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