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| Photo credit: Commonwealth Secretariat, Nigeria Elections 2019 | https://www.flickr.com/photos/comsec/47206819721/in/photostream/ |
Democracy, in its most familiar form—"one person, one vote"—rests on an intuitive appeal. It feels fair. Yet mathematicians and social choice theorists have spent decades demonstrating that this intuition is mathematically flawed. The claim that "democracy is impossible" is not a political opinion but a formal theorem. The claim that plurality voting—the system used in the United States and many other countries—is among the worst is also grounded in rigorous mathematical analysis. This article surveys the mathematical case against conventional democracy, reviews the empirical evidence on voting system performance, and examines mathematically superior alternatives.
Arrow's Impossibility Theorem and Its Implications.
The Core Problem.
The foundational argument that no voting system can be "perfect" was established by economist Kenneth Arrow in the 1950s. Arrow's Impossibility Theorem proves that when there are three or more choices, no preference aggregation rule can simultaneously satisfy a set of seemingly essential fairness criteria. These criteria include unanimity (if everyone prefers A to B, the group does too), transitivity (if the group prefers A to B and B to C, it prefers A to C), independence of irrelevant alternatives (the ranking of A and B should not depend on preferences for C), unrestricted domain (the system should handle any set of individual preferences), and non-dictatorship (no single voter should determine the outcome).
The theorem does not say democracy is impossible in practice. It says that a perfect democratic decision procedure is mathematically impossible. Any system will inevitably violate at least one of these intuitive fairness criteria under some conditions.
Statistical Evidence: Some Systems Are Worse Than Others.
While no system can be perfect, systems are not equally bad. Recent simulation studies have quantified how frequently different voting systems violate Arrow's conditions. In a 2026 study published in Theory and Decision, researchers simulated elections with up to 10,000 voters and between three and six candidates across thirteen voting systems. The results were striking: for all systems except Pairwise Majority, the frequency of jointly violating Arrow's criteria in elections with at least 30 voters and four alternatives exceeded 98%. In other words, almost every election run under most voting systems will produce a mathematically "unfair" outcome.
A related study, "Finding the Fairest Voting System using Likelihood Analysis" (2023), extended this analysis to nine fairness criteria with simulations involving up to 10 million voters. It found that Pairwise Majority was most likely to jointly satisfy all criteria, though it increasingly violated transitivity as the number of candidates increased.
Among systems that preserved transitivity, the Baldwin method performed best in three-candidate elections. Critically, as the number of candidates increased, the probability of satisfying all fairness criteria decreased rapidly for all systems.
Why Plurality Voting Is the Worst.
The Spoiler Effect and Wasted Votes.
Plurality voting—where the candidate with the most votes wins—is the system most commonly criticized by mathematicians and social choice theorists. Mathematician Keith Devlin has stated plainly that "pretty well the only thing everyone does agree on is that the present system—plurality voting—is the worst". The system is uniquely vulnerable to the spoiler effect, where a third-party candidate "splits" the vote with a similar major candidate, allowing a less popular candidate to win. This is not a bug but a mathematical feature of the system.
Game Theory and Polarization.
Beyond the spoiler effect, plurality voting creates perverse incentives analyzed through game theory. In a first-past-the-post system, voters face a strategic dilemma: they must vote for a candidate likely to win rather than their preferred candidate, or risk "throwing away" their vote. This incentive structure drives the consolidation of political systems into two polarized parties.
A game-theoretic model illustrates the problem. Consider voters distributed along an ideological spectrum. Under plurality voting, a right-leaning voter who prefers a centrist candidate may nevertheless vote for a far-right candidate, calculating that only the far-right candidate has a realistic chance of defeating the opposing far-left candidate.
The centrist voter, seeing both extremes as viable, may choose one arbitrarily. The result is that extreme candidates win while total voter satisfaction decreases. The system rewards candidates who cater to their base rather than seeking broad consensus, because alienating the base risks losing strategic voters to the opposition.
Mathematically Superior Alternatives.
Since perfection is impossible, the mathematical challenge is to identify systems that are "less bad"—that produce fairer outcomes more consistently. The empirical simulation studies provide guidance on which systems perform best.
Pairwise Majority and Condorcet Methods.
Pairwise Majority (also called Condorcet methods) selects the candidate who would beat every other candidate in a one-on-one race. Simulation studies consistently show this system is least likely to violate Arrow's fairness criteria. However, a Condorcet winner does not always exist, and the system can produce cycles (A beats B, B beats C, C beats A) that violate transitivity, especially as the number of candidates grows.
The Baldwin Method.
Among systems that preserve transitivity, the Baldwin method—a ranked voting system that iteratively eliminates candidates with the fewest Borda votes—performs best in three-candidate elections. Like all systems, its performance deteriorates as the number of candidates increases.
Approval Voting.
Approval Voting allows each voter to vote for every candidate they approve of; the candidate with the most approvals wins. The system is mathematically simple and effective at eliminating the spoiler effect, since voters can support both a preferred third-party candidate and a viable major-party candidate. An early real-world experiment occurred in the 1985 election of The Institute of Management Sciences (TIMS), providing empirical comparison with plurality voting.
However, approval voting has known vulnerabilities. It is manipulable by strategic voters, and it is not "statistically robust"—meaning the winner of a sample of ballots may not match the winner of the full profile. Nonetheless, its simplicity and resistance to the spoiler effect make it a significant improvement over plurality.
Range (Score) Voting and Majority Judgment.
Range Voting has voters score each candidate on a numerical scale (e.g., 0-10); the candidate with the highest total score wins. A special case is approval voting, where scores are restricted to 0 and 1. Computer simulations have found range voting to be among the fairest methods, though both range voting and its variant Majority Judgment (which uses median rather than average scores) are vulnerable to counterintuitive outcomes, including the possibility that a candidate preferred by only one voter can win.
The No-Show Paradox also plagues these systems: new voters can appear and cause their preferred candidate to lose.
Sortition: Random Selection.
A radical departure from voting altogether is sortition—selecting representatives by lottery, like juries. Mathematically, this ensures a statistically representative sample of the population, avoiding the distortions of campaigning, fundraising, and strategic voting.
Historically, ancient Athens and renaissance Venice used sortition precisely to prevent factional discord. A proportional lottery, where the probability of an option being selected is determined by the relative number of votes cast in its favor, has been analyzed as a defense against rent-seeking interest groups.
Modern applications include citizens' assemblies and pilot programs in various local contexts. Proponents argue sortition is the only system that can truly achieve representation, since elections inevitably produce elites and are vulnerable to manipulation.
Quadratic Voting.
Quadratic Voting allows voters to buy additional votes on issues they care about most, with the cost of each additional vote increasing quadratically (1, 4, 9, 16...). This mathematically allows voters to express the intensity of their preferences, addressing a key limitation of "one person, one vote," where a narrow majority can impose its will on an intense minority.
Recent research explores combining quadratic voting with sortition to ensure both minority representation and the expression of preference intensity.
Part IV: Experimental Evidence on Voter Preferences.
An experimental study published in the European Journal of Political Research (2026) examined whether voters actually prefer fairer electoral systems. Across four countries with different electoral rules, researchers found that people do want a fair electoral formula, supporting what political scientist Arend Lijphart called a "kinder, gentler" rule.
The study used vignettes presenting fictional election outcomes under different voting systems and measured respondents' satisfaction. The findings suggest that voters are not indifferent to electoral fairness; they recognize and prefer systems that produce outcomes proportionate to votes cast.
Conclusion.
Mathematics does not say democracy is impossible in practice. It says that a perfect democratic voting system is mathematically impossible. But it also says that some systems are vastly superior to others, and it identifies our most common system—plurality voting—as among the worst.
- Adopting Pairwise Majority or Condorcet methods, which simulation studies show are least likely to violate fairness criteria.
- Implementing Approval Voting, which eliminates the spoiler effect and is simple to administer and understand.
- Exploring Range Voting and Majority Judgment, which allow expression of preference intensity despite some known paradoxes.
- Experimenting with sortition and quadratic voting, which address representation and intensity in novel ways.
- Replacing plurality with proportional representation to reduce wasted votes and ensure that the composition of legislative bodies reflects the electorate.
The mathematical critique of democracy is not a counsel of despair. It is a call to take electoral design seriously—to recognize that the choice of voting system is as consequential as the choice of candidates, and to select systems that maximize fairness, representation, and legitimacy.
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