Lecture Notes: Is Democracy Mathematically Impossible?

Introduction

• Democracy may be mathematically impossible.
• This claim is not a value judgment or about human nature; it is a mathematical fact regarding current voting systems.
• The presentation discusses how groups make decisions and the flaws in our voting systems.

First Past the Post Voting

• Definition: Voters mark one candidate as their favorite; the candidate with the most votes wins.
• Historical Use: Used since the 14th century in England; currently used by 44 countries, including the US.
• Issues:
  • Majority may not elect the ruling party (e.g., British Parliament example).
  • Similar parties can split votes, leading to the Spoiler Effect (e.g., Ralph Nader's impact in the 2000 US election).
  • Encourages strategic voting, often resulting in a two-party system (known as Duverger's Law).

Alternative Voting Methods

• Majority Requirement: Candidates need 50% + 1 to win; if no majority, eliminate the candidate with the fewest votes and redistribute ballots.
• Instant Runoff Voting (IRV): Voters rank preferences; if no majority, eliminate candidates sequentially until one candidate achieves a majority.
  • Example: Minneapolis mayoral race in 2013 showed candidates behaved better due to the ranking system.
• Issues with IRV: Can lead to paradoxes, e.g., a candidate doing worse helping them to win (Condorcet's method).

Condorcet's Method

• Proposed by Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet in 1785.
• Concept: A candidate should win against every other candidate in head-to-head elections.
• Condorcet's Paradox: A situation where cyclical preferences exist, making it impossible to determine a clear winner.

Arrow's Impossibility Theorem

• Kenneth Arrow outlined five conditions for a fair voting system:
  1. Unanimity: If everyone prefers one option, it should be the group's choice.
  2. No Dictatorship: One person's vote shouldn't override others.
  3. Unrestricted Domain: The system must account for all preferences consistently.
  4. Transitivity: Preferences should be consistent (A > B, B > C should imply A > C).
  5. Independence of Irrelevant Alternatives: Adding new options shouldn't affect existing preferences.
• Arrow proved that no ranked voting system can meet all conditions with three or more candidates (Nobel Prize in Economics, 1972).

Optimistic Theorems

• Duncan Black's Theorem: If preferences are aligned along a single dimension, the median voter's choice reflects the majority decision.
• Approval Voting: A simpler method where voters tick candidates they approve of, reducing negative campaigning and the spoiler effect.

Conclusion

• Democracy, as currently practiced, may seem flawed; however, it remains crucial for civic engagement and societal change.
• Continuous interest and political engagement can drive improvements.
• Final Thought: "Democracy is the worst form of government, except for all the others that have been tried" (Winston Churchill).

Further Learning

• Encouragement to use Brilliant for learning and developing problem-solving skills.
• Offers courses on a variety of subjects, including probability and statistics, relevant to current issues.