A LIVING ENCYCLOPEDIA
Voting is the mechanism by which groups of individuals translate private preferences into collective decisions. It is at once the simplest and most complex of human inventions: simple because a single mark on a ballot can change the course of history; complex because no system of aggregating preferences is without paradox, limitation, or the potential for strategic manipulation.
This encyclopedia surveys the methods, history, and mathematical curiosities of voting -- from ancient Athenian ostraka to modern ranked-choice ballots, from Condorcet's paradox to Arrow's impossibility theorem. Each entry is a chapter in the ongoing story of humanity's attempt to govern itself through the counted voice.
Voting methods determine how individual preferences are converted into a collective outcome. The choice of method profoundly shapes results: the same set of voter preferences can produce different winners depending on the system used. No method is perfect -- this is not a flaw but a fundamental mathematical constraint first proven by Kenneth Arrow in 1951.
Each voter selects one candidate. The candidate with the most votes wins, regardless of whether they achieve a majority. Used in the United Kingdom, United States (most elections), Canada, and India.
Voters rank candidates in order of preference. If no candidate wins a majority, the last-place candidate is eliminated and their votes redistributed to second choices. The process repeats until a majority winner emerges.
Eliminates the spoiler effect in most cases
Voters may approve of as many candidates as they wish. The candidate with the most approvals wins. Simple, resistant to strategic voting, and proposed as a reform for single-winner elections worldwide.
No vote splitting Easy to understand
A family of methods that seeks the candidate who would win a one-on-one election against every other candidate. The Condorcet winner, when one exists, is considered the most broadly preferred. When no Condorcet winner exists, the result is the famous Condorcet paradox.
Condorcet winner may not always exist
The history of voting is the history of humanity's struggle to share power. From the pebbles dropped into Athenian urns to the secure digital ballots of the 21st century, the mechanisms change but the fundamental question endures: how should the many decide?
Cleisthenes establishes demokratia in Athens. Citizens vote by show of hands (cheirotonia) or by depositing pottery shards (ostraka) to exile dangerous leaders -- the origin of "ostracism."
English Parliament restricts voting rights to male landowners with property worth 40 shillings. The franchise becomes tied to wealth -- a restriction that would endure for centuries across Europe.
"The ballot is stronger than the bullet." -- Abraham Lincoln, 1858
New Zealand becomes the first self-governing nation to grant women the right to vote, following the petition led by Kate Sheppard signed by nearly 32,000 women -- almost a quarter of the adult female population.
The United States passes the Voting Rights Act, prohibiting racial discrimination in voting. The act follows years of civil rights activism, including the Selma to Montgomery marches.
Estonia becomes the first country to hold legally binding internet elections. Voters authenticate using national ID cards and cast ballots from home computers.
Perfection in voting is mathematically impossible. Every system that aggregates the preferences of three or more individuals into a collective ranking will, under certain conditions, violate at least one reasonable fairness criterion. This is not a failure of design but a feature of the problem itself.
Kenneth Arrow proved in 1951 that no ranked voting system can simultaneously satisfy all of: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. Every system must sacrifice at least one.
No perfect system exists.
Collective preferences can be cyclic even when individual preferences are not. Voter 1 prefers A > B > C, Voter 2 prefers B > C > A, Voter 3 prefers C > A > B. The majority prefers A to B, B to C, and C to A -- a logical circle with no clear winner.
In plurality voting, a third candidate who cannot win can nonetheless change the outcome by splitting the vote of a similar candidate. The "spoiler" draws support from the stronger of the two similar candidates, handing victory to their mutual opponent.
Plagues plurality systems Addressed by ranked choice
A reference compiled for citizens, scholars, and the curious.
Est. MMXXVI.