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 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 pass 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 100 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 pass 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 pass 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 what we need is the upsurge of Citizen concern
people concerned poor Rich or middle class to counteract the power of the special interest
and he got almost 100,000 votes in Florida I just don't know if I can with a conscience um
vote for uh 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 pass post system they had
no way of expressing that preference because you could only vote for one candidate so first pass
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 do verger's law so first pass the post isn't a great option
so what else could we do 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
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 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 c cidate 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 rank 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 35 candidates and so you would think if there's 35 candidates you'd want to dunk on
someone you'd want to 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 k k oh Lord 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 going to 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 help get them elected let's say we have three candidates Einstein curee and bore
now Einstein and bore have very conflicting views while C is ideologically in the center so let's
say Einstein gets 25% of the vote cirri gets 30 and bore 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 c as their second choice well C ultimately gets elected but now imagine that
bour 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 curee that gets eliminated
and because she's more moderate half of her voters select Einstein and the other half
select bore in the second round and this leads to boore winning so bore doing work in the first
round actually leads to him winning the election clearly this isn't something that we want
in a voting system this is what the french mathematician Condor also thought Condor
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 condor'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 board account 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 condr hated Border's idea he wrote that it was bound to lead
to error because it relies on irrelevant factors for its judgments so in 1785 Condor published an
essay in which he proposed a new voting system one he thought was the most Fair 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
[Music] method this voting system was actually discovered 450 years earlier by Raymond lull a
monk who was looking at how church leaders were chosen but L's ideas didn't make an impact because
his book ours electionus the art of Elections was lost and only rediscovered in 2001 so the voting
system is named after cond and not lol but will there always be a winner in this way let's try
condor'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 than 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 2: 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 condor's Paradox Condor 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 Le monang came to
power he was deemed a traitor for criticizing the regime specifically their new constitution
the next year he was arrested and died in jail over the next 150 years dozens of
mathematicians were proposing their own voting systems or modifications to Condor
or bord ideas one of those mathematicians was Charles Dodson 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 Condor Loops or other candidates
that had no chance of winning would affect the outcome of the election in 1951 Kenneth Arrow
published his PhD thesis and in it he outlined five very obvious and reasonable conditions
that AR 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 want to go through a version of his proof based on a formulation by
GN acus so let's say there are three candidates running for election Aristotle bore and C but
we'll refer to them as a b and c and we have a collection of Voters that will 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 now the first thing we want to 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 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 we'll call this
setup profile 0 now we'll create profile one which is identical to profile Z 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 of clipping 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 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 a 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 so is democracy doomed well
arrows 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 [Music] up 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 voters 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 minus 10
strongly disapprove of to plus 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 real world testing is likely required so is
democracy mathematically impossible well yes if we use rank 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 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 luckily
there's an easy way to be ready for whatever the future holds by expanding your knowledge and
critical thinking skills a little bit every day and you can get started doing that right now
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