What the Arrow Impossibility Theorem does not prove

As some of you out there may have noticed, I haven't been blogging for months. No, this hasn't been due to physical illness or terminal blogger fatigue. At one point I just got distracted by a combination of factors, further distractions multiplied, and it's taken me a while to get back into the game. (Also, when you've left an important subject alone for a while, it's not always easy to catch up.) But I do plan to start up again soon on a regular basis, and I have a backlog of things I'd like to blog about.

Meanwhile, Norman Geras was kind enough to guest-post a (slightly amended) e-mail message of mine on normblog ... which I've cross-posted below. Among other things, this gives me another opportunity to put in a plug for Gerry Mackie's very fine and valuable book Democracy Defended.

—Jeff Weintraub

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Guest-posted on the weblog of Norman Geras (normblog)
August 8, 2011
What the Arrow Impossibility Theorem does not prove (by Jeff Weintraub)

[The piece below is the slightly amended text of an email sent to me by Jeff a few days ago and is posted here with his permission - NG.]

This follows up your two recent posts questioning the nihilistic/futilitarian interpretation of the political implications of the so-called Arrow Impossibility Theorem. According to a typical formulation by Chris Dillow that you quoted, Kenneth Arrow allegedly demonstrated that 'under reasonable assumptions, it is impossible to aggregate individual interests or preferences'. According to David Broomhead, Arrow 'proved' that 'No fair voting system exists if there are three or more parties'. And so on. Too many people, even ones who are otherwise intelligent, well informed, intellectually serious and analytically acute, seem to believe this sort of thing.

I agree with you that claims of this type, especially when they're put so strongly, largely fall into the category of pseudo-sophisticated fallacies. I would add that the theoretical and ideological outlook underlying the inclination to accept this conclusion and its anti-democratic implications is usually some version of atomistic utilitarianism, which nowadays often calls itself 'rational choice'. That outlook tends to be characteristic of economists as a discipline, with occasional exceptions, but it's far from restricted to them. I recognize, of course, that not all people who invoke this fallacy about the Arrow Impossibility Theorem fully appreciate the extent of its anti-democratic implications (though others do). But the fact is that this futilitarian interpretation of the AIT, in any strong form, is anti-democratic in effect even if not in explicit intent.

I also agree with you that a more reasonable way to characterize the implications of the Arrow Impossibility Theorem would be something along the following lines:

Now, on my understanding of Arrow's Theorem, it proves that if you make certain reasonable assumptions about the requirements a method for aggregating individual preferences should meet, there is no conceivable method that won't, in certain circumstances, generate apparently anomalous results. I won't detail all of Arrow's postulated requirements, but just give a couple of illustrative examples. [....]

But note, third and finally, that nothing here entails that all electoral methods must be equally fair or that there can be no fair method. That, anyway, is my own understanding, and I don't know why David Broomhead makes the claim he does to the contrary. True, there can be no fair method that will be free in all circumstances of the sort of occasional anomalies I've illustrated above. However, I don't see why Arrow's Theorem would prevent us from adjudicating on grounds of fairness between a one-person-one-vote system and, say, a system that awarded more votes to the rich than to the poor. Again, one argument for proportional systems is that it's unfair for a party obtaining 25% of the popular vote to have only 5% representation in the legislative assembly. Whatever counter-arguments there may be to this, I don't see how Arrow's Theorem rules out judgements of comparative fairness.

That's an admirably careful and temperate way to put it. But we can take this criticism further and deeper.

It so happens that an excellent book by the political theorist Gerry Mackie, Democracy Defended, undertakes a systematic, exhaustive, and (in my view) totally devastating critique of these very widespread futilitarian and anti-democratic interpretations of the Arrow Impossibility Theorem and its practical implications. Mackie's critique is so thorough that some readers may find it not just exhaustive but exhausting. But when a potentially harmful fallacy is so widespread and deeply rooted, and so plausibly seductive for many intelligent and otherwise sophisticated people, painstaking thoroughness can be a virtue. I am inclined to agree with Jon Elster's assessment (in a back-cover blurb):

This brilliant counterrevolutionary book makes a frontal attack on the widely accepted claim that Kenneth Arrow's impossibility theorem for social choice shows democracy to be impossible, arbitrary, and meaningless. In delightfully direct and jargon-free language, Mackie demolishes the theoretical and empirical bases for this claim, notably in the strong version defended by William Riker and his students. His careful and exhaustive re-examination of all the instances on which Riker based his arguments is particularly valuable. At the same time, he puts up a strong defense - two cheers at least - for the institutions of representative democracy.

I think the next time someone repeats those lazy (but temptingly provocative) clichés about Arrow having 'proved' that democratic collective decision-making is inherently impossible and/or fraudulent, it might be helpful to refer them to Mackie's demolition of this claim. And if they don't want to take the trouble to read through and consider Mackie's critique, then they can stop parroting the claim.

Incidentally, I also share your disagreement with Chris Dillow's Benthamite claim that any notion of the 'public interest' is inherently 'vacuous' and 'meaningless'. But that's a topic for a longer discussion.

Yours for democracy,
Jeff Weintraub