Notes on Arrow's Impossibility Theorem

Arrow's impossibility theorem has been so worn-out by pretentious youtubers that it is hard to take it seriously anymore. Sure, democracy is mathematically impossible, whatever. At the same time, it is weird that we are still using plurality systems for most elections when their limitations are so well documented.

I have gotten into Arrow's theorem a couple of times, but after a while the details get fuzzy. I have never written about it though, so here are some notes hoping that they will help me (and you!) to remember the most interesting bits.

What does it say?

The theorem is about social choice. For example, given two options A and B, where some people prefer A over B and some others prefer B over A, how can we aggregate all those individual preferences into a single order that represents the will of the people? So basically voting.

Arrow used an axiomatic approach where he defined some conditions that a fair social choice mechanism should comply with. He then showed that this specific set of conditions is self-contradicting, i.e. no mechanism can meet all of them at the same time.

The axioms are:

• Unanimity: If everyone prefers A over B, the result should also rank A above B
• Independence of Irrelevant Alternatives (IIA): If we add another option C, it should not have an impact on the ranking between A and B
• Non-dictatorship: There is no single voter who controls the result

The proof then goes roughly like this:

1. Say we have a vote over the options A and B
2. By IIA, we can add another option C without changing the ranking of A and B
3. If no voter puts C in the middle, it can also not be ranked in the middle in the result
   • Since C is never in the middle, we can move around A and B without changing their relation to C
   • We can move A above or below B in the result by using unanimity
   • If C were in the middle in the result, moving A above or below B would also affect its position relative to C, which is not allowed according to IIA.
4. If C is always ranked at the bottom, it must also be at the bottom in the result (unanimity). If voters one by one move C to the top, there must be a first voter for which C switches to the top in the result. We call this voter the pivotal voter.
5. Now the pivotal voter puts C in the middle
   • If we ignore the bottom choice (by IIA), this case looks exactly like the case where the pivotal voter had C at the bottom, so their top choice must be at the top of the result
   • If we ignore the top choice (by IIA), this case looks exactly like the case where where the pivotal voter had C the top, so their bottom choice must be at the bottom of the result
6. Just as IIA allowed us to add C in step 2, we can also remove it again. We are left with the exact same vote we started with. However, we now know that the pivotal voter is a dictator over A and B.
7. If there are more than two option, it can be shown that the same pivotal voter is a dictator over all of them

Expressiveness

We cannot provide an accurate representation of the will of the people if we ignore a sizable chunk of the information. In Arrow's framework, a vote is an ordered preference of all options. It therefore contains strictly more information than a common plurality vote, where only the first choice is included.

Surprisingly, restricting or expanding the amount of information we model can help avoid Arrow's theorem, without having to let go of any of the axioms. For example, if we only allow voters to prefer a left-wing candidate over a center one if they rank the right-candidate lowest (see spatial voting), we do not get a dictator. This is still more expressive than plurality vote, just slightly restricted.

However, the obvious approach is to use an even more expressive model: A voting system where voters express how much they prefer one option over another by assigning score and where we simply add up those scores fulfills all of Arrow's axioms. Arrow himself was of the opinion that it is meaningless to compare "psychic magnitudes in voter's minds". But come on, Likert scales are a thing. Score voting is commonly used in practice, e.g. by judges in dance competitions or in google reviews — not so much for elections though.

Strategic Voting

Apart from the issue that a voting system does not capture all information, another issue occurs if voters provide inaccurate information about their preferences. For example, if a voter assumes that most other voters are indifferent to A and B, but they themselves have a slight preference of A over B, they might exaggerate that preference in order to move the result closer to their actual opinion.

The biggest issue with strategic voting is if it is used unevenly: If all proponents of option A vote honestly and all proponents of option B vote strategically, B has an unfair advantage.

Plurality vote is pretty bad when it comes to strategic voting. It is quite common to have to vote for a "lesser evil" to prevent an even worse option from winning. This is better in Arrow's model, because you can express both your first preference and your "lesser evil" preference. This removes a major inventive for strategic voting. However, it is still possible to gain an advantage by strategizing.

If everyone exaggerates, score voting effectively becomes approval voting, where scores are binary. Thinking about it another way: Score voting is approval voting with the option to add some nuance if you want to. The incentive to strategize can also be removed by aggregating by median instead of sum. But then we ignore the magnitude of many of the votes, which has issues of its own.

A related concept to strategic voting is strategic nomination, where additional options are introduced to strengthen or weaken another one. This is common in plurality votes. However, any voting mechanism that has the independence of irrelevant alternatives property is immune.

More Properties

There are many more desirable properties than the ones proposed by Arrow. We have already discussed resistance to strategic voting in the previous section. Here are some more:

• Anonymity is a stronger form of non-dictatorship that requires that all voters are treated the same (i.e. voters can be permutated without changing the result). Similarly, neutrality requires the same for candidates.
• Monotony means that if I improve my opinion on option A, it may only have a positive or neutral effect on the position of A in the result, never negative.
• The Condorcet criterion is a variant of unanimity that only requires a majority. It can lead to Condorcet cycles in cases such as ABC, BCA, CAB, where A > B, B > C, and C > A.
• Arrow's theorem is not the only impossibility theorem in social choice theory. Many other combinations of properties have similar issues. But even then, this usually just means that there are edge cases where the mechanisms behaves badly. So another property we can look out for is how common those edge cases are.
• There are also less quantitative properties. For example, voting mechanisms need to be understandable (i.e. simple) to have any kind of legitimacy. A common offender is the Schulze method.

Conclusion

Plurality voting is still bad, and finding a better mechanism is still hard.

In reality, we do not just aggregate static preferences, but talk to each other (if that is possible). Quantitative methods can help us structure discussions, e.g. by finding which options are even worth discussing, and they can help us to come to a result in the absence of consensus. But they are always embedded in a larger structure.

I don't know which voting mechanism I would recommend. But I would really like if we would start gathering more data (e.g. scores for all candidates) so we can study what would happen if we were to use a different mechanism.