The Mathematics of Democracy
No voting system is perfect. Arrow’s impossibility theorem proved it mathematically in 1951. But that doesn’t mean all systems are equally flawed. Explore 20 interactive simulations that reveal how plurality voting fails, how ranked-choice can elect consensus candidates, how gerrymandering distorts representation, and why the Condorcet paradox means a democracy can prefer A to B, B to C, and C to A—simultaneously. The math of fairness turns out to be deeply, beautifully unfair.
Different ways to aggregate preferences produce shockingly different winners from the same votes.
First-past-the-post: the most common system, and arguably the worst. Watch vote splitting hand victory to a minority candidate.
Rank candidates 1st, 2nd, 3rd. Eliminate the lowest, redistribute their votes. Watch rounds of elimination find a majority winner.
Give N-1 points to 1st, N-2 to 2nd, etc. The consensus candidate wins. But add a loser to the race and the winner can change!
Vote for as many candidates as you like. Simple, resistant to spoilers, and tends to elect broadly acceptable candidates.
Same voters, same preferences, 6 different methods. Watch them produce 6 different winners. Which system is “right”?
Mathematics proves that perfect fairness is impossible—and reveals the specific ways every system fails.
A beats B, B beats C, C beats A. Democracy is circular! Explore when and why collective preferences become intransitive.
No ranked voting system satisfies all fairness criteria simultaneously. Toggle each axiom and watch which systems survive—and which break.
Every non-dictatorial voting method can be gamed. Find the strategic vote that changes the outcome and see why honesty isn’t always rewarded.
Add a losing candidate and the winner changes. Nader 2000, Perot 1992—third parties don’t just lose, they decide who wins.
Sometimes voting for your favorite makes them LOSE. Staying home would have been better. IRV and other methods have this shocking flaw.
Place voters and candidates in ideological space and watch geometry determine election outcomes.
In a 1D political spectrum, the candidate closest to the median voter wins. Drag voters and candidates to see why politics converges to the center.
Color each point in 2D political space by who wins if the population is centered there. Different voting methods paint shockingly different maps.
Draw district boundaries on a map of voters. Win 80% of seats with 50% of votes. See how geometry can steal elections legally.
D’Hondt, Sainte-Laguë, and largest remainder methods. Same votes, different seat allocations. See which small parties get shut out.
Plurality voting naturally evolves toward two parties. Run generations of strategic voters and watch third parties vanish.
Who really has power in a voting system? These simulations measure influence, fairness, and the mathematics of representation.
Shapley-Shubik and Banzhaf power indices reveal who really controls weighted voting. A 49% shareholder can have 0% power!
How to divide 435 House seats among 50 states? Hamilton, Jefferson, and Webster methods disagree—and create bizarre paradoxes.
Rate each candidate 0-5 stars. STAR adds a runoff between the top two. Expressive, strategic-resistant, and gaining real momentum.
Win the presidency with 22% of the popular vote. Explore how winner-take-all states create swing-state tyranny and wasted votes.
Delegate your vote to someone you trust—who can delegate further. A network of trust that blends direct and representative democracy.