the alternative options into one of two categories, winners and losers (indeed, in most elections in the US, Canada and the UK only one candidate needs to be designated as a winner, and we don't really need any information about the ranking of the rest). Arrow's insistence that the outcome of a "perfect" election system gives an individual and distinct ranking to all alternatives is unrealistic.

Arrow's stipulation of the independence of irrelevant alternatives (that if A beats B in a choice between the two, A should still beat B when C is also an option) is mathematically neat, but does not reflect real human behaviour all that well. It is often the case that the availability of a third option makes us look at the first two in a different way, possibly even reversing our preferences.

In the real world, when a candidate wins against numerically superior but divided opposition, our instinct is to blame the opposition leadership for being divided rather than blaming the system for penalising them for their division or the voters for failing to unite around one of them; and that instinct seems right to me. (Yes, I know that preferential voting 'solves' that particular problem in most circumstances, and I am very much in favour of it, but not for that reason.)

Arrow's fifth stipulation is of monotonicity (that A getting more votes should not lead to his getting a worse result, or B getting fewer votes should not lead to her getting a better result). This is the one most often used as a criticism of preferential voting systems like STV. I am not at all convinced that this is a real-world problem. A number of years ago I campaigned in a city council by-election in Belfast where the votes cast were as follows:

Alliance (my candidate) 3646
UUP 2805
DUP 2445
Green 89

Local government elections in Northern Ireland use STV (because, unlike in the rest of the UK, our elections have to be *fair*). The votes of the DUP and Green candidates were therefore redistributed between us and the UUP:

Alliance 3646 +214 = 3860
UUP 2805 +1783 = 4588

Frustratingly, but not surprisingly, the UUP candidate won, with DUP transfers. Had the UUP and DUP first preference tallies been reversed, our candidate would have won as the UUP voters' second preferences were much more evenly split between her and the DUP. (Votes in these elections are physically tallied in such a way that this information can be gathered by the keen observer, and you can bet we were observing keenly!)

This is the sort of situation where the fans of monotonicity can have great fun. If 400 Alliance voters had instead tactically supported the DUP with first preferences, the argument goes, the UUP would have been eliminated after the first round and Alliance would have achieved a better election result (probably winning) despite getting fewer votes. This is all very well, but quite irrelevant to the real world; we had no advance knowledge of the likely gap between the UUP and DUP candidates, and certainly not enough control over our own supporters to get one eighth of them to vote for their least favoured alternative rather than for us. The best way of improving your chance of winning under any preferential voting system is to *get* *more* *votes*, which is as it should be: to exploit any theoretical lack of monotonicity in the system requires superhuman knowledge, which is not available to most election candidates.

Anyway, I'm sure that Kenneth Arrow is a great mathematician and economist; I just question whether his theorem is quite the knock-out blow to fans of democratisation (and especially of preferential voting systems) as some seem to think.


( 7 comments — Leave a comment )
Jul. 1st, 2008 07:29 am (UTC)
I was always willing to accept Arrow's theorem as right, but irrelevant to the discussion at hand: creating an improved voting system. The (dishonest) aim of the appeal to Arrow's Theorem is "no system is prefect! therefore all attempts at improvement are futile!"

That makes no sense. Just because you can't achieve perfection doesn't mean you can't achieve improvement.
Jul. 1st, 2008 08:03 am (UTC)
Your criticism of the requirement for universality is in fact a criticism of representative democracy as a system of governance. When we vote, we're not actually aiming to have a specific person sitting in a specific chair in a building in London (or wherever), we're aiming to have our government achieve certain goals, and getting the person into the chair is the best method available to achieve those goals. But it's far from perfect as a way of doing so, and one of the ways in which it is imperfect is that it doesn't allow us to say what's most important to us from their proposed policies, what's less important, and what we're actually opposed to.

Your objection to the monotonicity requirement is also dubious. A flaw in an electoral system does not have to be exploitable to be unfair. If all the votes were thrown into a bin and the winning candidate selected by cutting cards, no one could exploit this to improve their chances, but you would hardly call it a suitable system.

The irrelevant alternatives requirement is the real weakness in Arrow's theorem, as it explicitly disallows negotiation and trade-offs to maximise everyone's satisfaction with the outcome.
Jul. 1st, 2008 08:28 am (UTC)
Your criticism of the requirement for universality is in fact a criticism of representative democracy as a system of governance. Well, I just intended it as a criticism of the relevance of Arrow's theorem to discussion of election systems! But even following your argument through, you don't need a ranking of policies, you need to know which one policy will be pursued; whether an alternative is the second-best or second-worst has no impact on reality if the best one has been chosen and implemented.

If all the votes were thrown into a bin and the winning candidate selected by cutting cards, no one could exploit this to improve their chances, but you would hardly call it a suitable system. Actually I think that there is a lot to be said for sortition! But my objection to monotonicity is not that it is Wrong in principle - it is obviously Right - but that it is undetectable in real-world conditions.

The irrelevant alternatives requirement is the real weakness in Arrow's theorem. I agree. Not only for the reason you rightly identify, but because it utterly disallows one of the single most readily observable real voter behaviours - coming out to vote for candidate A because of the mere presence of candidate B on the ballot, when had there been no candidate B they would have stayed at home.
Jul. 1st, 2008 03:40 pm (UTC)
> coming out to vote for candidate A because of the mere presence of candidate B

Fine, go for compulsory voting, as long as there is a "None of the above" and/or "Please don't redistribute my vote to X".
Jul. 2nd, 2008 12:43 am (UTC)
I'm in favour of that, but it is a minority view!
Jul. 8th, 2008 12:22 pm (UTC)
Your example shows a serious problem with monotonicity and alternatives; underlying Arrow is an assumption of political equivalence. Nobody could say about your constituency that there were more voters who inclined towards your good self than otherwise; the political distance from the UUP to the DUP is much less than from the UUP to the Alliance or the Greens. There being a large plurality of Unionists of one stripe or other, the outcome was far more democratic than an Arrow-compliant one would have been.
Jul. 1st, 2008 05:48 pm (UTC)

Yes, like a lot of these mathematical political/economic theories, they start from a not particularly practical set of premises...

Some of them make more sense in slightly weaker forms. Like Monotonicity could be replaced by the principle that getting one more vote always improves (or leaves unchanged) your position all other votes being equal, which is satisfied by STV. The "Independence of irrelevant alternatives" could be replaced by a conditional one that the outcome between A and B is unchanged by the presence of C given unchanged individual preferences. Which I suspect is what is actually being assumed anyway. Of course the presence of C generally will affect preferences between A and B as you say, but the principle makes more sense that way.

The complete ordering thing appears far stronger than necessary - but it is actually equivalent to the more comprehensible principle that the system ought to pick a winning option that is preferred to any other option in a pairwise vote. This, IIRC, is the effect sought by some voting system that sashajwolf has referred to, but whose name I forget. This sounds weaker - but if this principle holds, then if you apply it to all options except the winner, then it must pick a unique second placed option which is pairwise preferred to all other options, except the winner, and so on.

In fact though, the condition that the winner should be pairwise preferred to any other is, although at first sight reasonable, rather questionable. It gives, in effect, an absolute preference to a compromise solution, even if no-one particularly likes that compromise. (And of course the Arrow example shows that this is impossible anyway - it is easy to have A majority-preferred to B majority-preferred to C majority-preferred to A).

I think the conclusion of Arrow's theorem is not that elections are pointless, but that any system is going to involve a trade-off between various desirable principles.

STV, IMO, gives a pretty good trade-off. In fact, it sacrifices the complete ordering even when that is possible - as your example shows, supposing the DUP had got 400 more votes, thus eliminating the UUP, leading to the Alliance victory; the UUP had the property of being pairwise-preferred to any other option, and yet would not have won. But it on the one hand requires the winner to be pairwise-preferred to the (STV-)second most popular option, but at the same time requires that the winner actually has a certain amount of support in itself. This is probably a good idea, because you're not just trying to pick a position on a spectrum, but an individual representative and a (hopefully) reasonably coherent policy programme. So choosing the least-hated compromise is not necessarily always the best idea.
( 7 comments — Leave a comment )

Latest Month

November 2019


Powered by LiveJournal.com
Designed by yoksel