One of the most famous results in voting theory is Arrow’s Theorem, which states that no voting system can satisfy five reasonable-sounding conditions.1 Among the five conditions of Arrow’s Theorem, the hardest one to satisfy is called the independence of irrelevant alternatives, frequently shortened to the cryptic abbreviation IIA.
This is a formalized cousin of the spoiler effect: In a nutshell, an “irrelevant alternative” is a candidate who will lose. If a losing candidate can change which of two front-runners wins, the losing candidate can be said to “spoil” the election in favor of one of those two candidates.
In this article, I will discuss the formalization of IIA in Arrow’s paper, the paths forward from Arrow’s famous impossibility result from 1950, and the reason why we should expect candidates with low levels of support to be relevant to elections even if the electoral process complies with Arrow’s famous criterion.
The formal definition
Formally, Arrow defines voting systems as functions that take “individual orderings” - that is to say, individual expressed preferences - and turn them into a collective ordering. This implies that a voting system doesn’t just give us a single winner; it also gives us a second-place ranking, a third-place ranking, et cetera.2
Arrow’s motivating example is this: What happens if a candidate drops out of the race after voters have already cast their ballots? Shouldn’t this give us the same result as if the candidate dropped out before the election took place?3
Condition 3: Let R1, R2, and R1’, R2’ be two sets of individual orderings. If, for both individuals i and for all x and y in a given set of alternatives S, xRiy if and only if xR1’y, then the social choice made from S is the same whether the individual orderings are R1, R2, or R1’, R2’.
Unpacking the formal language, what does this say? Well, it says that if we have two different sets of candidates, such as {Alex, Bob, Carrie} and {Alex, Bob}, and none of the voters change their minds about whether they prefer Alex to Bob, then both elections should place Alex and Bob the same way relative to each other. This definition doesn’t actually require the extra candidates to be losing candidates.
If Alex defeats Bob in an election where Carrie participates, Alex should also defeat Bob in an election where Carrie doesn’t participate. If Carrie wins, with Alex placing second, Arrow’s IIA criterion is not violated. On the other hand, if Bob wins, with Carrie placing second and Alex placing third, Arrow’s IIA criterion is violated.
Why almost every voting system violates IIA
A positive result closely related to Arrow’s theorem is May’s theorem. Formalized by Kenneth May in 1952, May’s theorem tells us that in an election with only two candidates, a simple majority vote is the voting system that meets Arrow’s criteria.4 A voting system meeting all of Arrow’s criteria must rank two candidates based on pairwise majority vote between those two candidates.
However, pairwise majority votes can form a cycle instead of a coherent ranking. For example, three people can prefer spaghetti to lasagna, lasagna to pizza, and pizza to spaghetti. In general, if options are sufficiently distinct and similarly popular, these kinds of cycles tend to occur around 6-9% of the time, as shown in Chapter 3 of my dissertation.
The existence of these cycles is known as the Condorcet paradox, thus named because they were formally described in a 1785 paper by Marie Jean Antoine Nicolas de Caritat.5 The Arrovian IIA condition is deeply and inextricably linked to Condorcet’s work.
Elections carried out with a Condorcet method violate Arrovian IIA in situations where a Condorcet paradox takes place. Elections carried out with other election methods additionally violate Arrovian IIA whenever they violate either of the Condorcet criteria by failing to elect a Condorcet winner or by electing a Condorcet loser; essentially, Arrovian IIA implies that the final ranking of options must be the same as with each individual pairwise result.
Relaxing the independence of irrelevant alternatives
What soon became readily apparent is that Arrow’s independence of irrelevant alternatives is too strong of a requirement. As formally defined by Arrow, the criterion is simply too strict, even if the idea is useful. So, how should we think about IIA if it can’t be used meaningfully as a strict pass / fail criterion?
The answer is that we can (and should) think about the severity of the failure, and the type of conditions under which the failure occurs. There are four popular approaches:
Assert that Condorcet paradoxes will not occur in real-world situations.
Reframe IIA in terms of cardinal intensity of preference.
Reframe IIA in terms of ordinal intensity of preference.
Consider the probability of violating pairwise criteria across a range of electorates.
Denying that the problem exists is usually a popular approach to solving problems, but it tends to be risky. In this case, denying the existence of the problem makes it easy to endorse Condorcet methods as mathematically perfect.
In general, reframing IIA in terms of intensity of preference means that the best performers are approval voting or another score voting system if a cardinal (rating-based) approach is taken, or a Borda count if an ordinal (ranking-based) approach is preferred.
If an approval or score vote is reduced to two candidates, this means effectively disenfranchising sincere voters who had nuanced preferences between candidates, whose votes are effectively either reduced (in the case of a score vote) or eliminated (in the case of an approval vote).
Conversely, a Borda count ensures that all voters’ ballots still express any relative preference between any two candidates. So do other voting systems that ask voters to rank candidates, including Condorcet methods and ranked choice voting. Importantly, however, a Borda count will not violate IIA very frequently outside of a Condorcet paradox.6
While the margin for uncertainty is slightly higher for an approval vote, the frequency of IIA violations in approval votes is not expected to be much higher than the frequency of IIA violations for a Borda count.7
IIA violations are much more likely to take place with systems that use a plurality vote, including the systems widely known as ranked choice voting / instant runoff voting. And this, I think, is the important level of reasoning to consider; not whether IIA will be violated, but how frequently it will happen. Among voting systems that ask voters to rank candidates, the Borda count, Nanson’s method, and more complex Condorcet methods are far more consistent if candidates are removed.
The paradox of individual choices
When analyzing voting systems, it’s traditional to assume that voters have consistent preferences. For example, a voter who preferred Biden to Clinton to Trump in a three-way contest ought to prefer Clinton to Trump in a two-way contest.
Let me change frames to talk about advertising and sales. Suppose there are two versions of a product: A basic functional model, and a deluxe model that costs about 20% more with several genuinely useful extra features functions. Many customers will stick to the basic model. This is reasonable, because it does the thing they want it to do but at a lower price.
However, something interesting happens if you introduce a third model, a luxury version priced at over twice the cost of the deluxe model. The luxury version has the features of the deluxe model, plus one or two extra features that really aren’t that useful. In some cases, the difference might just be cosmetic styling.
Suddenly, consumers who would never consider the luxury model flock to the deluxe model. Why? Because in a two-way comparison, the deluxe model looks like it’s a bad value. Both products have the same basic functionality, one costs more. In a three-way comparison, the luxury model looks overpriced … but in comparison, the 20% extra cost of the deluxe model looks like a small price to pay when you’re getting almost all the same features as the luxury model.
Picking the option with nearly the basic price and nearly the luxury features is usually a good choice. It’s a decent heuristic, a “rule of thumb” that can be used effectively in the absence of good information. And yet the use of that heuristic shows that individuals routinely violate the IIA criterion when making decisions by themselves.
This is a reason to take the IIA criterion with a grain of salt. While consistency of results after a candidate dies or is disqualified is desirable, even individual preferences are not always consistent, so it is no surprise that group preferences are not consistent.
The conditions can be described as treating all candidates fairly (i.e., all candidates can win), treating all voters fairly (i.e., no single voter is a dictator), being deterministically based on the ballots (i.e., ballots from voters are the only thing that matter), and being monotone. In this context, monotonicity means that a candidate won’t do worse if they gain popularity (or, conversely, that they won’t do better by losing popularity).
It may be that only the first-place rank outcome really matters.
This might seem unrealistic, but it happens. While Arrow wasn’t thinking of candidates dying suddenly, this is one of the situations in which a candidate may suddenly withdraw while the election is in progress. In the 1872 election, for example, presidential candidate Horace Greeley died at a very awkward point in the electoral calendar.
If there are only two possible candidates, satisfying IIA is trivial.
Who held the title of Marquis of Condorcet. No prior Marquis of Condorcet is considered historically noteworthy by modern standards, and the title became extinct when he died of political causes in 1794.
In my dissertation, I produced an “upper floor” estimate of IIA violations outside of Condorcet paradoxes of about 10%.
The difference is in behavioral assumptions. Generous behavioral assumptions lead to very Borda-like results for an approval vote. Ungenerous assumptions involve more plurality-like results.