social-science

Voting & Democracy

Electoral systems, gerrymandering, and impossibility theorems — the mathematics that reveals why perfect democracy is harder than it seems.

votingdemocracyelectionsgerrymanderingArrow's theoremsocial choiceranked choice

Democracy seems simple: let the people choose. But the mathematics of social choice reveals deep paradoxes. Arrow's impossibility theorem proves that no voting system can satisfy all reasonable fairness criteria simultaneously. Different voting methods — plurality, ranked choice, approval — can produce different winners from the same voters. And gerrymandering shows how drawing district lines can predetermine election outcomes.

These simulations explore the fascinating mathematics of democratic choice. Watch how the same election produces different winners under different voting rules, see gerrymandering create unfair outcomes from fair votes, discover why Condorcet cycles make 'the will of the people' incoherent, and explore how proportional representation changes political landscapes.

5 interactive simulations

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Arrow's Impossibility Theorem

See how the same voters produce different winners under plurality, Borda, Condorcet, ranked-choice, and approval voting — proving Arrow's impossibility theorem visually

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Electoral College vs Popular Vote

Simulate how a candidate can win the presidency while losing the popular vote — explore the mathematics of the Electoral College

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Gerrymandering Geometry

Visualize how drawing district boundaries on the same electorate can manufacture wildly different election outcomes — the dark geometry of gerrymandering

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Political Polarization Model

Watch a moderate population split into extremes as echo chambers, media bias, and radicalization drive opinions apart over generations

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Proportional vs Winner-Take-All

Compare D'Hondt, Sainte-Lague, and largest-remainder seat allocation methods — see how electoral thresholds and rounding rules reshape legislatures