The Meaning of the Arrow Theorem

Recently I finished reading the book Gaming the Vote by William Poundstone. I also assigned part of it to my Ethics & Economics Challenge students. It’s a fun and informative read, draping heavy-duty political science in engaging story-telling. (My post at e3ne.org on the topic is here.)

The book’s central thesis is that the American electoral system is irrational, and that range and approval voting methods provide obviously superior alternatives to the plurality rule for “single-winner” elections. Along the way, Poundstone discusses the Arrow Theorem and why it provides no obstacle to making a comparative judgment among voting rules.

Arrow’s impossibility theorem says that no social choice rule (method for coming up with a social preference ordering over possible alternatives) can satisfy the criteria of non-dictatorship (no one person can make the decision for the whole group, irrespective of the preferences of the rest), universal domain (no preference orderings are simply ruled out of order), Pareto optimality (if everyone prefers X to Y, so should the group), and independence of irrelevant alternatives (changing your relative ranking of X and Z shouldn’t affect your choice between X and Y). In simple language, the Arrow theorem says that there’s no such thing as a “will of the people”: only individuals have preferences.

Poundstone takes issue with that interpretation of the theorem, arguing that the “independence of irrelevant alternatives” criterion should be relaxed or removed. In essence, Poundstone believes that we can make interpersonal comparisons of utility (utility as cardinal, not just ordinal), and that once we do so, we can come up with some social choice rules that are objectively superior to others, because they result in more aggregate human welfare.

The assumption of cardinal, interpersonally comparable utility lies behind the case for range (or score) and approval voting as alternatives to plurality rule. The former methods are said to result in less “Bayesian regret” when used either sincerely or strategically. For instance, plurality voting leads to the “spoiler effect” (Nader causing Gore to lose to Bush) and lots of tactical voting (Nader supporters voting Kerry instead). Sometimes it can even result in victory for a candidate that would lose by a majority to every other candidate, or simply fails to choose the candidate that would beat every other candidate (Poundstone discusses how Stephen Douglas likely would have won a pairwise majority vote in 1860 rather than Abraham Lincoln).

From page 239 in the Poundstone book comes this graphic based on plausible simulations of different elections under various voting rules:

Bayesian regrets under alternative voting rules

Lower scores here are better, and thus you can see that range voting leads to overall least “regret” when voters are sincere. When voters are 100% strategic, range and approval tie (fully strategic range voters simply cast strategic approval-like votes: full marks to their favorite and the preferred candidate of the two with the best chance of winning, none to the rest).

Of course, the whole exercise depends on the notion that you can sum up regrets across voters. In some parts of life, we make rough-and-ready interpersonal comparisons of utility. When we speak of those “less fortunate,” we clearly have in mind the idea that the poor are less happy than the rich. The possibility of empathy seems to require a view that others are “more or less similar” to ourselves, including in their capacity for happiness. At the same time, the possibility of individuality seems to require that we acknowledge that others are “in some ways quite dissimilar” to ourselves. I can’t know what’s best for you, because your happiness has a large idiosyncratic, unmeasurable component.

Where does that leave us? The Bayesian regret calculations, it seems to me, give us good reason to favor range and approval voting over the current system, simply because in the absence of any other numeraire for making cost-benefit calculations of policies, the assumption of interpersonal comparisons of happiness, with everyone capable of the same amount of happiness (no utility monsters), is better than the alternative of throwing up our hands. But we still can’t get away from the fundamental insights drawn from the Arrow theorem: that only individuals have preferences and act on them, and that trying to maximize social welfare at the expense of respect for individuality is not only possibly unjust, but also irrational.