What is Condorcet's Paradox?

Condorcet's Paradox describes a situation in voting where voter preferences create a cycle (A is preferred to B, B to C, and C to A), meaning there's no single candidate that a majority prefers over all others in pairwise comparisons.

What is Arrow's Impossibility Theorem?

Arrow's Impossibility Theorem proves that no ranked voting system with three or more candidates can simultaneously satisfy several basic fairness criteria, such as unanimity, non-dictatorship, and transitivity.

Are there any voting systems that avoid Arrow's Paradox?

Yes, rated voting systems like approval voting are considered more robust. In approval voting, you simply vote for all the candidates you approve of, which avoids many of the paradoxes found in ranked-choice systems.

What is the median voter theorem?

Duncan Black's median voter theorem suggests that if voters and candidates align along a single political dimension (like liberal to conservative), the median voter's preference will often determine the election outcome, aligning with the majority.

Is democracy mathematically impossible?

While ranked-choice voting systems face mathematical paradoxes highlighted by Arrow's theorem, democracy isn't entirely impossible. Alternative systems like approval voting and considering the median voter offer more stable outcomes, and pragmatically, democracy remains the best system despite imperfections.

Key Moments

Why Democracy Is Mathematically Impossible

Q: How does ranked-choice voting (instant runoff) work?

In ranked-choice voting, voters rank candidates by preference. If no candidate wins a majority initially, the candidate with the fewest votes is eliminated, and their votes are redistributed based on the voters' second preferences, repeating until a majority winner emerges.

Veritasium

Education4 min read24 min video

Aug 27, 2024|8,975,045 views|307,994|22,921

veritasium science mathematics 2024 election democracy voting election theory voters political science democracy explained does democracy work

Save to Pod

Key Moments

TL;DR

Democracy faces mathematical impossibilities with ranked voting, but approval voting offers a viable solution.

Key Insights

First-past-the-post voting systems, while common, can lead to minority rule and the 'spoiler effect' where small parties influence election outcomes.

Ranked-choice voting, or instant runoff, aims to improve upon first-past-the-post but can still suffer from counterintuitive results and strategic manipulation.

Condorcet's method, where a winner must beat every other candidate head-to-head, is theoretically fair but can lead to 'Condorcet's Paradox' (voting cycles).

Arrow's Impossibility Theorem demonstrates that no ranked voting system with three or more options can simultaneously satisfy five seemingly reasonable criteria (unanimity, non-dictatorship, unrestricted domain, transitivity, independence of irrelevant alternatives).

While ranked-choice voting faces mathematical hurdles, systems like approval voting (where voters select all candidates they approve of) are proposed as more robust solutions.

Despite mathematical challenges, democracy remains the best available form of governance, emphasizing the importance of engagement and critical thinking.

THE FLAWS OF FIRST-PAST-THE-POST VOTING

The video begins by dissecting the prevalent 'first-past-the-post' (FPTP) voting system, common in many countries including the US and former British colonies. FPTP, where the candidate with the most votes wins, suffers from significant drawbacks. It frequently results in a party holding power despite not winning the majority of the popular vote. Furthermore, it creates the 'spoiler effect,' where third-party candidates can draw votes away from a major candidate they are ideologically closer to, inadvertently helping their least preferred candidate win. This system also incentivizes strategic voting and can lead to an entrenched two-party system, as predicted by Duverger's Law.

INSTANT RUNOFF AND ITS CHALLENGES

In response to FPTP's issues, ranked-choice voting (RCV), also known as instant runoff voting, is introduced. This system allows voters to rank candidates in order of preference. If no candidate secures a majority, the candidate with the fewest votes is eliminated, and their ballots are redistributed based on the voters' next preferences. While designed to mitigate the spoiler effect and encourage more civil campaigns, RCV is not without its own paradoxes. The example of a candidate performing poorly in the first round that paradoxically leads to their victory in the second round due to strategic eliminations highlights its potential for counterintuitive outcomes.

CONDERCET'S METHOD AND THE PARADOX OF VOTING

Condorcet's method, proposed by mathematician Jean-Charles de Borda's contemporary, offered another theoretically fair approach. It dictates that a candidate must win a head-to-head election against every other candidate. This is achieved by analyzing voters' ranked preferences. However, Condorcet's system can fall prey to cyclical preferences, known as Condorcet's Paradox. This occurs when voter preferences create a loop, such as A is preferred to B, B to C, and C back to A, leaving no clear winner and highlighting a fundamental challenge in aggregating group preferences.

ARROW'S IMPOSSIBILITY THEOREM AND THE BREAKDOWN OF RANKED VOTING

The core mathematical challenge to democratic voting systems is articulated through Kenneth Arrow's Impossibility Theorem. Arrow proved that for any voting system with three or more options, it is impossible to satisfy five basic, desirable criteria simultaneously: unanimity, non-dictatorship, unrestricted domain, transitivity, and independence of irrelevant alternatives. This groundbreaking work, which earned Arrow a Nobel Prize, mathematically demonstrates that no ranked voting system can perfectly and consistently aggregate individual preferences into a collective decision without violating at least one of these fundamental principles.

APPROVAL VOTING: A MORE PRACTICAL ALTERNATIVE

While ranked-choice voting faces significant theoretical obstacles according to Arrow's Theorem, the video introduces approval voting as a more viable and potentially more democratic solution. In approval voting, voters simply select all candidates they approve of. This system avoids the spoiler effect, can reduce negative campaigning, and is simpler to tally. A variation allows for degrees of approval. Though not widely implemented in large-scale elections, its historical use in electing the Pope and UN Secretary-General, and its theoretical advantages, suggest it is a promising alternative for aggregating voter preferences more effectively.

DEMOCRACY'S ENDURING VALUE DESPITE IMPERFECTIONS

Ultimately, the video reconciles the mathematical impossibilities with the practical reality of governance. It acknowledges that no voting system is perfect, and Arrow's theorem highlights inherent difficulties in ranked systems. However, it emphasizes that democracy, despite its flaws, remains superior to other forms of government, echoing Winston Churchill's sentiment. The key lies in continuous engagement, critical thinking, and a willingness to explore and adopt better systems, like approval voting, to improve collective decision-making and its outcomes.

Mentioned in This Episode

●Companies

●Organizations

●Concepts

●People Referenced

Common Questions

First-past-the-post is a system where the candidate with the most votes wins. It's flawed because it can lead to a party holding power without a majority of the popular vote and can cause a 'spoiler effect' where a third-party candidate causes the less preferred major candidate to lose.

Topics

Voting Systems Social Choice Theory First-past-the-post Ranked-choice Voting Instant Runoff Condorcet's Paradox Arrow's Impossibility Theorem Approval Voting Median Voter Theorem Election Integrity Mathematical Impossibility

Mentioned in this video

personDuncan Black

A mathematician who proposed a more optimistic theorem regarding median voter outcomes in single-dimensional political landscapes.

personRaymond Lull

A monk who developed a voting system similar to Condorcet's method centuries earlier, but his work was lost and later rediscovered.

conceptFirst-Past-The-Post

A voting system where the candidate with the most votes wins, often leading to minority rule and the spoiler effect.

conceptCondorcet's Paradox

A situation in voting where preferences form a cycle (A > B, B > C, C > A), leading to no clear winner under pairwise comparison.

personLewis Carroll

The pen name of Charles Dodgson, who, besides writing Alice in Wonderland, also studied and proposed voting systems.

personKenneth Arrow

An economist who developed Arrow's Impossibility Theorem, proving that no ranked voting system with three or more options can satisfy a set of desirable fairness conditions.

conceptRanked-Choice Voting

A voting system where voters rank candidates in order of preference, used to mitigate issues with first-past-the-post.

personJean Charles de Borda

A French mathematician who proposed a point-based voting method, criticized by Condorcet for its potential to be swayed by irrelevant candidates.

conceptArrow's Impossibility Theorem

A fundamental theorem in social choice theory stating it's impossible to design a fair ranked voting system for three or more alternatives that meets basic fairness criteria like unanimity, non-dictatorship, transitivity, and independence of irrelevant alternatives.

conceptDuverger's Law

A principle stating that first-past-the-post voting systems tend to lead to a two-party system.

personRalph Nader

A Green Party candidate in the 2000 US presidential election whose votes were seen as a spoiler, leading to George W. Bush's victory over Al Gore.

conceptApproval Voting

A rated voting system where voters approve of as many candidates as they wish, shown to increase turnout, decrease negative campaigning, and prevent the spoiler effect.