Veritasium On Why Democracy Is Mathematically Impossible

The video explains why there are always 2 major contestants in an election and why inherent biases lead to the voting choices we have today. If is filled with examples from History and considers/advocates for a ranked voting system.

Watch the video to find out.

The video explores the mathematical challenges and paradoxes inherent in different voting systems, illustrating why a “perfect” democracy is mathematically impossible when using ranked-choice methods.

The Flaws of “First Past the Post”

• How it Works: Voters pick one favorite candidate, and the person with the most votes wins [00:46].
• The Problems: This system frequently results in a winner who did not receive the majority of the popular vote [01:48]. It also causes the “spoiler effect,” where similar candidates steal votes from each other (e.g., Ralph Nader in the 2000 US Presidential Election). This effectively incentivizes strategic voting over honest preference, which eventually leads to a two-party system—a phenomenon known as Duverger’s Law [03:17].

Ranked Choice and the Condorcet Paradox

• Instant Runoff (Ranked Choice Voting): Voters rank their preferences. If no one gets a majority, the lowest candidate is eliminated and their votes go to those voters’ second choices [04:51]. While it encourages more cordial campaigns, it can create a bizarre mathematical anomaly where a candidate performing worse in the first round can actually cause them to win the overall election [06:59].
• Condorcet’s Method: Proposed in 1785, this method suggests the winner should be the candidate who beats every other candidate in a head-to-head match-up [09:31].
• The Paradox: This leads to a rock-paper-scissors scenario known as Condorcet’s Paradox. For example, a group might prefer Burgers over Pizza, and Pizza over Sushi, but also prefer Sushi over Burgers, resulting in an endless loop with no clear winner [11:09].

Arrow’s Impossibility Theorem

In 1951, Kenneth Arrow outlined five fundamental and reasonable conditions for a fair ranked voting system [12:19]:

1. Unanimity: If everyone prefers A over B, society must prefer A over B.
2. No Dictatorship: No single person’s vote should override the preferences of everyone else.
3. Unrestricted Domain: The system must consistently produce a valid conclusion for society based on all ballots.
4. Transitivity: If A beats B, and B beats C, then A must beat C.
5. Independence of Irrelevant Alternatives: Introducing a new option shouldn’t change the relative ranking of the existing options.

Arrow mathematically proved that satisfying all five of these conditions in a ranked voting system with three or more candidates is impossible, a discovery that earned him a Nobel Prize in Economics [14:06].

A Potential Solution: Rated Voting Systems

• Arrow’s theorem only applies to ordinal (ranked) voting systems.
• A mathematical workaround is using rated voting systems, such as Approval Voting. Instead of ranking candidates, voters simply tick all the candidates they approve of [19:50].
• Research shows this method increases voter turnout, prevents the spoiler effect, and successfully avoids the paradoxes outlined by Arrow’s theorem [20:13].