Three candidates, three voter groups, three reasonable voting methods. The candidates: A, B, C. Plurality picks the most first-place votes; Borda weights ranks; Condorcet picks who beats every other in head-to-head. Sometimes all three methods agree. Sometimes they fight. Sometimes Condorcet has no answer: the dreaded cycle A→B→C→A.
Ballots
Plurality (first-past-the-post)
Borda count (2 / 1 / 0 points)
Condorcet — pairwise majority comparisons
⚠ Condorcet paradox: there is no candidate who beats all others. A→B→C→A or similar cycle exists.
Arrow's impossibility theorem
Three reasonable voting methods, three ways they can produce different winners — and one way they all fail (the Condorcet cycle). Try the "Condorcet cycle" preset: 3 voter groups, each preferring A>B>C, B>C>A, C>A>B. Pairwise: A beats B 2-1, B beats C 2-1, C beats A 2-1. Cycle. No "majority winner" exists.
Arrow's impossibility theorem (1951): with three or more candidates, no voting method can satisfy all five desirable properties:
• Unanimity — if all voters prefer A to B, A is ranked above B
• Non-dictatorship — no single voter determines the outcome regardless of others
• Transitivity — if society prefers A to B and B to C, it prefers A to C
• Independence of Irrelevant Alternatives — A vs B should depend only on voters' A-vs-B preferences
• Universal domain — works for all possible ballot configurations
You can satisfy any four, but not all five. Real elections are forced to choose which property to sacrifice. Plurality drops IIA (spoiler effects). Borda is also not IIA. Approval voting works on different ballot structure. Ranked-choice voting drops monotonicity. There is no perfect democratic method — only methods with chosen failure modes.