Why Democracy Is Mathematically Impossible - Veritasium

The Mathematical Impossibility of Rational Democracy

Democracy is mathematically impossible and the methods currently used to elect leaders are fundamentally irrational. Specifically, if there are three or more candidates to choose from, there is no ranked-choice method to rationally aggregate voter preferences without requiring concessions. The ultimate conclusion of Arrow’s Impossibility Theorem is that democracy is doomed if ranked-choice methods are utilized. Despite these flaws, democracy remains "the best thing we've got," as it is the "worst form of government except for all the other forms that have been tried".

Flaws of First Past the Post Voting

First Past the Post (FPTP) is a common voting system, used by 44 countries and involving voters marking only one favorite candidate, with the candidate receiving the most votes winning. However, this method presents significant problems:

• Minority Rule: FPTP frequently leads to situations where the party holding power was not voted for by the majority of the country. In the British Parliament over the last hundred years, a single party held a majority of seats 21 times, but in only two of those instances did that party receive a majority of the actual votes.
• Spoiler Effect: Similar parties end up stealing votes from each other. In the 2000 US presidential election, many voters for third-party candidate Ralph Nader (who generally preferred Al Gore) inadvertently helped George W. Bush win, as they had no way to express their preference hierarchy under FPTP.
• Strategic Voting and Two-Party Systems: FPTP incentivizes voters to vote strategically (not for their true favorite if they are small). This winner-takes-all approach leads to a concentration of power in larger parties, eventually causing a two-party system, a phenomenon known as Duverger's Law.

Problems with Ranked-Choice Voting

Ranked-choice voting (RCV) asks voters to rank candidates. If no candidate achieves a majority of first choices, the candidate with the fewest votes is eliminated, and their votes are redistributed based on voters' second preferences until one candidate has a majority. This system is mathematically identical to holding repeated elections.

• Improved Candidate Behavior: RCV encourages candidates to be cordial and polite to each other, as they are "desperate for second and third choices" from other voters' ballots, reducing traditional partisan mudslinging.
• Perverse Outcomes: RCV can produce undesirable results where a candidate who performs worse in the first round (e.g., losing some first-preference votes) can actually end up winning the election.

Early Efforts and the Condorcet Paradox

Social choice theory, the branch of mathematics that rigorously studies voting systems, was founded by the French mathematician Condorcet.

• The Borda Count: Proposed by Jean-Charles de Borda, this system awards points based on candidate rankings. Condorcet criticized it because the outcome could be affected by the introduction of extra people who have no chance of winning.
• The Condorcet Method: Proposed by Condorcet in 1785 (though discovered earlier by Ramon Llull), this system holds that the winner must beat every other candidate in a head-to-head election, based on voters' rankings.
• Condorcet's Paradox: Condorcet's own method runs into a loop where collective preferences are non-transitive, meaning a group can prefer A over B, B over C, yet also prefer C over A, resulting in no clear winner.
• Lewis Carroll's Efforts: Charles Dodgson (pen name Lewis Carroll) also tried to find a fair system but encountered similar issues, like Condorcet loops and outcomes being affected by non-winning candidates.

Arrow's Impossibility Theorem

Kenneth Arrow published his Impossibility Theorem in 1951, proving the fundamental difficulty of rational voting aggregation. He outlined five reasonable conditions a rational voting system should meet:

1. Unanimity: If every individual prefers one option over another, the group outcome must reflect that.
2. No Dictatorship: No single person's vote should override the preferences of everyone else.
3. Unrestricted Domain: The system must produce a conclusion for society based on all ballots, every time, without ignoring or randomly guessing results.
4. Transitivity: If the group prefers option A over B and B over C, they must also prefer A over C.
5. Independence of Irrelevant Alternatives: The ranking between two options should not be affected by the introduction of a third, irrelevant option.

Arrow proved that satisfying all five of these conditions in a ranked voting system with three or more candidates is impossible. The proof demonstrates that any system that meets the other criteria must ultimately result in a dictator—a single pivotal voter whose preferences determine society's preference between two candidates regardless of how other voters arrange their choices.

Rated Voting as a Solution

Arrow's Impossibility Theorem only applies to (ordinal) ranked voting systems. An alternative is rated voting systems.

• Approval Voting: The simplest rated system, where voters just tick the candidates they approve of. Other versions allow indicating strength of approval (e.g., -10 to +10).
• Benefits: Research indicates approval voting increases voter turnout, decreases negative campaigning, and prevents the spoiler effect. Voters can express approval without worrying about the size of the party they are voting for.
• Support and Usage: Kenneth Arrow eventually agreed that rated voting systems were likely the best method. Approval voting was used to elect the Pope between 1294 and 1621 and is currently used to elect the United Nations Secretary General.

Additionally, Duncan Black found a more optimistic theorem: if voters and candidates align along a single political dimension (e.g., liberal to conservative), the preference of the median voter will reflect the majority decision, avoiding Arrow's paradoxes and inconsistencies.