民主制度在数学上为何不可能？阿罗不可能定理揭示投票系统的困境

Democracy might be mathematically impossible.

This isn't a value judgment, a comment about human nature, nor a statement about how rare and unstable democratic societies have been in the history of civilization.

Our current attempt at democracy, the methods we're using to elect our leaders, are fundamentally irrational.

And this is a well-established mathematical fact.

This is a video about the math that proved that fact and led to a Nobel Prize.

It's a video about how groups of people make decisions and the pitfalls that our voting systems fall into.

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One of the simplest ways to hold an election is to ask the voters to mark one candidate as their favorite on a ballot.

And when the votes are counted, the candidate with the most votes wins the election.

This is known as "first past the post" voting.

The name is kind of a misnomer though.

There is no post that any of the candidates need to get past.

The winner is just the candidate with the most votes.

This method likely goes back to antiquity.

It has been used to elect members of the House of Commons in England since the 14th century, and it's still a common voting system with 44 countries in the world using it to elect its leaders.

30 of these countries were former British colonies.

The US, being a former British colony, still uses first past the post in most of its states to elect their representatives to the electoral college.

But first past the post has problems.

If you are selecting representatives in a parliament, you can, and frequently do, get situations where the majority of the country did not vote for the party that ends up holding the power.

In the last a hundred years, there were 21 times a single party held a majority of the seats in the British Parliament, but only two of those times did the majority of the voters actually vote for that party.

So a party, which only a minority of the people voted for, ends up holding all of the power in government.

Another thing that happens because of first past the post is that similar parties end up stealing votes from each other.

The 2000 US presidential election, which was an election essentially between Al Gore and George W. Bush.

At that point, every state in the nation used first past the post to determine the outcome of the election.

Bush had more votes in Florida, but by a ridiculously slim margin.

It was fewer than 600 votes.

But there was another candidate on the ballot, Ralph Nader.

Nader was a Green candidate.

He was certainly to the left of either Gore or Bush.

- And what we need is the upsurge of citizen concern, people concern, poor, rich, or middle class, to counteract the power of the special interests.

And he got almost a hundred thousand votes in Florida.

- I just don't know if I can, with a conscience, vote for Bush or Gore.

- I will vote for Ralph Nader.

Most of those voters were devastated that by voting for Nader rather than Gore, they ended up electing Bush.

This is what is called a spoiler effect.

Almost all Nader voters preferred Gore to Bush, but in a first past the post system, they had no way of expressing that preference because you could only vote for one candidate.

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So first past the post incentivizes voters to vote strategically.

Say there are five parties, one of them will be the smallest one, and so they won't win.

Why would you vote for them?

This is also true if you have four parties or three parties.

This winner-takes-all voting system leads to a concentration of power in larger parties, eventually, leading to a two-party system.

This effect is common enough that it has a name: Duverger's Law.

So first past the post isn't a great option.

So what else could we do?

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Well, we can say that a candidate can only win an election if they get a majority, at least 50% plus one of the vote.

But what if we hold an election and no one gets a majority?

We could go to the people who voted for the candidate with the fewest votes and ask them to vote again, but choose a different candidate and we could repeat this process over and over eliminating the smallest candidate until one candidate reaches a majority.

and we could repeat this process over and over eliminating the smallest candidate until one candidate reaches a majority.

But holding many elections is a big hassle

so instead we could just ask voters to rank their preferences from their favorite to their least favorite.

And if their favorite candidate gets eliminated, we go to their second preferences.

When the polls close, you count the voters' first choices.

If any candidate has a majority of the votes, then they're the winner.

But if no candidate has a majority, the candidate with the fewest votes gets eliminated and their ballots are distributed to those voters' second preferences,

and this keeps happening until one candidate has a majority of the votes.

This is mathematically identical to holding repeated elections, it just saves the time and hassle so it's referred to as instant runoff,

but the system is also known as preferential voting or ranked-choice voting.

An instant runoff doesn't just affect the voters, it affects how the candidates behave towards each other.

It was the Minneapolis mayor's race, 2013, they were using ranked-choice voting.

The incumbent mayor had stepped down and there were all of these people came out from the woodwork wanting to be mayor.

There're 35 candidates.

And so you would think if there's 35 candidates you'd want to dunk on someone, you'd wanna like kind of elbow yourself into the spotlight.

That's not what happened.

These 35 candidates, all of them were really nice to each other.

They were all super cordial, super polite, to the degree that at the end of the final mayoral debate, they all came together and they sang "Kumbaya" together.

♪ Kumbaya, my Lord, kumbaya ♪ ♪ Oh, Lord, kumbaya ♪

The amount of vitriol and anger and partisan, you know, mudslinging that we're all used to, to see this vision of an actual "Kumbaya."

It's not even a joke.

All of these people getting along so desperate for second and third choices from other people that they're like, "I'm gonna be the picture-perfect, kindest candidate possible."

But there's also a problem with instant runoff.

There can be cases where a candidate doing worse can actually help get them elected.

Let's say we have three candidates: Einstein, Curie, and Bohr.

Now, Einstein and Bohr have very conflicting views while Curie is ideologically in the center.

So let's say Einstein gets 25% of the vote, Curie gets 30, and Bohr gets 45.

No one got a majority.

So it goes to the second round with Einstein being eliminated

and because people who voted for Einstein put down Curie as their second choice, well, Curie ultimately gets elected.

But now imagine that Bohr has a terrible campaign speech or proposes a very unpopular policy so bad that some of his voters actually switch over to Einstein's side.

Well now it's Curie that gets eliminated

and because she's more moderate, half of her voters select Einstein and the other half select Bohr in the second round,

and this leads to Bohr winning.

So Bohr doing worse in the first round actually leads to him winning the election.

Clearly, this isn't something that we want in a voting system.

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This is what the French mathematician Condorcet also thought.

Condorcet was one of the first people applying logic and mathematics to rigorously study voting systems making him one of the founders of a branch of mathematics known as social choice theory.

He was working during the time of the French Revolution, so fairly determining the will of the people was having a cultural moment right then.

In 1784, Condorcet's contemporary at the French Royal Society of Science, Jean-Charles de Borda, proposed a voting method.

You ask the voters to rank the candidates.

If there are five candidates, ranking someone first gives that candidate four points, ranking them second would give them three, and so on, with zero points being awarded for last place.

But the Borda count has a problem because the number of points given to each candidate is dependent on the total number of candidates.

Adding extra people that have no chance of winning can affect the winner.

Because of this, Condorcet hated Borda's idea.

He wrote that it was "bound to lead to error because it relies on irrelevant factors for its judgments."

So in 1785, Condorcet published an essay in which he proposed a new voting system, one he thought was the most fair.

(soft music)

Basically the winner needs to beat every other candidate in a head-to-head election.

But with more than two candidates, do you need to hold a large number of head-to-head elections to pick the winner?

Well, no.

Just ask the voters to rank their preferences just like in instant runoff and then count how many voters rank each candidate higher than each other candidate.

This feels like the most fair voting method.

This voting system was actually discovered 450 years earlier by Ramon Llull, a monk who was looking at how church leaders were chosen,

but Llull's ideas didn't make an impact because his book, "Ars eleccionis," the art of elections, was lost and only rediscovered in 2001.

So the voting system is named after Condorcet and not Llull.

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But will there always be a winner in this way?

Let's try Condorcet's method for choosing dinner between you and two friends.

There are three options: burgers, pizza or sushi.

You really like burgers, so that's your first preference, your second choice is pizza, and you put sushi last.

Your friend prefers pizza, then sushi, then burgers,

and your other friend prefers sushi, then burgers, then pizza.

Now if you choose burgers, it can be argued that sushi should have won instead since two of you prefer sushi over burgers and only one prefers burgers to sushi.

However, by the same argument, pizza is preferred to sushi and burgers are preferred to pizza by a margin of two-to-one on each occasion.

So it seems like you and your friends are stuck in a loop.

Burgers are preferred to pizza, which is preferred to sushi, which is preferred to burgers and so on.

This situation is known as Condorcet's paradox.

Condorcet died before he could resolve this problem with his voting system.

He was politically active during the French Revolution writing a draft of France's constitution.

In 1793 during the Reign of Terror when La Montagne came to power, he was deemed a traitor for criticizing the regime, specifically their new constitution.

In the next year, he was arrested and died in jail.

(gentle music)

Over the next 150 years, dozens of mathematicians were proposing their own voting systems or modifications to Condorcet's or Borda's ideas.

One of those mathematicians was Charles Dodgson, better known as Lewis Carroll.

When he wasn't writing "Alice in Wonderland," he was trying to find a system to hold fair elections.

But every voting system had similar kinds of problems, You'd either get Condorcet loops or other candidates that had no chance of winning would affect the outcome of the election.

(lively jazz music)

In 1951, Kenneth Arrow published his PhD thesis and in it he outlined five very obvious and reasonable conditions that a rational voting system should have.

Condition number one: if everyone in the group chooses one option over another, the outcome should reflect that.

If every individual in the group prefers to eat sushi over pizza, then the group as a whole should prefer sushi over pizza.

This is known as unanimity.

Condition two: no single person's vote should override the preferences of everyone else.

If everyone votes for pizza except one person who votes for sushi, the group should obviously choose pizza.

If a single vote is decisive, that's not a democracy, that's a dictatorship.

Condition three: everyone should be able to vote however they want and the voting system must produce a conclusion for society based on all the ballots, every time.

It can't avoid problematic ballots or candidates by simply ignoring them or just guessing randomly, it must reach the same answer for the same set of ballots every time.

This is called unrestricted domain.

Condition four: the voting system should be transitive.

If a group prefers burgers over pizza and pizza over sushi, then they should also prefer burgers over sushi.

This is known as transitivity.

Condition five: if the preference of the group is sushi over pizza, the introduction of another option, like burgers, should not change that preference.

Sure, the group might collectively rank burgers above both or in the middle or at the bottom, but the ranking of sushi over pizza should not be affected by the new option.

This is called the independence of irrelevant alternatives.

But here's the thing, Arrow proved that satisfying all five of these conditions in a ranked voting system with three or more candidates is impossible.

This is Arrow's Impossibility Theorem, and it was so groundbreaking that Arrow was awarded the Nobel Prize in Economics in 1972.

So I wanna go through a version of his proof based on a formulation by Geanakoplos.

So let's say there are three candidates running for election: Aristotle, Bohr, and Curie, but we'll refer to them as A, B, and C,

and we have a collection of voters that we'll line up in order.

So we have voter 1, 2, 3, and so on all the way up to N.

Each of these voters is free to rank A, B and C however they like.

I'll even allow ties.

And the first we wanna show is that if everyone ranks a particular candidate first or last, then society as a whole must also rank that candidate first or last.

Let's arbitrarily pick candidate B.

If say half of the voters rank B first and half rank B last, then the claim is our voting system must put B either first or last and we'll prove it by contradiction.

So say this is how everyone voted.

If our system does not put B first or last, but rather in the middle, say A is ranked above B, which is above C, then we'll get a contradiction.

Because if each of our voters moved C above A, then by unanimity, C must be ranked above A.

However, because we didn't change the position of any A relative to B, A must still be ranked above B

and because we didn't change the position of any C relative to B, C must still be ranked below B,

and by transitivity, if A is preferred to B and B is preferred to C, then A must be ranked above C.

But this contradicts the result by unanimity and that proves that if everyone ranks a candidate first or last, then society must also rank them first or last.

Now let's do a thought experiment where every voter puts B at the bottom of their ranking, we'll leave the ranking of A and C arbitrary.

Well then, by unanimity, we know that B must be at the bottom of society's ranking and we'll call this setup Profile 0.

Now we'll create Profile 1 which is identical to Profile 0 except the first voter moves B from the bottom to the top.

This, of course, doesn't affect the outcome, but we can keep doing this creating Profiles 2, 3, 4, and so on with one more voter flipping B from the bottom to the top each time.

If we keep doing this, there will eventually come a voter whose change from having B at the bottom to B at the top will first flip society's ranking, moving B to the top.

Let's call this voter the pivotal voter and we'll label the Profile p.

Profile o is then the profile right before the pivotal change happens.

Let's now create a Profile q, which is the same as p, except the pivotal voter moves A above B.

By independence of irrelevant alternatives, the social rank must also put A above B.

Since for all of our voters, the relative position of A and B is the same as it was in Profile o,

and B must be ranked above C because the relative positions of B and C are the same as they were in Profile p, where our pivotal voter moved B to the top.

By transitivity A must be ranked above C in the social ranking.

This is true regardless of how any of the non-pivotal voters rearrange their positions of A and C, because these rearrangements don't change the position of A relative to B or C relative to B.

This means the pivotal voter is actually a dictator for determining society's preference of A over C.

The social rank will always agree with the pivotal voter regardless of what the other voters do.

We can run a similar thought experiment where we put C at the bottom and prove that there is again, a dictator, who in this case determines the social preference of A over B.

And it turns out this voter is the same one who determines the social preference for A over C.

The pivotal voter is therefore a complete dictator.

(dark music)

So is democracy doomed?

Well, Arrow's impossibility theorem seems to say so.

If there are three or more candidates to choose from, there is no ranked-choice method to rationally aggregate voter preferences.

You always need to give something up.

(hopeful music)

But the mathematician, Duncan Black, found a much more optimistic theorem which might actually represent reality better.

If voters and candidates are naturally spread along a single dimension, say ranging from liberal on the left to conservative on the right, but this could apply to any other political dimension.

Well, then Black showed that the preference of the median voter will reflect the majority decision.

The median voter's choice will often determine the outcome of the election, a result that aligns with the majority of voters, avoiding the paradoxes and inconsistencies highlighted by Arrow.

And there's more good news.

Arrow's Impossibility Theorem only applies to ordinal voting systems, ones in which the voters rank candidates over others.

There is another way: rated voting systems.

The simplest version is known as approval voting where instead of ranking the candidates, the voters just tick the candidates they approve of.

There are also versions where you could indicate how strongly you like each candidate, say from -10, strongly disapprove of, to +10, strongly approve.

Research has found that approval voting increases voter turnout, decreases negative campaigning and prevents the spoiler effect.

Voters could express their approval for a candidate without worrying about the size of the party they're voting for.

It's also simple to tally, just count up what percentage of the voters approve of each candidate and the one with the highest approval wins.

Kenneth Arrow was initially skeptical of rated-voting systems, but toward the end of his life, he agreed that they were likely the best method.

Approval voting is not new.

It was used by priests in the Vatican to elect the Pope between 1294 and 1621.

It's also used to elect the Secretary General of the United Nations, but it hasn't been widely used in large-scale elections.

And so more real-world testing is likely required.

(mellow music)

So is democracy mathematically impossible?

Well, yes, if we use ranked choice methods of voting, which is what most countries in the world use to elect their leaders.

And some methods are clearly better at aggregating the people's preferences than others, the use of first past the post voting feels quite frankly ridiculous to me, given all of its flaws.

But just because things aren't perfect doesn't mean we shouldn't try.

Being interested in the world around us, caring about issues, and being politically engaged is important.

It might be one of the few ways we can make a real difference in the world.

Like Winston Churchill said, "Democracy is the worst form of government except for all the other forms that have been tried."

Democracy is not perfect, but it's the best thing we've got.

The game might be crooked, but it's the only game in town.

(static buzzes and whines)

The world is changing.

How it works today is no guarantee of how it'll work tomorrow from how we elect presidents to how we do our jobs.

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