[MUSIC PLAYING] Different voting systems can produce radically different election results.

So it's important to ensure the voting system we're using has certain properties, that it fairly represents the opinions of the electorates.

The impressively counter-intuitive Arrow's impossibility theorem demonstrates that this is much harder than you might think.

[THEME MUSIC] 13 00:00:31,250 --> 00:00:33,650 Last episode, we tried to determine the best color among the candidates green, blue, purple, red, and orange based on the opinions of 55 voters.

But each of the four different rank ballot voting systems, plurality, two-round runoff, instant runoff, and Borda count, produced a different winner.

Furthermore, none of the voting systems met the Condorcet criterion.

If a candidate wins a head-to-head election against any other candidate, then that candidate should be the overall winner.

Here's how we represented ballots in the last episode.

To simplify things for later, we'll represent the same ballot like this.

Now, the leftmost color is first, the next color is second, and so on.

Let's note that each system can also be used to produce a complete group ranking.

First, determine the group winner using your chosen voting system.

Then kick them off the ballot and re-rank the remaining candidates.

Using the same voting system, find the new group winner.

That's the overall second-place candidate, repeating this procedure until everyone is ranked.

Let's see how this works in the two-round runoff system.

Blue won and green appears to be in second.

But if we kick out blue and repeat the two-round runoff procedure on the re-ranked ballots, green and orange go to the second round, but orange wins the overall.

Repeating this, the complete group ranking is blue, orange, red, purple, green.

Green is actually ranked last.

Clearly, it's important which voting system we use to count ballots.

The result depends on the system.

In order to decide which voting system to use for an election, we should study its properties.

What are the kinds of general properties that we would like our voting system to have?

Here's two.

Property one, unanimity, also known as the Pareto principle-- if every individual ranks color A above color B, then our voting system needs to rank color A above color B.

For example, if these are the only three ballots in an election, blue should obviously win.

Everyone ranks it first.

And purple should beat red.

Everyone likes purple more than red.

And property two, independence of irrelevant alternatives-- the way our voting systems rank color A relative to color B should only depend on how each individual ranks color A relative to color B.

Altering the ranking of color C shouldn't alter the relative positions of color A and color B.

For example, let's assume this is the result of an election where three voters submitted these ballots.

If voter two changes their mind and decides they hate purple, that shouldn't change the fact that green is ranked higher than red.

Or if orange drops out of the race, that shouldn't change anyone else's relative ranking.

Here's an unpopular suggestion for a voting system with both desired properties, dictatorship.

A voting system doesn't have to count everyone's vote equally.

It doesn't even have to count some votes at all.

If the outcome of the election is determined by a single individual's vote, the dictator, then both properties are satisfied.

Moreover, economist Kenneth Arrow proved that a dictatorship is the only rank voting system with both unanimity and independence of irrelevant alternatives.

In particular, none of the four voting systems we looked at satisfy these two properties.

Phrased another way, here's Arrow's impossibility theorem, also known as Arrow's paradox or Arrow's theorem.

There is no rank voting system with the following three properties, unanimity, independence of irrelevant alternatives, and non-dictatorial.

It's worth highlighting that Arrow's theorem only applies to rank voting systems, which are the kind we've been discussing.

It doesn't apply to voting systems with other types of ballots, like cardinal voting, where each candidate is assigned any number.

For example, this ballot shows somebody who loves green and purple, hates orange, and has mixed feelings about blue and red.

We won't be working with these voting systems, at least not in this episode.

Let's outline the proof of Arrow's theorem.

We'll assume we're using a rank voting system that satisfies unanimity and independence of irrelevant alternatives.

Using just those two properties, we want to prove that the rank voting system we're using must be a dictatorship.

First, we need to introduce a new term.

Define a "polarizing candidate" as any candidate that is ranked first or last by every voter.

We need to introduce a polarizing candidate because they'll help us test the extremes of any voting system.

In these sample ballots, green is a polarizing candidate.

Here's a fact we'll need in the middle of the proof.

In any voting system with unanimity and independence of irrelevant alternatives, a polarizing candidate must be ranked first or last in the overall ranking.

Your challenge problem for the week is to prove this fact using just those two properties.

To simplify matters, we'll have just five people vote among four colors, green, blue, purple, and red.

We'll think of the voting system as a function.

The input is a collection of ballots and the output is a group ranking of the candidates.

But we don't know specifically what that ranking is.

In other words, the function is fairly mysterious.

We don't know specifics, but we do know two of its properties, unanimity and independence of irrelevant alternatives.

The proof will have two phases, an experimental phase where we test what happens when we apply the voting system in different sample elections and a second phase where we use the results of these tests to prove that one voter is a dictator.

Here's the first phase.

To test this mysterious voting system function, we'll go through these six rounds of sample elections.

Specifically, we're testing the case where purple is a polarizing candidate since we know that means purple will be ranked first or last in the overall ranking.

Notice that the only difference between rounds is that each time, one voter switches purple from their last choice to their first choice.

These are like controlled experiments we're running on a mysterious voting system.

Hopefully, with these tests, we can gain information about the way the voting system function works.

In the final vote, every voter ranks purple first, so the property of unanimity implies that it must be first in the overall ranking.

Similarly, in the initial vote, purple must be ranked last in the overall ranking.

And for every round of voting, purple is a polarizing candidate.

So from the challenge problem, we know that it's either first or last in the overall ranking.

At some point, as we moved through rounds of sample votes, purple flipped from being the lowest ranked to the highest ranked.

In this example, that happened between round two and round three.

The only difference between those two rounds is that voter two moved purple from the bottom to the top of their ballot.

There was just one voter-- in this case, number two-- who changed their individual opinion of purple and it changed purple's status in the overall ranking.

Our test elections reveal that voter two is important.

I claim that voter two is the dictator.

No matter how voter two ranks the candidates, the overall ranking will always be exactly the same as voter two's individual preferences.

Only their opinion matters.

To prove this claim, we'll show that if voter two rank's color one above color two, then color one must be about color two in the overall ranking.

No matter how the same five voters submit their ballots, voter two is the only ballot that matters.

Let's assume these are the results of the election.

We're now moving into the second phase, where we use the sample test results to prove that voter two is the dictator, that only their vote matters.

We'll do this with a specific election, a specific collection of sample ballots.

But it's important to notice that the following proof will work for any collection of ballots.

To prove voter two is the dictator, we want to show that the overall ranking is the same as voter two's ranking, blue first, green second, purple third, and red last.

To start, we'll prove that the overall ranking also ranks blue above green.

This step involves comparing our current ballot to several others, including our previous test ballots.

Let's make a modification of our current collection of ballots so that we can later compare them.

We'll move purple to the front of voter one's rankings and to the back of voter three, four, and five's rankings.

Voter two ranks blue first, purple second, and then leaves the other colors in the same order.

We make these modifications because they'll allow us to easily compare this modified collection of ballots to the second round and third round of test voting from before.

We're choosing to compare the current ballot to these two rounds because they are the important rounds.

They are when purple flipped from being at the bottom of the overall ranking to the top.

In the second round of test voting, blue must be ranked above purple in the overall ranking since purple is ranked last.

This section has a lot of tricky logic about rankings, but here's the crucial step.

The modified ballot and the second round of test voting are the same when you only look at blue and purple.

Each voter gives them the same relative ranking.

So the independence of irrelevant alternatives implies that blue and purple must have the same relative overall ranking.

So blue is also above purple in the overall ranking for the modified ballot.

Similarly, in the third round of test voting, purple is ranked above blue in the overall ranking since purple is first.

The modified ballot and the third round of test votes are the same when you only consider purple and green.

Each voter gives them the same relative ranking.

So the overall ranking of the modified vote also has purple above green.

Now we know that in the overall ranking for the modified ballots, blue is above purple and purple is above green.

Collectively, this implies that blue is above green.

And finally, we compare the modified collection of ballots to the current election.

They both assign blue and green the same relative rankings, so the independence of irrelevant alternatives implies that blue is above green in the overall ranking for the current collection of ballots.

I know that was a lot of ballots and a lot of comparisons, but the overall conclusion is this.

In the current election, we used only the fact that voter two ranked blue above green to show that blue must be above green in the overall ranking.

The other voters' opinions were irrelevant.

Similar logic will show that the overall ranking always reflects voter two's preferences, that anytime voter two ranks color one above color two, the overall ranking must also have color one above color two.

Since this is true for all the colors, they must be identical rankings.

We've omitted some details from the proof, but that's the core idea.

Compare ballots to the test cases and use the assumed properties, unanimity and independent and irrelevant alternatives, to identify that there must be a dictator.

The conclusion of Arrow's theorem is powerful.

Only a dictatorship satisfies these two desirable properties.

But the strength of the conclusion comes from the strength of the premises.

In particular, the assumption that a rank voting system has independence of irrelevant alternatives might be too much.

All four of the voting systems we reviewed fail to have this property.

Here's an example with plurality voting.

Most people prefer blue to green.

If they were the only candidates, blue would win.

But once purple is introduced, many of the blue voters switch to purple and green is now ranked above blue.

Purple is an irrelevant alternative.

It shouldn't impact the relative rankings of green and blue, but it does.

As Kenneth Arrow said of his own impossibility theorem, "Most systems are not going to work badly all of the time.

All I proved is that all can work badly at times."

ea t purccohammseen.ts on our episode about voting systems.

Johan Richter made an interesting point about the difference between idealized voting systems and the real world.

They noted that "in real cases of a two-round runoff, voters can usually change their vote in the second round.

Even if the candidate they placed first in round one went through to the second round, they can vote for his competitor.

For strategic voters, this might make a difference."

In fact, the Gibbard-Satterthwaite theorem says that any of the types of voting systems we've been looking at are susceptible to tactical voting, meaning that people might have an incentive not to vote honestly.

And thanks to Nat Tuck for making me laugh out loud.

Here's their comment.

"An ordinal ranking, even if honest, throws away a bunch of useful information.

Consider ranking the choices get $11, get $10, get $1, get punched in the face.

That'll be your ordered ranking, but the ranking throws away the fact that you prefer getting $11 over getting punched in the face only slightly more than you prefer getting $10 over getting punched in the face."

Ordinal voting or ranked voting is the kind we've been talking about.

It just provides a ranking.

But in cardinal voting, you can also assign each candidate a score.

For example, my score for get $11 is plus 11, for get $10 is plus 10, for get $1 is plus one, and for get punched in the face is minus a million.

There's a bunch of different ways to implement cardinal voting, each with their own set of pros and cons.