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July 1st, 2008

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.

July Books 1) Children of the Atom

1) Children of the Atom, by Wilmar H. Shiras

On that list of the 100 most influential sf books that was going round a year or so ago, this was the only one whose author I simply had never heard of. It is set twenty years in the future (ie 1973), and revolves around the assembling of a group of children whose parents all died after a nuclear accident in 1959, and who all display exceptional intelligence. At the end of the book, the children decide that they must integrate into the mainstream of society.

It's obviously at least in part a parable of fandom / geekdom, but a rather effective one. Definitely an under-rated classic.

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