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Why Democracy Is Mathematically Impossible — Transcript

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0:00- Democracy might be mathematically impossible.
0:02(serious music)
0:04This isn't a value judgment, a comment about human nature,
0:08nor a statement about how rare
0:09and unstable democratic societies have been
0:11in the history of civilization.
0:14Our current attempt at democracy,
0:16the methods we're using to elect our leaders,
0:19are fundamentally irrational.
0:21And this is a well-established mathematical fact.
0:27This is a video about the math
0:28that proved that fact and led to a Nobel Prize.
0:32It's a video about how groups of people make decisions
0:35and the pitfalls that our voting systems fall into.
0:39(subdued music)
0:41One of the simplest ways to hold an election
0:43is to ask the voters to mark one candidate
0:45as their favorite on a ballot.
0:47And when the votes are counted,
0:49the candidate with the most votes wins the election.
0:52This is known as "first past the post" voting.
0:56The name is kind of a misnomer though.
0:58There is no post
0:59that any of the candidates need to get past.
1:01The winner is just the candidate with the most votes.
1:06This method likely goes back to antiquity.
1:08It has been used to elect members
1:10of the House of Commons in England since the 14th century,
1:13and it's still a common voting system
1:15with 44 countries in the world using it
1:18to elect its leaders.
1:1930 of these countries were former British colonies.
1:23The US, being a former British colony,
1:25still uses first past the post in most of its states
1:28to elect their representatives to the electoral college.
1:32But first past the post has problems.
1:36If you are selecting representatives in a parliament,
1:38you can, and frequently do,
1:40get situations where the majority of the country
1:42did not vote for the party that ends up holding the power.
1:48In the last a hundred years,
1:49there were 21 times a single party held
1:52a majority of the seats in the British Parliament,
1:54but only two of those times did the majority of the voters
1:58actually vote for that party.
2:01So a party, which only a minority of the people voted for,
2:04ends up holding all of the power in government.
2:08Another thing that happens because of first past the post
2:10is that similar parties end up stealing votes
2:13from each other.
2:15- The 2000 US presidential election,
2:18which was an election essentially
2:20between Al Gore and George W. Bush.
2:23At that point, every state in the nation
2:25used first past the post
2:27to determine the outcome of the election.
2:28Bush had more votes in Florida,
2:31but by a ridiculously slim margin.
2:34It was fewer than 600 votes.
2:36But there was another candidate on the ballot, Ralph Nader.
2:41Nader was a Green candidate.
2:43He was certainly to the left of either Gore or Bush.
2:48- And what we need is the upsurge
2:50of citizen concern, people concern,
2:52poor, rich, or middle class,
2:54to counteract the power of the special interests.
2:57- And he got almost a hundred thousand votes in Florida.
3:00- I just don't know if I can, with a conscience,
3:03vote for Bush or Gore.
3:05- I will vote for Ralph Nader.
3:07- Most of those voters were devastated
3:11that by voting for Nader rather than Gore,
3:14they ended up electing Bush.
3:16This is what is called a spoiler effect.
3:19Almost all Nader voters preferred Gore to Bush,
3:23but in a first past the post system,
3:25they had no way of expressing that preference
3:30because you could only vote for one candidate.
3:33(inquisitive music)
3:34- So first past the post incentivizes voters
3:37to vote strategically.
3:39Say there are five parties,
3:41one of them will be the smallest one, and so they won't win.
3:45Why would you vote for them?
3:47This is also true if you have four parties or three parties.
3:51This winner-takes-all voting system leads
3:54to a concentration of power in larger parties,
3:57eventually, leading to a two-party system.
4:01This effect is common enough
4:03that it has a name: Duverger's Law.
4:08So first past the post isn't a great option.
4:11So what else could we do?
4:13(subdued music)
4:15Well, we can say that a candidate can only win an election
4:17if they get a majority, at least 50% plus one of the vote.
4:22But what if we hold an election and no one gets a majority?
4:26We could go to the people who voted for the candidate
4:28with the fewest votes and ask them to vote again,
4:31but choose a different candidate
4:33and we could repeat this process over and over
4:36eliminating the smallest candidate
4:38until one candidate reaches a majority.
4:41But holding many elections is a big hassle
4:44so instead we could just ask voters
4:46to rank their preferences
4:48from their favorite to their least favorite.
4:50And if their favorite candidate gets eliminated,
4:52we go to their second preferences.
4:55When the polls close, you count the voters' first choices.
4:58If any candidate has a majority of the votes,
5:01then they're the winner.
5:02But if no candidate has a majority,
5:04the candidate with the fewest votes gets eliminated
5:06and their ballots are distributed
5:08to those voters' second preferences,
5:10and this keeps happening
5:11until one candidate has a majority of the votes.
5:15This is mathematically identical
5:17to holding repeated elections,
5:19it just saves the time and hassle
5:21so it's referred to as instant runoff,
5:24but the system is also known
5:25as preferential voting or ranked-choice voting.
5:29An instant runoff doesn't just affect the voters,
5:32it affects how the candidates behave towards each other.
5:36- It was the Minneapolis mayor's race, 2013,
5:39they were using ranked-choice voting.
5:41The incumbent mayor had stepped down
5:44and there were all of these people
5:46came out from the woodwork wanting to be mayor.
5:48There're 35 candidates.
5:50And so you would think if there's 35 candidates
5:53you'd want to dunk on someone,
5:54you'd wanna like kind of elbow yourself into the spotlight.
5:58That's not what happened.
5:59These 35 candidates,
6:00all of them were really nice to each other.
6:03They were all super cordial, super polite,
6:07to the degree that at the end of the final mayoral debate,
6:13they all came together and they sang "Kumbaya" together.
6:19♪ Kumbaya, my Lord, kumbaya ♪
6:24♪ Oh, Lord, kumbaya ♪
6:30- The amount of vitriol and anger
6:32and partisan, you know, mudslinging that we're all used to,
6:36to see this vision of an actual "Kumbaya."
6:40It's not even a joke.
6:41All of these people getting along
6:43so desperate for second and third choices
6:45from other people that they're like,
6:46"I'm gonna be the picture-perfect,
6:48kindest candidate possible."
6:52- But there's also a problem with instant runoff.
6:55There can be cases where a candidate doing worse
6:58can actually help get them elected.
7:02Let's say we have three candidates:
7:04Einstein, Curie, and Bohr.
7:06Now, Einstein and Bohr have very conflicting views
7:10while Curie is ideologically in the center.
7:13So let's say Einstein gets 25% of the vote,
7:16Curie gets 30, and Bohr gets 45.
7:19No one got a majority.
7:21So it goes to the second round
7:22with Einstein being eliminated
7:24and because people who voted for Einstein
7:27put down Curie as their second choice,
7:29well, Curie ultimately gets elected.
7:32But now imagine that Bohr has a terrible campaign speech
7:36or proposes a very unpopular policy
7:39so bad that some of his voters
7:41actually switch over to Einstein's side.
7:44Well now it's Curie that gets eliminated
7:47and because she's more moderate,
7:49half of her voters select Einstein
7:51and the other half select Bohr in the second round,
7:54and this leads to Bohr winning.
7:57So Bohr doing worse in the first round
8:00actually leads to him winning the election.
8:04Clearly, this isn't something
8:05that we want in a voting system.
8:08(serious music) This is what
8:09the French mathematician Condorcet also thought.
8:12Condorcet was one of the first people applying logic
8:15and mathematics to rigorously study voting systems
8:18making him one of the founders
8:20of a branch of mathematics known as social choice theory.
8:24He was working during the time of the French Revolution,
8:26so fairly determining the will of the people
8:29was having a cultural moment right then.
8:34In 1784, Condorcet's contemporary
8:37at the French Royal Society of Science,
8:39Jean-Charles de Borda, proposed a voting method.
8:44You ask the voters to rank the candidates.
8:47If there are five candidates,
8:48ranking someone first gives that candidate four points,
8:51ranking them second would give them three, and so on,
8:54with zero points being awarded for last place.
8:59But the Borda count has a problem
9:01because the number of points given to each candidate
9:03is dependent on the total number of candidates.
9:06Adding extra people that have no chance of winning
9:09can affect the winner.
9:11Because of this, Condorcet hated Borda's idea.
9:14He wrote that it was "bound to lead to error
9:17because it relies on irrelevant factors for its judgments."
9:21So in 1785, Condorcet published an essay
9:24in which he proposed a new voting system,
9:27one he thought was the most fair.
9:29(soft music)
9:30Basically the winner needs
9:31to beat every other candidate in a head-to-head election.
9:35But with more than two candidates,
9:37do you need to hold a large number
9:38of head-to-head elections to pick the winner?
9:41Well, no.
9:42Just ask the voters to rank their preferences
9:44just like in instant runoff
9:46and then count how many voters rank each candidate higher
9:49than each other candidate.
9:51This feels like the most fair voting method.
9:56This voting system was actually discovered
9:58450 years earlier by Ramon Llull,
10:01a monk who was looking at how church leaders were chosen,
10:04but Llull's ideas didn't make an impact
10:06because his book, "Ars eleccionis," the art of elections,
10:10was lost and only rediscovered in 2001.
10:13So the voting system is named after Condorcet and not Llull.
10:18(gentle music)
10:19But will there always be a winner in this way?
10:22Let's try Condorcet's method for choosing dinner
10:25between you and two friends.
10:27There are three options: burgers, pizza or sushi.
10:30You really like burgers, so that's your first preference,
10:33your second choice is pizza, and you put sushi last.
10:37Your friend prefers pizza, then sushi, then burgers,
10:40and your other friend prefers sushi,
10:42then burgers, then pizza.
10:44Now if you choose burgers, it can be argued
10:47that sushi should have won instead
10:49since two of you prefer sushi over burgers
10:52and only one prefers burgers to sushi.
10:55However, by the same argument, pizza is preferred to sushi
10:58and burgers are preferred to pizza
11:01by a margin of two-to-one on each occasion.
11:03So it seems like you and your friends are stuck in a loop.
11:07Burgers are preferred to pizza, which is preferred to sushi,
11:10which is preferred to burgers and so on.
11:14This situation is known as Condorcet's paradox.
11:20Condorcet died before he could resolve
11:22this problem with his voting system.
11:24He was politically active during the French Revolution
11:26writing a draft of France's constitution.
11:30In 1793 during the Reign of Terror
11:32when La Montagne came to power,
11:35he was deemed a traitor for criticizing the regime,
11:37specifically their new constitution.
11:40In the next year, he was arrested and died in jail.
11:45(gentle music)
11:46Over the next 150 years, dozens of mathematicians
11:49were proposing their own voting systems
11:51or modifications to Condorcet's or Borda's ideas.
11:56One of those mathematicians was Charles Dodgson,
11:59better known as Lewis Carroll.
12:01When he wasn't writing "Alice in Wonderland,"
12:03he was trying to find a system to hold fair elections.
12:08But every voting system had similar kinds of problems,
12:11You'd either get Condorcet loops
12:13or other candidates that had no chance of winning
12:16would affect the outcome of the election.
12:18(lively jazz music)
12:20In 1951, Kenneth Arrow published his PhD thesis
12:24and in it he outlined five very obvious
12:27and reasonable conditions
12:28that a rational voting system should have.
12:31Condition number one:
12:32if everyone in the group chooses one option over another,
12:35the outcome should reflect that.
12:37If every individual in the group
12:39prefers to eat sushi over pizza,
12:41then the group as a whole should prefer sushi over pizza.
12:44This is known as unanimity.
12:47Condition two: no single person's vote
12:50should override the preferences of everyone else.
12:52If everyone votes for pizza
12:54except one person who votes for sushi,
12:56the group should obviously choose pizza.
12:59If a single vote is decisive, that's not a democracy,
13:02that's a dictatorship.
13:04Condition three: everyone should be able
13:07to vote however they want
13:08and the voting system must produce a conclusion for society
13:11based on all the ballots, every time.
13:14It can't avoid problematic ballots
13:16or candidates by simply ignoring them
13:18or just guessing randomly, it must reach the same answer
13:22for the same set of ballots every time.
13:25This is called unrestricted domain.
13:28Condition four: the voting system should be transitive.
13:31If a group prefers burgers over pizza
13:33and pizza over sushi,
13:35then they should also prefer burgers over sushi.
13:38This is known as transitivity.
13:40Condition five: if the preference of the group
13:43is sushi over pizza,
13:44the introduction of another option, like burgers,
13:47should not change that preference.
13:49Sure, the group might collectively rank burgers above both
13:52or in the middle or at the bottom,
13:54but the ranking of sushi over pizza
13:56should not be affected by the new option.
13:59This is called the independence of irrelevant alternatives.
14:04But here's the thing, Arrow proved that satisfying all five
14:07of these conditions in a ranked voting system
14:09with three or more candidates is impossible.
14:12This is Arrow's Impossibility Theorem,
14:15and it was so groundbreaking
14:16that Arrow was awarded the Nobel Prize in Economics in 1972.
14:20So I wanna go through a version of his proof
14:23based on a formulation by Geanakoplos.
14:26So let's say there are
14:27three candidates running for election:
14:30Aristotle, Bohr, and Curie,
14:33but we'll refer to them as A, B, and C,
14:36and we have a collection of voters
14:38that we'll line up in order.
14:39So we have voter 1, 2, 3, and so on all the way up to N.
14:44Each of these voters
14:45is free to rank A, B and C however they like.
14:48I'll even allow ties.
14:50And the first we wanna show is that if everyone ranks
14:53a particular candidate first or last,
14:55then society as a whole
14:57must also rank that candidate first or last.
15:00Let's arbitrarily pick candidate B.
15:03If say half of the voters rank B first
15:06and half rank B last, then the claim is
15:08our voting system must put B either first or last
15:13and we'll prove it by contradiction.
15:16So say this is how everyone voted.
15:19If our system does not put B first or last,
15:22but rather in the middle, say A is ranked above B,
15:25which is above C, then we'll get a contradiction.
15:29Because if each of our voters moved C above A,
15:33then by unanimity, C must be ranked above A.
15:38However, because we didn't change the position
15:40of any A relative to B,
15:43A must still be ranked above B
15:46and because we didn't change the position
15:48of any C relative to B, C must still be ranked below B,
15:52and by transitivity,
15:54if A is preferred to B and B is preferred to C,
15:57then A must be ranked above C.
16:00But this contradicts the result by unanimity
16:03and that proves that if everyone ranks
16:05a candidate first or last,
16:07then society must also rank them first or last.
16:13Now let's do a thought experiment
16:14where every voter puts B at the bottom of their ranking,
16:18we'll leave the ranking of A and C arbitrary.
16:21Well then, by unanimity,
16:23we know that B must be at the bottom of society's ranking
16:27and we'll call this setup Profile 0.
16:30Now we'll create Profile 1 which is identical to Profile 0
16:34except the first voter moves B from the bottom to the top.
16:38This, of course, doesn't affect the outcome,
16:40but we can keep doing this
16:42creating Profiles 2, 3, 4, and so on
16:45with one more voter flipping B
16:47from the bottom to the top each time.
16:49If we keep doing this, there will eventually come a voter
16:53whose change from having B at the bottom to B at the top
16:56will first flip society's ranking, moving B to the top.
17:01Let's call this voter the pivotal voter
17:03and we'll label the Profile p.
17:06Profile o is then the profile
17:08right before the pivotal change happens.
17:11Let's now create a Profile q, which is the same as p,
17:15except the pivotal voter moves A above B.
17:19By independence of irrelevant alternatives,
17:22the social rank must also put A above B.
17:26Since for all of our voters,
17:28the relative position of A and B
17:30is the same as it was in Profile o,
17:33and B must be ranked above C
17:36because the relative positions of B and C are the same
17:40as they were in Profile p,
17:42where our pivotal voter moved B to the top.
17:45By transitivity A must be ranked above C
17:49in the social ranking.
17:51This is true regardless
17:52of how any of the non-pivotal voters rearrange
17:55their positions of A and C,
17:58because these rearrangements don't change
18:00the position of A relative to B or C relative to B.
18:06This means the pivotal voter is actually a dictator
18:09for determining society's preference of A over C.
18:13The social rank will always agree with the pivotal voter
18:16regardless of what the other voters do.
18:19We can run a similar thought experiment
18:21where we put C at the bottom
18:23and prove that there is again, a dictator,
18:26who in this case determines
18:28the social preference of A over B.
18:31And it turns out this voter is the same one
18:33who determines the social preference for A over C.
18:38The pivotal voter is therefore a complete dictator.
18:42(dark music)
18:44So is democracy doomed?
18:46Well, Arrow's impossibility theorem seems to say so.
18:49If there are three or more candidates to choose from,
18:52there is no ranked-choice method
18:54to rationally aggregate voter preferences.
18:56You always need to give something up.
19:02(hopeful music)
19:03But the mathematician, Duncan Black,
19:05found a much more optimistic theorem
19:07which might actually represent reality better.
19:11If voters and candidates are naturally spread
19:13along a single dimension,
19:14say ranging from liberal on the left
19:16to conservative on the right,
19:18but this could apply to any other political dimension.
19:21Well, then Black showed that the preference
19:24of the median voter will reflect the majority decision.
19:28The median voter's choice
19:30will often determine the outcome of the election,
19:32a result that aligns with the majority of voters,
19:35avoiding the paradoxes
19:36and inconsistencies highlighted by Arrow.
19:40And there's more good news.
19:42Arrow's Impossibility Theorem only applies
19:44to ordinal voting systems,
19:45ones in which the voters rank candidates over others.
19:49There is another way: rated voting systems.
19:53The simplest version is known as approval voting
19:56where instead of ranking the candidates,
19:58the voters just tick the candidates they approve of.
20:01There are also versions where you could indicate
20:03how strongly you like each candidate,
20:05say from -10, strongly disapprove of,
20:08to +10, strongly approve.
20:11Research has found
20:12that approval voting increases voter turnout,
20:15decreases negative campaigning
20:17and prevents the spoiler effect.
20:19Voters could express their approval for a candidate
20:21without worrying about the size
20:23of the party they're voting for.
20:25It's also simple to tally,
20:27just count up what percentage of the voters
20:29approve of each candidate
20:30and the one with the highest approval wins.
20:33Kenneth Arrow was initially skeptical
20:35of rated-voting systems,
20:37but toward the end of his life,
20:38he agreed that they were likely the best method.
20:41Approval voting is not new.
20:43It was used by priests in the Vatican
20:45to elect the Pope between 1294 and 1621.
20:49It's also used to elect
20:51the Secretary General of the United Nations,
20:53but it hasn't been widely used in large-scale elections.
20:57And so more real-world testing is likely required.
21:01(mellow music)
21:02So is democracy mathematically impossible?
21:04Well, yes, if we use ranked choice methods of voting,
21:07which is what most countries in the world use
21:10to elect their leaders.
21:11And some methods are clearly better
21:13at aggregating the people's preferences than others,
21:15the use of first past the post voting
21:18feels quite frankly ridiculous to me,
21:20given all of its flaws.
21:21But just because things aren't perfect
21:23doesn't mean we shouldn't try.
21:25Being interested in the world around us,
21:27caring about issues,
21:28and being politically engaged is important.
21:31It might be one of the few ways
21:33we can make a real difference in the world.
21:35Like Winston Churchill said,
21:36"Democracy is the worst form of government
21:40except for all the other forms that have been tried."
21:43Democracy is not perfect, but it's the best thing we've got.
21:47The game might be crooked, but it's the only game in town.
21:50(static buzzes and whines)
21:55The world is changing.
21:57How it works today is no guarantee
21:59of how it'll work tomorrow
22:00from how we elect presidents to how we do our jobs.
22:03Luckily, there's an easy way to be ready
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