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Ranked Choice Voting is Arbitrarily Bad

Recently, there’s been headway in adopting Ranked-Choice Voting, used by several states in the 2020 US Democratic presidential primaries and to be adopted by New York City in 2021.

For all its virtues, Ranked Choice Voting contains a number of risks, largely due to tactical voting and democratic illegitimacy.

First, a quick primer on existing systems.

The one we’re used to is called Plurality Voting, and is by far the simplest: Each voter casts a vote for one candidate, and the candidate with the most votes wins.

Though clear and intuitive, there are several problems best illustrated by example:

Tactical Voting for “Realistic Candidates”
Say public polls report:

  • 45% of voters prefer Alice
  • 45% of voters prefer Bob
  • 10% of voters prefer Carol

No matter how strongly voters support Carol, on election day, they would rather vote for Alice or Bob than “waste” a vote on a candidate who won’t win.

It’s worth asking why Carol was polling so low in the first place, but a common explanation is perpetuation through a party system. If Alice and Bob’s parties have won historically, the electorate may be locked into a perpetual two-party system, no matter how compelling a particular third-party candidate happens to be.

Loss of Popular Moderate Candidates
In another race, voters have real preferences such that:

  • 50% of voters prefer Alice > Carol > Dave > Bob
  • 50% of voters prefer Bob > Carol > Dave > Alice

Alice and Bob are both despised by half the population, yet one of them is guaranteed to win. Meanwhile, Carol has universal appeal, but would receive 0 votes in a plurality election, no matter how polarizing the other candidates are.

In a more extreme case, we might have:

  • 25% of voters prefer Alice > Bob > Carol > Dave
  • 25% of voters prefer Bob > Alice > Carol > Dave
  • 30% of voters prefer Dave > Alice > Carol > Bob
  • 20% of voters prefer Carol > Alice > Bob > Dave

Alice is the clear intuitive choice, but according to plurality rules, Dave ends up winning despite being despised by 70% of the electorate.

Tactics in Ranked Choice Voting
In theory, these are precisely the problems solved by RCV, but the general form I’ve outlined above can still apply.

Given real preferences:

  • 33% of voters prefer Alice > Bob > Dave > Carol
  • 33% of voters prefer Bob > Alice > Dave > Carol
  • 34% of voters prefer Dave > Alice > Bob > Carol

Each cohort knows that Carol is not a realistic threat to their preferred candidate, and will thus rank her second, while ranking their true second choice last. For any individual, this is a good strategy to maximizing the odds of their preferred candidate, but in aggregate, it leads to:

  • 33% of voters prefer Alice > Carol >  Bob > Dave
  • 33% of voters prefer Bob > Carol > Dave > Alice
  • 33% of voters prefer Dave > Carol > Alice > Bob

Leading to a victory for Carol, even though she was universally despised.

What’s Wrong with Tactics?
At some point, we might bite the bullet and say that all voting is tactical, in the sense that it incorporates outside information rather than merely expressing one’s preferences.

But as we saw in the last case, tactical voting doesn’t just skew results, it can have arbitrarily bad outcomes, like the election of a universally despised candidate, or the failure to elect a universally liked candidate.

Self-defeating Tactics Over Time
It is also worth introducing the concept of self-fulfilling and self-defeating tactics.

Self-fulfilling manipulation is what we’re used to. A candidate attempts to portray themselves as having a real chance of winning that is not yet guaranteed, thus giving voters the impression that their vote is likely to matter.

Self-defeating manipulation is weirder, and possibly much worse. Take our RCV scenarios again with equally sized cohorts:

  • Cohort 1 prefers: Alice > Bob > Carol
  • Cohort 2 prefers Bob > Alice > Carol

2 weeks before the election, a polling organization correctly reports these  preferences, thus creating the perverse reaction described earlier, and a shift to:

  • Cohort 1 states a preference for Alice > Carol > Bob
  • Cohort 2 states a preference for Bob > Carol > Alice

1 week before the election, the polling organization reports these new stated preferences. Voters see that their preferred candidate is no longer at risk of losing to their second favorite, and shifts back to to their real preferences:

  • Cohort 1 prefers: Alice > Bob > Carol
  • Cohort 2 prefers Bob > Alice > Carol

Thus restarting the cycle of polling > new tactics > new polling up until election day.

At this point, each voter is confused and uncertain, and votes according to tactical necessity rather than actual preferences. In this scenario, Alice, Bob or Carol could win, and it just depends on the particular rhythm of polling, speed of reporting, date of the election, and other factors largely incidental to voter preferences.

Condorcet Paradox of Circular Preferences
Finally, consider the case where ranked choice voting is simply incoherent:

  • 33% of voters prefer Alice > Bob > Carol
  • 33% of voters prefer Bob > Carol > Alice
  • 33% of voters prefer Carol > Alice > Bob

No matter which candidate is elected, there is an alternative preferred by 66% of the population, such that there appears to be no stable solution. If Alice is elected, the majority of voters would prefer that Carol be elected instead.

Note that this is not a coincidental feature of perfectly balanced distributions, we can just as easily have:

  • Cohort 1 (20%) prefers Alice > Bob > Carol
  • Cohort 2 (35%) prefers Bob > Carol > Alice
  • Cohort 3 (45%) prefers Carol > Alice > Bob

Where in the best case scenario, Carol wins, but 55% of voters would prefer Bob.

(In general, the worst case scenario depends on the number of candidates, such that in a 5 candidate race, an alternative could be prefered by up to 80% of voters.)

One might object that this is simply unrealistic. Are such circular preferences coherent?

We can generate intuition using a simple model where candidates and voters are mapped on 2-dimensional space, and rank candidates based on proximity. In the above example:

Circularity in Space

The key thing to note is that circularity exists across cohorts, but not within them. Each voter or cohort has a simple linear ranked order, it is only in aggregate that the paradox appears.

To be clear, this paradox exists equally in Plurality Voting, it just remains invisible. Even without stated preferences, real preferences could contain the same circular result.

But since voting serves as a coordination mechanism, this invisibility is a feature. With a single vote per person, simple plurality feels like a fair result. In contrast, Ranked Voice Voting risks the instability of a majority-preferred alternative, and undermines democratic legitimacy.