A brief exploration of the Independence of Irrelevant Alternatives (IIA) condition in voting systems, and why its absence makes voting systems behave irrationally.
I am currently writing a post on Arrow’s Impossibility Theorem, but I wanted to write a quick post about the killer property that makes that theorem possible: Independence of Irrelevant Alternatives (IIA).
IIA is the ultimately innocent-sounding condition that we would absolutely want in a voting system. But it is actually so restrictive that it is essentially impossible to satisfy, and thus leads to the impossibility result.
Rather than start with a heavy definition, I’d like to explain the loose intuitive idea with a famous joke about Sidney Morgenbesser and then build up to the formal definition and implications:
Morgenbesser, ordering dessert, is told by a waitress that he can choose between blueberry or apple pie. He orders apple. Soon the waitress comes back and explains cherry pie is also an option. Morgenbesser replies “In that case, I’ll have blueberry.”
At the risk of ruining the joke by explaining it, the humor comes from the fact that Morgenbesser clearly prefers apple to blueberry. Therefore, it would make little sense for him to ever pick blueberry if apple is on the menu. How could the presence of cherry pie cause him to change his mind about apple vs. blueberry?
At the individual level, it’s easy enough to maintain coherent preference orders. However, the key takeaway from Arrow’s theorem is that it becomes basically impossible to maintain such rationality when we start aggregating multiple coherent individual preferences together. While Morgenbesser would never do this consciously on his own, let us see how leaving the decision of dessert up to a group of rational dinner guests can lead to the same kind of irrationality that Morgenbesser’s joke illustrates. For some reason, the cruel host of the dinner party wants everyone to vote on the one dessert that everyone must eat.
Suppose the 100 guests at the dinner party are split into two groups as follows:
| Number of Guests | Preferences |
|---|---|
| 61 | Apple > Blueberry |
| 39 | Blueberry > Apple |
Simple enough, Apple wins by quite a lot (61% of the vote). However, suppose now the host of the party gives our guests the option to pick Cherry. Cherry is very polarizing, and Blueberry supporters despise it. On the other hand, while some Apple supporters also find Cherry detestable, most of them instead find Cherry to be a very exciting and delectable dish. Thus, they decide to support it instead of Apple, despite still preferring Apple to Blueberry. The new rankings are as follows:
| Number of Guests | Preferences |
|---|---|
| 30 | Apple > Blueberry > Cherry |
| 31 | Cherry > Apple > Blueberry |
| 39 | Blueberry > Apple > Cherry |
Taking only the first choices into account, Blueberry gets first place with 39% of the vote, Cherry comes in second with 31%, and Apple now comes in third with 30%. Unfortunately for Apple, Cherry’s entry into the race has caused it to lose. Blueberry wins with 39% of the vote, while Apple only gets 30%. Not a single voter changed their mind about how they feel about Apple relative to Blueberry, but the mere presence of Cherry has caused Apple to get fewer votes than Blueberry. We can also see that Cherry had no real chance of winning, since 69% of voters (over two thirds) preferred both Apple and Blueberry to Cherry. But Cherry was able to bleed off enough votes from Apple to cause it to come in last place.
And, unfortunately, not even Ranked Choice Voting (RCV) can save Apple here, because Apple came in last place. In RCV, until a candidate gets a majority, the candidate with the fewest votes is eliminated and their votes are transferred to the next available choice on those ballots. In this case, Apple, the only candidate who could defeat Blueberry, will be eliminated first. The Apple voters will then have their votes transferred to their second choice, which is Blueberry.
Thus, Cherry will lose in the final round to Blueberry 31 to 69. By voting honestly for Cherry first, the Cherry-first-Apple-second voters effectively ordered the dessert they liked least, Blueberry, instead of their second choice, Apple. This is the spoiler effect in action. That majority of 61% of voters still prefer Apple to Blueberry, but Blueberry wins because of the presence of Cherry.
And, worst of all, if just one of the Cherry-first voters had instead ranked Apple first and Cherry second, Apple would have survived the first round. Cherry would be eliminated instead, with the votes of the Cherry-first voters transferring to Apple, giving Apple a majority 61% to 39% over Blueberry. Apple would have won and ultimately defeated Blueberry in the final round, despite no voter changing their relative preference of Apple and Blueberry. Let that sink in: this voter still ranks Blueberry below both Apple and Cherry. But the preference between Apple and Cherry decides if Blueberry defeats Apple.
This is not some abstract or theoretical violation either. This has happened in real ranked-choice voting elections! Alaska’s 2022 Special Election and Burlington’s 2009 Mayoral Election had similar dynamics where spoiler candidates that had no chance to win caused a consensus option, who would have defeated all other candidates, to be eliminated first. Despite claims by RCV supporters that RCV solves the spoiler effect, and allows voters to rank their true preferences without fear of “wasting” their vote, that is simply not true. In those real elections, this spoiler effect was very real, and caused honest ballots cast by voters to be weaponized against their own preferences. For the honest truth about RCV, see my post on it.
This is loosely what IIA says should not happen. If we want to know how Apple compares to Blueberry in the final tally, we should be able to look purely at how voters feel about Apple relative to Blueberry, and not have to worry about how voters feel about Cherry. I like to call IIA a “stability” property. If we mess around with irrelevant candidates, the standing of the relevant alternatives that we didn’t adjust should stay “stable”.
Definition: (Independence of Irrelevant Alternatives) If every voter maintains the same relative ranking of $x$ and $y$ when moving between two profiles, then society should maintain the same relative ranking of $x$ and $y$ as well, even if we move around any or all other candidates.
There are systems which try to satisfy IIA directly, called “Condorcet” methods, which induce a ranking via direct majority comparisons. For instance, with the above pie scenario, we have that Apple defeats Blueberry and Cherry head to head. So society should rank Apple over Blueberry, and Apple over Cherry. Further, Blueberry defeats Cherry, so society should rank Blueberry over Cherry. This gives a societal ranking of Apple > Blueberry > Cherry, and Apple wins. However, this is not guaranteed to work!
Suppose the Blueberry supporters realize that Apple is the true consensus option, and is going to win. Thus, they manipulate the ballot by ranking Apple last, so that Cherry will defeat it in the head-to-head comparison. Once again, they are changing their preference between Apple and Cherry, but maintaining their preference of Blueberry over both options.
| Number of Guests | Preferences |
|---|---|
| 30 | Apple > Blueberry > Cherry |
| 31 | Cherry > Apple > Blueberry |
| 39 | Blueberry > Cherry > Apple |
Now, Apple beats Blueberry 61% to 39%, Blueberry defeats Cherry 69% to 31%, but Cherry defeats Apple 70% to 30%. This is a “Condorcet cycle” (think rock-paper-scissors), which means we can no longer simply induce a ranking via who wins by a majority. Condorcet methods try to remedy this by adding tiebreaking rules. A system like “Ranked Pairs” locks in the societal ranking via the strongest wins. The two strongest wins are Blueberry > Cherry and Cherry > Apple, so Ranked Pairs would declare the societal ordering Blueberry > Cherry > Apple, and Blueberry wins again!
This is an IIA violation because the only thing we changed between profiles was how a group of voters felt about Cherry and Apple. We didn’t change how any voters felt about Apple or Blueberry, but the societal order flipped from Apple winning to Blueberry winning. Not even a Condorcet method can escape an IIA violation because we can get contradictory majority comparisons.
Some systems that satisfy IIA include dictatorship, where a single voter determines the outcome, and non-responsive systems such as “Elect Alphabetically”, where the outcome is always the same regardless of the ballots. However, these systems are not at all desirable, and Arrow’s theorem really tells us that responsiveness is essentially incompatible with IIA. In my forthcoming post on Arrow’s theorem, I will also talk about an exception where you can satisfy IIA with infinitely many voters!
Technically, cardinal voting systems–where you score candidates independently and aggregate the scores–also satisfy IIA, in a sense. If you assume that scores are purely independent, then of course you get IIA. If a candidate is an “objective” 3/5 to me, then why would I change that score just because some irrelevant candidate enters the race?
However, if we interpret IIA as an ordinal property, then (most) cardinal systems violate IIA. Here’s an example:
| Number of voters | Score for A | Score for B |
|---|---|---|
| 4 | 5 | 4 |
| 1 | 4 | 5 |
| Total | 24 | 21 |
If the single voter maintains their preference for B over A, but changes their score for A to be 0, then A would only get 20 points and lose to B’s 21, even though the relative ranking of A and B is unchanged for all voters. This violates the strict notion of IIA, in terms of ordinal preferences. Nobody changed their ordinal preference between A and B, but the outcome changed from A winning to B winning.
So long as there is at least one score you can give between the minimum and maximum score, then you can generate an ordinal IIA violation like this. But what if you only have two scores? I encourage you to give it a try and see if you can find a violation (or explain why there cannot be one)!
Notice this change would also give that singular voter a beneficial strategy. They clearly prefer B to A, but they needed to exaggerate their score for A to be much lower than they might genuinely feel to get B to win. This is a violation of strategyproofness, which we will talk about in the next section.
I framed some of the above examples of IIA violations as an attempt to manipulate the election through strategy. In fact, this is not a coincidence. It turns out that IIA and strategyproofness are fundamentally linked. In a future post, I will be talking about this more in-depth in the specific context of Approval voting, but I’d like to share a brief preview of the connection here.
In a 2007 paper by Vorsatz, he proves a connection between strategyproofness and IIA. In any system which is “neutral” (no candidate bias) and strategyproof, we have that the system must satisfy IIA. This is a very strong result, because it means that where there are IIA violations, there are also strategyproofness violations. However, this is a particular statement of a much more general result by Blair and Muller that actually proves a sort of bi-directional connection between strategyproofness and IIA (though you need a few more conditions to get the converse direction), which we will not get into here.
The precise definition of strategyproofness is a bit technical, but I explain it in my post on the Gibbard-Satterthwaite Theorem. Suffice it to say, the gist of it is that changing your ballot should never lead to a better result according to your “true” ballot. For example, if your “true” ballot would say “I like Apple more than Blueberry”, then you shouldn’t be able to (or, from another perspective, need to) change your ballot somehow in a way that subsequently changes the winner from Blueberry to Apple. If you could do that, then you would have an incentive to lie, and a beneficial strategy.
Definition: (Strategyproofness) A voting system $f$ is strategyproof if for any voter, any two preference rankings $R_1$ and $R_2$, and any two profiles $P_1$ and $P_2$ where the only difference between $P_1$ and $P_2$ is that the voter has ranking $R_1$ in $P_1$ and ranking $R_2$ in $P_2$, then the outcome of $P_1$ must be weakly preferred to the outcome of $P_2$ according to the voter’s original ranking $R_1$.
\[f(P_1) \geq_{R_1} f(P_2)\]In other words, you should never be able to change your ballot in a way that gets you a better outcome according to your previous preference.
Lemma: (Vorsatz Lemma 1) If a voting system is neutral and strategyproof, then it satisfies Independence of Irrelevant Alternatives (IIA).\label{stratproof-iia}
Intuitively, if you could change the outcome between $x$ and $y$ by changing how your ballot says you feel about some irrelevant candidate $z$, then that itself creates opportunity for a strategy where you can get a better outcome depending on how you feel about $x$ and $y$. This is what we saw in the above pie example: by only changing how voters placed Cherry in their ranking, different groups of voters were able to switch the winner between Apple and Blueberry for their benefit.
This means that you can actually prove a system is not strategyproof by showing that it violates IIA, which is generally easier. Or, if you happen to have a system that you know is strategyproof, then you can immediately conclude that it satisfies IIA as well, so long as it is also neutral.
This also somewhat connects Arrow’s theorem with Gibbard’s theorem. Arrow tells us that IIA is essentially impossible to satisfy, and Gibbard tells us that strategyproofness is similarly impossible to satisfy. Since the impossibility of IIA implies the impossibility of strategyproofness, we get that Arrow’s theorem implies Gibbard’s theorem (very loosely speaking).
You’re on my blog, you knew Approval voting was going to come up! Approval voting is a simple system where voters approve or disapprove of each candidate, and the candidate with the most approvals wins. Approval voting is actually quite interesting because it does technically satisfy IIA in the ordinal sense, despite being a cardinal system.
In their seminal 1978 paper on Approval voting, Brams and Fishburn show that Approval voting is strategyproof on dichotomous preferences. Since Approval is also neutral, then by Lemma \ref{stratproof-iia} it must satisfy IIA on dichotomous preferences as well! Therefore, Approval’s technical exception to Arrow’s theorem and Gibbard’s theorem is not just a technicality or coincidence, but actually shows a very real and meaningful connection.
But what does it even mean to satisfy IIA on dichotomous preferences? Does that have any impact on how real voters use Approval voting? As a matter of fact, yes.
Intuitively, Approval satisfies IIA because the societal ordering of candidates is determined by the number of approvals they get. For candidate $x$ to get more approvals than candidate $y$, it must be the case that more voters approve of $x$ and not $y$ than approve of $y$ and not $x$. If we change how voters feel about some irrelevant candidate $z$, we change the number of approvals for $z$, but we don’t change the number of approvals for $x$ or $y$. Thus, we don’t change the societal ordering of $x$ and $y$.
For voters who approve of both or neither, we can also change their approvals of $x$ and $y$ so long as we move them in the same direction. For example, if a voter approves of both $x$ and $y$, we can change their ballot to approve of neither, and that would not change the relative ordering of $x$ and $y$ since that does not change the difference in their approval counts. Similarly, if a voter approves of neither $x$ nor $y$, we can change their ballot to approve of both, and that would not change the relative ordering of $x$ and $y$ either.
This is true at the ballot level, but not necessarily at the level of general preferences unless approvals are truly independent. For example, if Cherry pie enters the race, that doesn’t have to change how I feel about Apple or Blueberry, and whether or not I approve of them.
However, maybe Cherry is so good that I no longer approve of Apple, because compared to Cherry, Apple is no longer good enough to be worth approving. In that case, the presence of Cherry would change how I feel about Apple, and thus change my approval of Apple, which could change the societal ordering of Apple and Blueberry. This is a violation of IIA at the level of general preferences, but not at the ballot level, since I actually changed how I expressed my preference of Apple relative to Blueberry on my ballot when I removed my approval for it (from strict preference to indifference).
That said, the fact that Approval does satisfy IIA at the ballot level is still a nice assurance, because it guarantees that so long as you maintain the same preferences between $x$ and $y$–such as an Approval of $x$ and not $y$–you will never accidentally cause $y$ to suddenly win over $x$ just because you change your approval of some irrelevant candidate $z$.
This can happen in a system like RCV, STAR, non-Approval SCORE, Condorcet methods, and many others, where you are already saying “I want $x$ more than $y$” but might need to change your ballot in some way to get $x$ to win over $y$, and have that preference actually be respected by the system. To never have to worry about such a case ever occurring is a nice stability property to have, and it is something I really appreciate about Approval voting.
To make this concrete, here is one plausible strategic equilibrium in Approval voting using the same dinner-party preferences:
| Number of Guests | Ranking | Approval Ballot |
|---|---|---|
| 30 | Apple > Blueberry > Cherry | Apple |
| 31 | Cherry > Apple > Blueberry | Cherry, Apple |
| 39 | Blueberry > Apple > Cherry | Blueberry |
Approval totals are Apple 61, Blueberry 39, and Cherry 31, so Apple wins.
In this scenario, which some perceptive readers may recognize as the equilibrium under Laslier’s Leader rule, no voter has an incentive to change their ballot. Every voter is currently doing everything they can to get the optimal possible outcome for themselves. No voter (or group) has some possible clever strategy to change the outcome to something they would prefer more. And the consensus option, Apple, wins.
There is, however, one more cute way to see that Approval satisfies IIA. We said earlier that Condorcet methods try to satisfy IIA directly by looking at majority head-to-head comparisons. The only time you get an IIA violation is in the case of a cycle. Since Approval is provably a two-tiered Condorcet method which never admits a cycle, Approval is technically the only “Condorcet” method that satisfies IIA (at the ballot level).
IIA is a controversial property, to say the least. On one hand, it seems so desirable! On the other hand, Arrow tells us it’s essentially impossible to satisfy. The fact is that IIA’s impossibility really exemplifies the basic idea that aggregating many coherent individual preferences together necessarily results in preferences that may no longer be coherent.
We have to live with the fact that in any reasonable voting system, an irrelevant candidate can change how society will treat relevant candidates.
If there’s a pass/fail criterion worth dying on a hill for, IIA is not it. But it’s still a very nice property to have, and the fact that Approval voting satisfies it–even just at the ballot level–is a comfy little guarantee. Not necessarily a reason to say Approval is the only good system, but it’s a nice feather in its cap.
Blair, D., & Muller, E. (1983). Essential aggregation procedures on restricted domains of preferences. Journal of Economic Theory, 30(1), 34-53. https://doi.org/10.1016/0022-0531(83)90092-3
Brams, S. J., & Fishburn, P. C. (1978). Approval Voting. The American Political Science Review, 72(3), 831-847. https://doi.org/10.2307/1955105
Fishburn, P. C., Arrow’s impossibility theorem: Concise proof and infinite voters, Journal of Economic Theory, Volume 2, Issue 1, 1970, Pages 103-106, ISSN 0022-0531, https://doi.org/10.1016/0022-0531(70)90015-3
Maniquet, F., & Mongin, P. (2015). Approval voting and Arrow’s impossibility theorem. Social Choice and Welfare, 44(3), 519–532. http://www.jstor.org/stable/43662604
Vorsatz, M. (2007). Approval voting on dichotomous preferences. Social Choice and Welfare, 28(1), 127–141. http://www.jstor.org/stable/41106808
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