If every voter in an election submits a ranked list of their preferences, there are no ties, and all the candidates are ranked, what is the best way to mathematically decide the top candidate, the winner of the Election With access to a complete set of ranked ballots, which means we know every person's opinion, it seems like a clear winner should emerge.

But it doesn't.

The outcome of the election depends critically on what process you use to convert all those individuals' preferences into a group preference.

For the sake of example, let's say we're voting to determine the best color.

Let's start simple.

If there's only two options, like green and blue, we'd probably select the winner based on majority rule.

If the majority of people prefer green to blue, green wins.

Now let's try it with three colors, green, blue, and purple.

Everyone will submit a ballot with ranked preferences.

Each person writes one next to their favorite color, two next to their second favorite color, and three next to their least favorite color.

Using everyone's individual ranked preferences, we want to create a group ranking.

For simplicity, let's pretend only three people voted.

Here's their individual ballots.

What's the group's favorite color?

Each color was exactly one person's favorite, so that's not so helpful.

Let's try something like our majority rule strategy here.

When comparing two colors, what do people most prefer?

Two people ranked green higher than blue, and only one ranked blue higher than green.

So green beats blue.

2/3 of people ranked blue higher than purple, so blue beats purple.

And for the same reason, purple beats green.

The pairwise preferences form a cycle.

That's called the Condorcet paradox.

Even though each individual has strictly ranked their preferences, there's no way to rank their collective preferences.

There's no group favorite color.

This method of comparing each pair of candidates fails.

Instead, we'll need a voting system that compares all the candidates at once.

To help us review four common election decision methods, we'll expand our sample election, determining the best color, to include five colors, green, blue, purple, red, and orange.

We have 55 ballots with the complete ranked preferences of each voter.

Our task will be to turn these ballots into a winner, an overall favorite color.

Here are the ballots we received.

For example, 18 people ranked green first, red second, orange third, purple fourth, and blue last.

12 people filled out their ballots like this, 10 filled them out like this, nine like this, four like this, and finally, two people filled out their ballots like this.

The numbers from these sample ballots are from a great blog from the American Mathematical Society linked in the description.

First we'll look at the plurality method, also known as first past the post.

It's probably the simplest mathematically.

The winner is the color with the most number one votes.

In our best color election, green is the plurality winner with 18 ballots ranking it number one.

In practice, voters can just check one box, since choices two through five don't matter.

A plurality is not a majority.

Out of five colors, the plurality winner could have just slightly more than 1/5 of the votes.

Here in the United States, we tend to say that we use a majority voting method, but that's just an artifact of our essentially two-party system.

In most types of elections, including presidents, governors, et cetera, we use the plurality method.

But many other countries elect their president with a two-round runoff, or majority runoff, voting method.

The first round of voting is similar to the plurality method, just count how many number one votes each candidate has.

If one candidate has a majority, more than 50% of the votes, they win.

If not, the top two candidates face off in a second round.

In practice, many countries hold two separate elections where voters select a single candidate each time.

But using a single ranked ballot, we can simulate two elections.

In the color election, green and blue are the round one victors, but neither obtained a majority.

During the second phase of voting, everyone who originally voted for green or blue keeps their vote.

Then we look at each other ballot and ask, if this person is forced to pick between green and blue, which color do they select?

In other words, did they rank green or blue higher?

In this case, they all prefer blue to green, and blue wins the second round majority.

The third method is instant runoff voting, used by the Oscars to determine the best picture.

This method moves through several rounds, each time eliminating the candidate with the lowest number of votes.

The votes for the eliminated candidate are then redistributed based on the rank order of the remaining candidates.

In our color election, orange is ranked number one by the least people, only six, so it's eliminated.

Now cross orange off each ballot and re-rank the remaining colors.

On four of the ballots, this will bump blue up to first place, and on two of the ballots this will bump purple up to first place.

Re-tallying the first place rankings in round two, red has the least votes and is eliminated.

All nine of its votes moved to purple in round three.

Blue is eliminated, giving its votes to purple.

With a majority in round four, purple wins.

Finally, the Borda Count method is frequently used by sports leagues to determine award winners.

Each ranking is assigned a point value, four points for first, three points for second, two points for third, one point for fourth, and zero points for fifth.

A candidate's score is determined by adding together the points it receives for its ranking on each ballot.

For example, green is ranked first on 18 ballots worth four points each, and fifth on the remaining 37 ballots worth zero points each.

So green's score is 18 times four, which is 72.

Blue gets first on four ballots, second on 14 ballots, and fourth on 11 ballots for a total of 101 points.

Similar calculations show that purple's score is 107, red's is 136, and orange's score is 134. r That makes red our Borda Count winner.

That was four different methods for counting votes-- plurality, two-round runoff, instant runoff, and Borda Count-- and four different winners.

I find this to be spectacularly counterintuitive.

We have all the possible information about each individual's preferences.

We know exactly which colors they like more than other colors.

But that doesn't clearly lead us to a group preference.

The best color depends entirely on how we tally the votes.

So there are many ways to convert a collection of ballots into an overall winning candidate, and each of these election decision methods will have different properties and biases.

We want to study those to determine which method is most appropriate for a given context.

An often desired property of a voting system is the Condorcet criterion.

The winning candidate should beat every other candidate in a head-to-head election.

That is, the winning candidate should win a runoff election, regardless of who they are competing against.

Here's the results of our pairwise elections.

Orange wins each of its pairs.

For example, the majority of people, 36, rank orange higher than purple, so orange beats purple, which makes orange the Condorcet winner.

But remember that orange lost the election using each of the other four methods.

Since the winner from those voting systems is different than the Condorcet winner, we say that those methods-- plurality, two-round runoff, instant runoff, and Borda count-- all failed to meet the Condorcet criterion.

Clearly, voting is complicated for a whole host of reasons.

How do you split up regions?

How do you ensure different groups and interests are represented?

How should power be distributed among political parties?

Answering these complex and divisive questions is both important and hard, and we've linked in the description to a series from CGP Grey outlining some of the approaches.

We've focused on a very small subset of these issues.

How do you select a winner from a collection of ranked ballots?

On next week's episode, we'll dive deeper into the properties these voting systems do and do not have, including the incredible Arrow's Impossibility Theorem.

domain and help support our show.

You guys how some really awesome comments in response to our linkages video, and here's a few responses to them.

So fossilfighters asked, what are the curvy parts of the linkage?

What are they doing?

That's a great question.

So sometimes we wanted to make a whole part of the linkage, like the whole side of the pantograph, rigid.

We don't want the pantograph to be able to bend in half on the side, and so in order to force that part of the linkage to stay all in one line, we had to add an extra edge on the outside.

And we made it curvy so that you could see that it was there.

Good question.

So Abi Gail said, surely the linkages don't map every point.

So their point is that if you have a linkage, it has a fixed size and it's only going to be able to, for example, multiply by two linked points that are within that size of the linkage.

You can't multiply a point way out here if your linkage is only this big.

That's a great point, and it's totally valid.

You have to make bigger and bigger linkages if you want to multiply bigger and bigger points on your complex plane.

And finally, Lucas said, I still want to see those linkages.

I don't care how complicated they are.

That's fair.

I'd like to see them, too.

I decided not to include them in the video because when I was drawing them on a piece of paper, they just got crazier and crazier, and it wasn't helpful for me to understand how they worked.

It didn't really give me any insight into it.

If you can draw these huge, complicated linkages in a way that gives you an insight into how they work, or anyone else an insight, I would love to see it.

That would be super cool.

But they involve combining nine, 12, 40 of these smaller linkages together just to make the ones that square a complex number or multiply two complex numbers together.

And that's a lot of lines on a paper.

I recommend checking out the resources in the description if you want to learn more about how you combine the small linkages to make a big one.