Why Democracy Is Mathematically Impossible
Summary
TLDRThis video delves into the mathematical challenges of democracy, particularly in voting systems. It critiques the 'first past the post' method, highlighting its tendency to produce majority governments that don't reflect the majority's will. The script explores alternatives like instant runoff voting and points out its own paradoxes, such as how a poorly performing candidate can influence election outcomes. It introduces Arrow's Impossibility Theorem, which states that no ranked voting system can satisfy all desired fairness conditions, and contrasts it with Black's Median Voter Theorem, suggesting a more optimistic view of democratic voting. The video concludes by advocating for approval voting as a potentially fairer method and encourages political engagement despite the imperfections of the democratic process.
Takeaways
- 😀 Democracy's current voting methods are fundamentally irrational, leading to potential instability and misrepresentation.
- 🏛️ 'First past the post' voting, used by 44 countries, can result in parties gaining power without majority support, as seen in the UK's parliamentary history.
- 🗳️ The spoiler effect, as in the 2000 US presidential election, can lead to unintended election outcomes due to vote-splitting among similar parties.
- 🔄 'Instant runoff' or ranked-choice voting can mitigate the spoiler effect and encourage more cooperative candidate behavior, as demonstrated in the 2013 Minneapolis mayoral race.
- 🔢 Arrow's Impossibility Theorem states that no ranked voting system can satisfy five reasonable conditions simultaneously, suggesting inherent limitations in democratic voting systems.
- 🤔 Condorcet's method, which involves pairwise comparison of candidates, can result in paradoxes where cyclic preferences prevent a clear winner from emerging.
- 📊 Black's Median Voter Theorem offers a more optimistic view, suggesting that in one-dimensional political spectra, the preference of the median voter can determine the majority decision.
- 👍 Approval voting, an alternative to ranked voting, allows voters to express approval for one or more candidates and can increase voter turnout and reduce negative campaigning.
- 🏆 Approval voting has historical precedents, such as electing the Pope in the Vatican and the Secretary-General of the United Nations.
- 💡 Despite theoretical challenges, democracy remains the best form of government available, and engagement in the political process is crucial for making a difference.
- 🌐 The script concludes with a call to expand knowledge and critical thinking skills to better understand and participate in an ever-changing world.
Q & A
What is the main argument presented in the video about democracy and mathematics?
-The video argues that the methods currently used to elect leaders in democratic societies are fundamentally irrational from a mathematical perspective, leading to potential instability and misrepresentation.
What is 'first past the post' voting and why does it have issues?
-'First past the post' is a voting system where the candidate with the most votes wins. It has issues because it can lead to situations where the majority of the country did not vote for the party that ends up holding power, and it can cause similar parties to 'steal' votes from each other, leading to a two-party system.
What is the 'spoiler effect' in elections?
-The 'spoiler effect' occurs when a candidate with similar views to another takes votes away, potentially causing the less preferred candidate to win. This effect discourages voters from voting for their true preference if it's not a leading contender.
What is the 'instant runoff' voting system and how does it work?
-The 'instant runoff' voting system, also known as preferential or ranked-choice voting, allows voters to rank candidates by preference. If no candidate has a majority after the first count, the candidate with the fewest votes is eliminated, and their votes are redistributed according to the voters' second preferences, repeating this process until a candidate has a majority.
How does the 'instant runoff' system affect candidate behavior?
-The 'instant runoff' system encourages candidates to be more cordial and polite to each other, as they are vying for second and third preferences from voters who support other candidates.
What is Arrow's Impossibility Theorem and what does it imply for voting systems?
-Arrow's Impossibility Theorem states that it is impossible to create a ranked voting system that satisfies five reasonable conditions when there are three or more candidates. This implies that no voting system can perfectly and rationally aggregate voter preferences without some form of compromise.
What are the five conditions outlined by Kenneth Arrow for a fair voting system?
-The five conditions are: Unanimity (if everyone prefers one option, the group should too), Non-dictatorship (no single vote should override all others), Unrestricted Domain (all voters can vote freely and the system must produce a conclusion), Transitivity (if the group prefers A over B and B over C, they should prefer A over C), and Independence of Irrelevant Alternatives (adding or removing a candidate should not affect the preference between two existing candidates).
What is Condorcet's method and how does it differ from other voting systems?
-Condorcet's method is a voting system where the winner is the candidate who would win in a head-to-head election against every other candidate. It differs from other systems by focusing on pairwise comparisons of candidates rather than aggregating total points or votes.
What is Condorcet's Paradox and how does it challenge the fairness of voting systems?
-Condorcet's Paradox occurs when there is a circular preference among three or more options, where each option is preferred to the next and the last is preferred to the first, creating a logical loop with no clear winner. This paradox challenges the idea that a voting system can always produce a fair and consistent outcome.
What alternative voting system is suggested in the video as a potential solution to the issues with ranked voting?
-The video suggests rated voting systems, such as approval voting, as a potential solution. In approval voting, voters indicate which candidates they approve of, and the candidate with the highest approval wins, which can increase voter turnout and decrease negative campaigning.
Outlines
🗳️ The Impossibility of Perfect Democracy
The script discusses the inherent mathematical challenges in achieving a perfect democratic voting system. It begins by critiquing the 'first past the post' voting method, which has historically led to situations where the majority's choice does not align with the ruling party. The script also touches on the 'spoiler effect' as seen in the 2000 US presidential election, where third-party candidates can inadvertently influence the outcome by splitting the vote. The discussion then shifts to alternative voting methods, such as 'instant runoff' or ranked-choice voting, which aim to address some of these issues but are not without their own complexities and potential for paradoxical outcomes.
🔄 The Evolution of Voting Systems
This section delves into the evolution of voting systems, from the Condorcet method to the Borda count, each with its own set of advantages and disadvantages. The Condorcet method, which involves pairwise comparison of candidates, is highlighted for its potential to determine a consensus candidate, yet it is also shown to be susceptible to paradoxes. Borda's system, which assigns points based on rankings, is criticized for being influenced by the number of candidates, which can lead to irrelevant factors affecting the outcome. The script also introduces the work of mathematicians like Charles Dodgson (Lewis Carroll) and Kenneth Arrow, who contributed to the understanding of voting systems and the inherent impossibilities within them.
🤔 Arrow's Impossibility Theorem
The script presents Kenneth Arrow's seminal work, the Impossibility Theorem, which states that it is impossible to create a ranked voting system that satisfies five reasonable conditions when there are three or more candidates. These conditions include unanimity, non-dictatorship, unrestricted domain, transitivity, and independence of irrelevant alternatives. The theorem is illustrated through a thought experiment involving voters' preferences for candidates, demonstrating that any such system will inevitably lead to a dictatorial influence on the outcome, thus challenging the very concept of democratic decision-making.
📊 Alternative Voting Systems and Their Impact
The script explores alternative voting systems, such as approval voting, which allows voters to express approval for multiple candidates rather than ranking them. This method is shown to have potential benefits, including increased voter turnout, reduced negative campaigning, and the prevention of the spoiler effect. It also touches on the historical use of approval voting in the Vatican and for electing the UN Secretary-General. The discussion highlights the need for further real-world testing of these systems to better understand their effectiveness in large-scale elections.
🌐 The Imperfections of Democracy and the Importance of Engagement
In the final paragraph, the script acknowledges the imperfections of democracy but emphasizes its value as the best form of government available, as famously stated by Winston Churchill. It encourages viewers to stay politically engaged and to continuously expand their knowledge and critical thinking skills to adapt to a changing world. The script concludes with a sponsorship message for Brilliant, an educational platform that offers courses in various subjects, including probability and statistics, to help viewers become better thinkers and problem solvers.
Mindmap
Keywords
💡Democracy
💡First Past the Post
💡Spoiler Effect
💡Strategic Voting
💡Instant Runoff
💡Condorcet's Paradox
💡Arrow's Impossibility Theorem
💡Median Voter Theorem
💡Approval Voting
💡Independence of Irrelevant Alternatives
Highlights
Democracy's methods of electing leaders are fundamentally irrational, a mathematical fact that led to a Nobel Prize.
First-past-the-post voting, used in 44 countries, can lead to parties with minority votes holding power.
The spoiler effect in first-past-the-post voting can result in the election of a less preferred candidate.
Strategic voting in first-past-the-post systems can concentrate power in larger parties, leading to a two-party system.
Instant runoff voting, or ranked-choice voting, saves time by eliminating the need for multiple elections.
Candidates in ranked-choice voting systems tend to be more cordial to secure second and third preferences.
Instant runoff voting can paradoxically elect a candidate despite them performing worse initially.
Condorcet's method, proposed by the French mathematician, aims to find the most fair voting system.
Condorcet's paradox shows that a voting system can end in a loop with no clear winner.
Arrow's impossibility theorem states that it's impossible to satisfy five reasonable conditions in a ranked voting system with three or more candidates.
Duncan Black's theorem offers a more optimistic view of democracy, suggesting the preference of the median voter reflects the majority.
Rated voting systems, like approval voting, can increase voter turnout and decrease negative campaigning.
Approval voting prevents the spoiler effect and allows voters to express approval without strategic concerns.
Arrow initially doubted rated voting systems but later acknowledged their potential as the best method.
Democracy, despite its flaws, is the best form of government compared to all other forms tried.
Expanding knowledge and critical thinking skills is crucial for adapting to the changing world.
Brilliant.org is highlighted as a tool for daily learning and critical thinking skill development.
Transcripts
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
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