theory.toys – Arrow's Theorem & Voting Paradoxes

Voting Method

Method:

Preference Profile

Scenario:

Custom Preferences

Three candidates: A, B, C

A > B > C: 10 A > C > B: 5 B > A > C: 8 B > C > A: 12 C > A > B: 7 C > B > A: 8

Strategic Voting

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Social Choice Theory

Social choice theory studies how to aggregate individual preferences into collective decisions. Arrow's Impossibility Theorem (1951) proves that no voting system can simultaneously satisfy a set of seemingly reasonable criteria when there are three or more alternatives.

Key Paradoxes

Condorcet Cycle: A beats B, B beats C, but C beats A in pairwise majority votes, creating a cycle with no clear winner.
Spoiler Effect: Adding a losing candidate changes the winner between other candidates (violates Independence of Irrelevant Alternatives).
Monotonicity Violation: In some systems (like IRV), ranking a candidate higher can cause them to lose.
Strategic Manipulation: Gibbard-Satterthwaite theorem proves that all reasonable voting systems are vulnerable to strategic voting.

Voting Methods Compared

Plurality: Each voter picks one favorite; most votes wins. Simple but vulnerable to vote splitting.
Borda Count: Candidates get points based on rankings (3 for 1st, 2 for 2nd, 1 for 3rd). Can elect Condorcet losers.
Instant Runoff (IRV): Eliminate last-place candidates and redistribute votes. Not monotonic.
Condorcet: Winner beats all others in pairwise contests. May not exist (cycles).
Approval: Vote for all acceptable candidates. Strategy-proof for dichotomous preferences.