Why Democracy Is Mathematically Impossible

First past the post is the simplest voting method where voters mark one favorite candidate and the candidate with the most votes wins. The name is a misnomer as there is no post to pass.

• Used to elect members of the House of Commons in England since the 14th century • Currently used by 44 countries worldwide to elect leaders • 30 of these countries were former British colonies • The US still uses it in most states for electoral college representation

• Majority of voters can support a party that doesn't hold power • In British Parliament over 100 years, only 2 of 21 single-party majorities reflected actual voter preference • Similar parties steal votes from each other, causing the spoiler effect • Incentivizes strategic voting rather than genuine preference expression

Winner-takes-all system leads to concentration of power in larger parties and eventually a two-party system, an effect known as Duverger's Law.

Voters rank candidates from favorite to least favorite. If no candidate gets a majority on the first count, the candidate with fewest votes is eliminated and their ballots distributed to voters' second preferences. This repeats until someone reaches a majority.

• Minneapolis mayor's race in 2013 had 35 candidates using ranked-choice voting • All candidates were exceptionally polite and cordial to each other • They ended the final debate singing together, seeking second and third choice votes • Contrasts with typical partisan mudslinging seen in other voting systems

Paradoxical outcome can occur where a candidate doing worse in the first round actually helps them win. When Bohr receives fewer votes due to poor campaigning, Curie gets eliminated instead, causing Bohr to win in the second round.

Instant runoff is mathematically identical to holding repeated elimination elections but saves time and effort, also called preferential voting or ranked-choice voting.

French mathematician Condorcet was one of the first people applying logic and mathematics to rigorously study voting systems, making him a founder of social choice theory. He worked during the French Revolution when determining the people's will was culturally important.

A candidate wins only by beating every other candidate in head-to-head elections. Voters rank preferences and you count how many rank each candidate higher than others. This voting system was actually discovered 450 years earlier by Ramon Llull, a monk studying church leader selection, but his work was lost and rediscovered in 2001.

• When choosing between burgers, pizza, or sushi with three voters: burgers beat pizza, pizza beats sushi, and sushi beats burgers • Creates a circular preference loop with no clear winner • Known as Condorcet's paradox, it challenged his voting system • Demonstrates that fair voting has inherent mathematical limitations

Condorcet died before resolving his paradox. During the French Revolution's Reign of Terror, he was deemed a traitor for criticizing the new constitution and was arrested, dying in jail in 1794.

• Unanimity: If everyone prefers one option, the group must also prefer it • No dictatorship: No single person's vote should override everyone else • Unrestricted domain: System must process all possible ballots consistently • Transitivity: If A beats B and B beats C, then A must beat C • Independence of irrelevant alternatives: Adding new options shouldn't change existing preferences

In 1951, Kenneth Arrow proved that satisfying all five conditions simultaneously in a ranked voting system with three or more candidates is mathematically impossible. You always must give something up.

• Shows that if everyone ranks a candidate first or last, society must also rank them first or last • Creates thought experiment moving a candidate B from bottom to top one voter at a time • Identifies a pivotal voter whose change flips society's ranking • Proves this pivotal voter becomes a dictator for all candidate preferences

Arrow's theorem was groundbreaking enough to earn him the Nobel Prize in Economics in 1972. It demonstrates that no ranked-choice method can rationally aggregate voter preferences with three or more candidates.

• If voters and candidates spread naturally along a single dimension like liberal to conservative • The median voter's preference reflects the majority decision • Avoids the paradoxes and inconsistencies highlighted by Arrow • Suggests democracy may work better in practice than theory suggests

Arrow's Impossibility Theorem only applies to ordinal voting systems where voters rank candidates. Rated voting systems offer an alternative where voters indicate approval or disapproval rather than ranking.

• Voters tick candidates they approve of instead of ranking them • Variants allow indicating strength of approval from -10 to +10 • Increases voter turnout and decreases negative campaigning • Prevents the spoiler effect by allowing approval without party size concerns • Simple to tally: highest approval percentage wins

• Used by Vatican priests to elect the Pope between 1294 and 1621 • Currently used to elect the UN Secretary General • Kenneth Arrow agreed late in life that rated voting was likely the best method • Not yet widely used in large-scale elections, requiring more real-world testing

Democracy is mathematically impossible if using ranked-choice methods, which most countries use to elect leaders. Some voting methods clearly aggregate preferences better than others, making first past the post feel ridiculous given its flaws.

Just because democracy isn't perfect doesn't mean we shouldn't try. Being interested in the world, caring about issues, and being politically engaged may be one of the few ways to make a real difference.

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 have.

The game might be crooked, but it's the only game in town. Despite mathematical impossibilities, democracy remains our best framework for collective decision-making.