Lecture Notes on Voting Systems and Democracy

Introduction to Voting System Issues

• Claim: Democracy might be mathematically impossible.
• Not a critique of human nature or the stability of democracy historically.
• Assertion: Our methods of electing leaders are fundamentally irrational (a mathematical fact).
• Focus: Math that led to a Nobel Prize and how voting systems have pitfalls.

First Past the Post Voting

• Definition: Voters mark one candidate; the candidate with the most votes wins.
• History: Used since the 14th century in England; used in 44 countries.
• Problems:
  • A party with minority votes can hold power (e.g., UK's Parliament history).
  • Similar parties can split votes, leading to undesired outcomes (e.g., 2000 US election).
  • Leads to strategic voting and eventually a two-party system (Duverger's Law).

Alternatives to First Past the Post

• Majority Requirement: A candidate must win an election with at least 50% +1 of votes.
• Instant Runoff Voting (Preferential/Ranked-Choice Voting):
  • Voters rank candidates.
  • Eliminated candidates' votes are redistributed until a majority is reached.
  • Impact: Encourages more polite campaigns (e.g., 2013 Minneapolis mayor’s race).
  • Problem: Candidates doing worse can sometimes get elected (non-intuitive outcomes).

Condorcet's and Borda's Contributions

• Condorcet Method:
  • Uses head-to-head elections.
  • Developed in opposition to Borda's method, which could be skewed by irrelevant factors.
  • Problem: Condorcet's Paradox (cyclical preferences).
• Borda Count: Points system, can be skewed by irrelevant candidates.

Arrow's Impossibility Theorem

• Five Conditions of Rational Voting System:
  • Unanimity, no dictatorship, unrestricted domain, transitivity, independence of irrelevant alternatives.
• Outcome: Impossible to satisfy all five in a ranked voting system with three or more candidates.
• Proof: Explained through profiles and pivotal voter/situations.

Other Voting Methods and Theorems

• Duncan Black's Optimism: Median voter preference tends to reflect the majority decision.
• Rated Voting Systems: Approval voting allows voters to approve multiple candidates without ranking them.
  • Benefits: Increases turnout, reduces negative campaigning, prevents spoiler effect.

Conclusion

• Democracy's Challenges: Arrow's theorem suggests imperfections in current systems.
• Potential Solutions: Rated voting systems like approval voting.
• Philosophical Note: Despite flaws, democracy is preferable to other systems.
• Encouragement: Continue being politically engaged and informed.

Additional Resources

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