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Language: en
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 P
the post voting the name is kind of a
misnomer though there is no post that
any of the candidates need to get ped
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 P 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 nater
nater 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 nater most of those voters were
devastated that by voting for nater
rather than Gore they ended up electing
Bush This is what is called a spoiler
effect almost all nater 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
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