Multi-round Voting
© 31 August 2021 Luther Tychonievich
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Alternatives to First Past the Post


In my last post I discussed first-past-the-post (FPP) voting as an oscillatory dynamic system that all-but ensures that a person with average views will never hold office. In this post I plan to skim over a few alternative voting systems.

As a reminder, FPP only works well when there are only two options. Political parties are ways of filtering many options into just two prior to the election.

Run-off systems

The basic principle of run-off elections is as follows:

  1. Run an election with many candidates

  2. If any candidate gets > 50% of the votes, declare them a winner

  3. Otherwise, remove some number of unpopular candidates and re-run the election with a smaller ticket.

Run-off elections can successfully narrow the pool down to just two candidates without the necessity of political parties. However, they are not very successful in practice because they require multiple rounds of voting, which is tedious enough that many voters only participate in some rounds. This, in turn, strongly encourages trying to get that 50% cut-off in the first round, which political parties are reasonably skilled at producing, leading to many of the same problems as FPP.

Run-off systems also have to decide which candidates to remove. A common approach is to remove all by the two most popular candidates or remove all who got less than some percentage of the vote. Both of these approaches have problems, as an experience from my university will illustrate:

Six years ago, the CS department was discussing changes to the Introduction to Programming course we taught and which was taken by students in a dozen different majors. We asked each department, “‍what programming language should we teach?‍” and got many answers: R, C, MatLab, LabView, Java, none had the majority. We could have tried to do a run-off election, but instead we asked “‍if you can’t have that language, what’s your second choice?‍” and almost everyone gave the same answer: Python. As an aside, six years later many of those same faculty now list Python as their first choice. Python wasn’t even on the first ticket and couldn’t have won any run-off, but when we switched to it everyone was reasonably happy.


Both challenges with run-offs are at least partially addressed by using ranked-choice ballots. In these ballots I don’t just mark my first choice: I rank all options from most-desirable to least-desirable.

Given a ranked-choice ballot, we can now perform “‍instant run-off‍” (IR):

  1. Collect ranked ballots with many candidates

  2. If any candidate gets > 50% over the first-choice votes, declare them a winner

  3. Otherwise, remove some number of unpopular candidates and redistribute ranks so that one each ticket the first-ranked remaining candidate is their new first-choice and so on; then repeat the check for a winner.

IR doesn’t remove the decision point on how to decide whom to remove, but it does give us more options. Like a standard run-off, we can remove the candidate that got the fewest first-choice votes, but we can also remove the candidate with the most last-place votes, or the candidate with the fewest votes in the top half of the ranking, or the candidate with the lowest average ranking, or …

Ranked choice can also be used to bypass districting: a process called “‍proportional representation by single transferable vote‍” (PR-STV) lets us have a single ranked-choice ballot for multiple positions and end up with candidates that represent the various views of the constituency in similar proportions to the constituency.

Better but imperfect

Instant run-off and single-transferable-vote are better than first past the post, but they aren’t perfect. And those are objective assertions, not merely opinion. Voting theorists have enumerated various desirably properties of a voting system and analyzed if various systems meet those properties. To see what I mean, let’s consider just three of the properties:

  1. Simplicity: any high-school graduate should be able to verify the outcome given the anonymous vote tallies, and if we can reduce the education level needed it’s better.

  2. Clones: in an election where two candidates are identical, one of the two should win if and only if one would win if the other dropped out.

  3. Honesty: you should never be happier with the outcome if you lied on the ballot (e.g. but marking your second-choice candidate as your first choice) than if you told the truth.

There are dozens of these properties, all clearly desireable, and first past the post satisfies very few of them while run-off, instance run-off, and single transferable vote satisfy many more.

However, no voting system is without flaws. In 1951 Kenneth Arrow proved what has become known as “‍Arrow’s Impossibility Theorem‍” which shows that no ranked-choice voting system satisfies both of two properties that jointly mean “‍if most people prefer X to Y, the system will not pick Y‍”. We can get around that theorem by adding more information to the ballot (perhaps trying to distinguish between “‍close second choice‍” and “‍distant second choice‍”, for example), but subsequent theorems such as Allan Gibbard’s 1973 “‍Gibbard’s theorem‍” shows the underlying conflict remains. No perfect voting system can exist.

However, these theorems and “‍better‍” judgements depend only on the election itself. My main problem with first past the post is that it creates two parties, both closer to the edges of political view than most constituents, who alternate power. If there have been similar analyses of the long-term impact of other voting systems, I have not yet seen them.


I opened the previous post by commenting on some campaign signs opposing single-party rule. I then explained why I believe alternating single-party rule is the steady state of any first-past-the-post representation democracy. And I’ve mentioned a few of the many voting systems that do not have those particular forces and thus likely won’t have that particular steady state.

What steady state would IR or PR-STV have on our government? I don’t know, but I’d love to find out. Unfortunately, I see no plausible path to changing our voting systems, which would require a constitutional amendment, meaning ⅔ approval of congress, meaning support from both of the parties I hope it will dethrone.

Practically, I do take a lesson from all of this. Being upset by what the government is doing at least half of the time is baked into the structure of our voting system. It does not mean that the people I disagree with are evil or stupid; it merely means they are representing an end of a continuum of thought that is unlike my position. And I find that a reminder that political opponents are rational and well-meaning is something I need often as I strive to be both kind and informed.

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