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Ranked-Choice Voting fail choosing a lunch spot

I'm excited in general about ranked-choice voting as a way to reduce polarization in the US. But recently I had the experience of trying to organize a lunch for a group of people where almost every method of vote tallying failed in some way. To protect the guilty, the indecisive, and the body politic of the United States, I've changed some of the details.

I really hate organizing lunch.

The setup

A group of people wanted to go to lunch. They whittled the options down to three:

  • Lefty's Beef. Big beefy sandwiches and fries cooked in beef fat.
  • Moderato's. Expansive menu of OK food, but we go here all the time and most people have gotten tired of it.
  • Righty Tighty Vegan. Lots of kale, coconut milk, and things with almond butter.

Sigh. Already you can see some of the issues. The final choice will disappoint and possibly enrage one or more people.

The ballots

Six people sent their preferences, with 1 being their top choice:

Place Allie Bob Carrie David Elaine Frank
Lefty's 1 2   1 2 3
Mod's 2 3 2 2 1 2
Righty   1 1 3   1

At first glance, Righty Tighty Vegan got a plurality of first-choice votes—but unfortunately not a majority. Given that Allie and Elaine didn't even put Righty on their ballots suggests some, ah, strong feelings about the place.

So the organizer decided to try a different method of counting.

Instant-Runoff method

Most states and localities in the US that use ranked-choice voting go by the instant-runoff method. Each round, the algorithm removes candidate getting the fewest votes, and then promotes the remaining 2nd-choice votes to 1st. Rinse and repeat.

In what has become our Mittagskampf, this means we eliminate Moderato's (which only got 1 vote) and promote the two 2nd-choice votes for Lefty's to first-choice. Now the results look like this:

Place Allie Bob Carrie David Elaine Frank
Lefty's 1 1   1 1 3
Righty   1 1 3   1

Lefty's wins, 4 to 3! Awright, let's get some BEEF!

Well, I don't have to tell you how Carrie and Frank feel about this, or how confusing Bob can get when picking lunch. (He might be a Libertarian.)

Plus, we haven't really solved the problem of polarization in American politics.

Promote everything?

OK, what if we promote all the second- and third-choice votes until we get a majority? After the first pass we get this:

Place Allie Bob Carrie David Elaine Frank
Lefty's 1 1   1 1 2
Mod's 1 2 1 1 1 1
Righty   1 1 2   1

Aha! Now we have Mod's with 5, Lefty's with 4, and Righty with 3. Except...still not a majority. And if we promote all the remaining 2s to 1s, we'll get Mod's 6, Lefty's 5, and Righty 4, which also doesn't seem fair.

To blazes with everything!

What happened here is that we the number of votes is a multiple of the number of options, so a deadlock is possible. Several other methods would guarantee a result of some kind, and incidentally favor Moderato's, but in no case would any of the choices break 50%. Basically, the Beef and Vegan camps will never agree on anything other than Moderato's, even though no one really gets excited about it.

However, unfortunately for some and to the delight of others, before we could figure out a fourth option, Gwen cast her vote and broke the tie. And then after Hank's and Irina's votes, we had another deadlock.

I really hate organizing lunch.

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