# Arrow's Impossibility Theorem Disproved?!

### Not Really

If you’re on Econ Twitter you may have seen this tweet making the rounds a few days ago:

With no reference link and just a blurry picture of half a slide, the reason behind this tweet is more than a little cryptic. It’s based on this working paper from Harvard economist Erik Maskin.

The tweet, like my title, is hyperbolizing for engagement. Maskin doesn’t claim to disprove Arrow’s theorem. Rather, he claims that weaking one of Arrow’s conditions drastically changes the result while still preserving the ‘attractiveness’ or ‘reasonableness’ of the conditions as good characteristics of a collective decision making process. Even in this restricted claim though, I think that Maskin misses a big part of the reason for the strictness of Arrow’s assumptions.

## Arrow’s Conditions

Arrow’s Impossibility Theorem is a result about social welfare functions (SWF). These SWFs are mappings from many individual rankings over a set of options to a single social ranking which ideally reflects everyone’s interests. He proposed four conditions as reasonable characteristic that any social welfare function should have. These are

Unrestricted domain: SWF is defined for all possible individual preferences

Pareto principle: If all individuals prefer x to y then the SWF prefers x to y

Non-dictatorship: There is no individual who’s preferences always determine the SWF

Independence of irrelevant alternatives (IIA): If the individual’s relative rankings of x and y do not change, then the SWFs relative rankings of x and y cannot change, even if the rankings of x and y with respect to other options change.

## Maskin’s Argument

Maskin points out that the first three assumptions are weak and satisfied by almost any SWF you can think of. Even first-past-the-post satisfies conditions 1-3 for any number of alternatives. It’s condition 4 that narrows down this wide space of possible SWFs to zero. So what’s the point of this 4th condition anyway?

Maskin says it’s to avoid vote-splitting. He gives this as an example:

In these two scenarios many SWFs, like first-past-the-post, violate the independence of irrelevant alternatives (IIA). The relative rankings of Trump and Kasich are the same in both scenarios. 65% of the electorate prefers Kasich to Trump. But Rubio splits the first place votes in scenario 1. So first-past-the-post ranks Trump above Kasich in scenario 1 but Kasich above Trump in scenario 2 even though no one changes their opinion about which they prefer out of the two.

IIA would prevent scenarios like this, but Maskin says it’s stricter than it needs to be to avoid ‘spoilers’ or vote-splitting like the above scenario and that this strictness mean that SWFs which follow IIA throw away valuable information about preference intensities. Masking proposes this modification to IIA (MIIA):

If

(i) each individual ranks x and y the same way in the first profile as in the second, and

(ii) each individual ranks the same number of alternatives between x and y in the first profile as in the second,

then the social ranking of x and y must be the same for both profiles.

The first part of MIIA is the same as the original, but the second part makes an exception for when individuals switch what they rank in-between two options. For example, say we have two voters 1 and 2 and four options {A, B, C, D}.

1 ranks {A, C, D, B} and 2 ranks {B, C, A, D}. Importantly 1 ranks A>B and 2 B>A. Say that our SWF sticks to the strict IIA and ranks A>B with these preferences. But now the voter’s preferences have changed and they want to hold another election.

1 changes their ranks to {A, B, C, D} and 2 changes to {B, C, D, A}. On one hand, IIA says that the pairwise rankings where 1 prefers A>B and 2 vice versa have not changed, so the ranking of these two options in our SWF must also stay the same. On the other hand, Maskin’s MIIA points out that B is a better compromise now than A was in the first election. More formally, Maskin says that when someone’s preferences change from {A, C, D, B} to {A, B, C, D}, you can infer that their intensity of preference between A and B has decreased, even though this isn’t directly measured by the ranking.

The rest of Maskin’s paper is a proof that the Borda count is the only voting rule which satisfies Arrow’s first three conditions and the MIIA.

## The Problems

Maskin’s paper is interesting and it was a surprise when the Borda count popped out of the conditions he set down, but there are a couple of problems with his argument that IIA is too strict.

First is that the Borda count doesn’t actually avoid vote splitting by most reasonable definitions or his own informal definition. His informal definition:

In everyday language, candidate A spoils the election for B if (i) B wins when A doesn’t run, and (ii) C wins when A does run

In footnote 7 he defines gives a formal definition of vote-splitting/spoiling, but it puts an arbitrary importance on first place votes that the Borda count does not share:

“Thus, formally, A is a spoiler for B if B beats A when all voters rank A at the bottom, but C beats B when some voters switch to ranking A above B

C (with no other changes to the preference profile).”and

In this scenario Z is not a spoiler by Maskin’s definition only because no one ranks it first. But Z still wins the election for X just by running even though it had no chance of winning itself. Y beats X when all voters rank Z at the bottom, but X beats Y when some voters switch their rank of Z. The results of the Borda count are still very susceptible to siphoning by irrelevant candidates even if they don’t split the first place votes.

The second problem is that the analysis ignores an important justification for the full IIA condition: resistance to agenda setting. This is a confusing issue in the context of IIA because formally, the condition is only about changing voter preferences over a fixed agenda of options. But in his exposition of the idea, Arrow himself explains it in a situation where one candidate dies and has to be taken off of the ballot causing confusion decades later. Still, it is true that if IIA is satisfied, and individual preferences do not change, that adding an alternative to the agenda should not change the pairwise rankings of alternatives in the social choice function.

With the Borda count, control over ballot setting rules would become extremely important to the outcome of elections. This control becomes even more fine-grained if we move out of the context of elections where multi-dimensional candidates roughly average out onto a 1-D spectrum, and into the context of legislation. Here, there are so many laws across so many different dimensions that the agenda setter could essentially totally control outcomes if IIA is not satisfied. For example, imagine a vote in congress over whether or not to increase farming subsidies. Pairwise, congress does not want to increase them. Since the Borda count does not satisfy IIA, the farming lobby need only add a strategic option to the agenda that gets second place from enough voters to change the outcome without changing anyone’s position on farming subsidies.

## Conclusion

Ultimately, the point of Arrows theorem is that no perfect voting system exists. Maskin’s result is very interesting in that it is a pretty minimal relaxation of Arrow’s conditions and it produces an unambiguous single recommendation rather than a large set of possible voting rules. Certainly all the criticisms I levelled on the Borda count apply at least as much to plurality rule. Plurality rule is essentially the worst non-pathological voting rule there is so Borda would definitely be an improvement. The tweet that started this post is definitely misleading. Arrow’s theorem is as strong as ever, but Maskin convincingly showed that the Borda count is about as close as we can get to an optimal collective decision making rule.

It's weird that in arguing for relaxing Arrow's theorem, he mistakenly tightens the definition of spoiling in order to conclude that Borda's satisifies the relaxed theorem.