Reflections on Arrow’s theorem and voting rules

Harvard students did not actually register for classes until the end of second week of the semester, so it was customary to shop around for courses during the first week or so.

I was not entirely self-taught; in 1969 I took a seminar on Formal Models in Politics taught by the newly arrived and very junior assistant professors Michael Leiserson and Robert Axelrod—probably one of the first such political science courses taught anywhere outside of Rochester.

For a catalog of problematic (or ‘paradoxical’) features of many voting rules, see Felsenthal (2012).

Thus I define a voting rule as incorporating tie-breaking and similar mechanisms, though I will avoid the problem of ties in these reflections. I also acknowledge that this definition excludes approval and range voting as voting rules, since they do not operate on ballot profiles as defined in the Arrovian setup.

Of course, many distinct voting rules—for example, plurality rule, the Borda rule, Instant Runoff Voting (IRV), and so on—are equivalent to majority rule in the special case of two alternatives.

The Borda rule satisfies all of these conditions except IIA. Another definition of neutrality (e.g., Sen 1970, p. 72) is stronger in the multi-alternative case and itself implies IIA. However, the Borda rule is neutral in the commonsensical way defined above, even though it violates IIA.

Since I state Arrow’s theorem in terms of preference aggregation rules, I (like Penn 2015) treat transitivity of social preference as an independent condition. On the other hand, I take the domain of a preference aggregation rule to be all logically possible ballot profiles, so I do not treat Arrow’s ‘universal domain’ condition as an independent condition.

Arrow’s conditions can be weakened further in various ways; for an accessible survey, see Penn (2015).

This point is illustrated forcefully in a working paper by Dougherty and Heckelman (2017). They generate large samples of simulated three-alternative ballot profiles with varying numbers of voters derived from an ‘impartial culture’, an ‘impartial anonymous culture’, and ANES thermometer scores for candidates in several US presidential elections with significant third-party candidates. They then apply seven common preference aggregation rules (plurality, anti-plurality, Hare/IRV, Nanson, Borda, Copeland and pairwise majority rule) to those profiles and determine how frequently each rule violates one or more of Arrow’s (1963) conditions. With very few exceptions, at most one condition is violated at each profile and the culprit is either IIA or transitivity. More specifically, majority rule occasionally violates transitivity, while the other rules frequently violate IIA.

The letter writer in fact advocated the use of IRV, which indeed comes closer to meeting requirements (2) and (3) than ordinary plurality rule does, but does not fully meet them and, moreover, fails to meet one other requirement (non-negative responsiveness) that plurality rule does meet.

Indeed, even if honest ballot rankings imply a clear majority rule winner, it may be open to one or more voters to ‘contrive a (top) cycle’ by misrepresenting their preferences over other alternatives, which may then be resolved in a way favorable to their honest preferences.

The susceptibility of US elections to ‘spoilers’ (such as Nader in 2000) is the dominant and recurring complaint in William Poundstone’s book on Gaming the Vote: Why Elections Aren’t Fair (2008). In more formal social choice theory, ‘strategic candidacy’ (e.g., Dutta et al. 2001) and ‘independence of clones’ (Tideman 1987) pertain to related problems.

However, these two problematic features may tend to counteract one another; in any case, that is true with respect to plurality rule (Dowding and Van Hees 2008).

In particular, the notorious McKelvey (1976, 1979) ‘global cycling theorem’ pertaining to majority rule over a space of two or more dimensions does not imply that a cycle exists (or is even likely to exist) over a small set of arbitrarily selected points in the space.