The Condorcet Paradox

The obvious first reaction is that something must be wrong with majority rule and a better rule would fix it. Condorcet's contemporary Jean-Charles de Borda had already proposed one, in a memoir read to the Académie royale des sciences in 1770: give points by rank position and add them up. The Borda count always produces a ranking, and that ranking never cycles.

Condorcet's objection to Borda was sharp and, as it turned out, permanent. The Borda count can elect an option that would lose a straight majority vote against a rival — a candidate the majority actively rejects. Worse, its winner can change when a hopeless candidate enters or drops out, because dropping a candidate reshuffles everyone's point totals. Borda's own reported reply was that his system was "only intended for honest men". Condorcet thought this missed the point: the flaw was in the arithmetic, not in the voters.

So the disagreement between them was not about which of two rules is better. It was about which defect you are willing to live with, and that framing is the one that survived.

Three ways out, each with a cost

Restrict what people are allowed to want. Black's single-peakedness works, but it is an assumption about the electorate, not a property of the rule. You cannot legislate it. And it fails precisely on multi-dimensional questions — budgets that trade off health against defence, policies that mix economic and social positions — which is to say, on most of politics.

Break the tie with a further rule. Many real procedures do this. Charles Dodgson — Lewis Carroll — rediscovered the cycle in the 1870s while writing about Oxford college elections, and proposed choosing the option that needs the fewest swaps in voters' ballots to become a Condorcet winner. It is a defensible answer. It is also, as computer scientists established a century later, computationally hard to determine in general. Other tie-breakers exist; each is defensible and none is uniquely right.

Use more information. The cycle appears because pairwise voting is told only who is ahead, never by how much. Methods that collect strength of feeling — scores, grades, approvals — do not cycle in this way. That is a real escape, and it is the main route out of Arrow's theorem too. The cost is that such methods depend on how honestly and how comparably people use the scale, which reopens a different set of problems.

How often does this actually happen?

Under the standard theoretical benchmark called impartial culture, where every voter picks one of the possible rankings uniformly at random, cycles are common enough to matter — around 9% of the time with three options and a large electorate, rising steeply as options are added.

But impartial culture is close to a worst case. Real electorates are correlated: people cluster along ideological lines, which is exactly the structure that makes single-peakedness likely. Andranik Tangian, William Gehrlein and others have argued this at length, and Adrian Van Deemen's 2014 survey of the empirical literature found that documented cycles in real elections and committee votes are rare — though he also stressed that the data are thin, because the ranked ballots needed to detect a cycle are usually never collected.

That last point deserves weight. Most voting systems in use cannot detect a cycle even in principle, because they never ask voters for a full ranking. The frequency of cycles in real life is genuinely uncertain, and the uncertainty runs in the direction of underestimation.

1299 and 1433 — the medieval anticipations. The Catalan philosopher Ramon Llull described an election method based on pairwise comparisons in works including Ars electionis (1299). Nicholas of Cusa proposed a rank-scoring method in De concordantia catholica (1433), five centuries before Borda. Both manuscripts were lost to the mainstream and only brought back into the social choice literature by Iain McLean and Arnold Urken in the 1990s. Whether Llull noticed the cycle is not established.

1770 — Borda. Jean-Charles de Borda, a naval officer and mathematician, read a memoir to the Académie royale des sciences proposing the rank-scoring method now named after him. It was published in 1781 as Mémoire sur les élections au scrutin.

1785 — Condorcet. Marie Jean Antoine Nicolas de Caritat, Marquis de Condorcet, published the Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix. Most of the book is about something else: his jury theorem, which shows that if each voter is more likely than not to be right, large majorities are almost certainly right. The cycle appears as an obstacle he runs into while working out how to extend that result to more than two options, and he treats it as a serious problem rather than a curiosity. The Essai is famously difficult — dense, disorganised, and printed with errors. It was admired more than it was read.

1794 — the author. Condorcet was among the most consistent liberals of the Revolution: an early opponent of slavery, an advocate of women's suffrage when almost no one else was, and the author of a proposed republican constitution. He voted against executing the king. Under the Terror a warrant was issued for his arrest; he hid for months in Paris, writing his Esquisse d'un tableau historique des progrès de l'esprit humain, was captured, and was found dead in a cell at Bourg-la-Reine in March 1794. The cause of death has never been settled.

1870s — Dodgson. Charles Dodgson, mathematics lecturer at Christ Church, Oxford, wrote three pamphlets on election methods for college business and rediscovered the cycle independently. There is no evidence he had read Condorcet.

1948 and 1958 — Black. The Scottish economist Duncan Black published "On the Rationale of Group Decision-making", introducing single-peaked preferences and the median voter result. His 1958 book The Theory of Committees and Elections went back to the archives, recovered Condorcet, Borda and Dodgson, and put their work at the foundation of what became public choice theory. Almost everything written on the subject since traces through Black.

1951 — Arrow. Kenneth Arrow, then a graduate student, proved that the trouble Condorcet found in majority rule is unavoidable in any rule of its type. Arrow had not read Condorcet when he proved it; he learned of the connection afterwards.

It killed the idea of "the will of the people" as a well-defined object. Before Condorcet it was natural to assume that a group has preferences the way a person does, and that voting is a measuring instrument for reading them off. The cycle shows there may be nothing there to measure. When preferences cycle, "what the group wants" is not merely hard to determine — it does not exist. Any procedure that returns an answer has, in that case, invented it.

It made procedure a substantive political question. If the running order can determine the outcome, then rules of order are not neutral plumbing. This is now standard in the study of legislatures: the power to schedule, to bundle, to decide what is amended against what, is real power. William Riker built an entire research programme on it in the 1980s, arguing under the name heresthetics that skilled politicians win by restructuring the choice rather than by changing minds, and documenting cases — the 1956 US school construction bill among them — where an amendment appears to have been introduced specifically to break a majority apart.

It launched social choice theory. Condorcet's cycle is the seed of the field that produced Arrow's theorem, the Gibbard–Satterthwaite theorem on strategic voting, Sen's work on welfare and liberty, and the modern comparison of electoral systems. Duncan Black, Kenneth Arrow, Amartya Sen and William Riker all treated the 1785 Essai as their starting text.

It is the reason "which voting system?" is a real question. Plurality, runoff, instant-runoff, Borda, Condorcet methods and score voting are not interchangeable ways of finding the same answer. On the same ballots they can produce different winners, and the cycle shows there need not be a fact of the matter about which is correct. Choosing a voting system means choosing which properties you value, in advance, knowing you cannot have all of them.

Committees and boards. Any body that decides by successive pairwise votes — motions against amendments, options against alternatives — is exposed. Standard parliamentary procedure, which amends a motion and then votes the amended motion up or down, is precisely a two-stage knockout, and the chair's control of the order is the agenda power in the paradox.

Ranked-choice reform debates. As jurisdictions adopt ranked ballots, the Condorcet criterion — does the system elect an option that beats every other head-to-head, when one exists? — is a live criterion in the argument, because instant-runoff voting does not always satisfy it. Ranked ballots also, for the first time, make cycles detectable in real elections.

Recommendation and ranking systems. Merging several ranked lists into one — search results from multiple signals, products ordered by several criteria, sports teams ranked by different polls — is formally the same problem. Rank aggregation algorithms cycle for exactly Condorcet's reason, and the literature on them, going back to Kemeny in 1959, is social choice theory under another name.

Machine learning evaluation. When model A beats model B on one benchmark, B beats C on another, and C beats A on a third, there is no "best model" to find. Head-to-head preference comparisons of language models, now a standard evaluation method, produce genuine non-transitivity, and leaderboards built from them inherit Condorcet's problem along with his warning about who chooses the match-ups.

Let $N = {1,\dots,n}$ be the voters and $X$ the alternatives. Each voter $i$ has a preference relation $\succ_i$ on $X$ that is complete, asymmetric and transitive. The pairwise majority relation $\succ_M$ is defined by

$$x \succ_M y \iff |{i \in N : x \succ_i y}| > |{i \in N : y \succ_i x}|$$

$\succ_M$ is always complete for odd $n$, but it need not be transitive. The three-voter profile above gives $A \succ_M B \succ_M C \succ_M A$.

A Condorcet winner is an $x$ with $x \succ_M y$ for all $y \ne x$. The paradox is precisely the possible non-existence of one.

How bad can it get

McGarvey's theorem (1953). For any complete asymmetric relation $R$ on a finite set $X$, there is a preference profile whose majority relation is exactly $R$. McGarvey's construction uses at most $|X|(|X|-1)$ voters; later work reduced this bound. The consequence is severe: majority rule imposes no structure at all on the collective ranking beyond completeness. Any pattern of cycles you can draw is realisable.

How likely is it

Under impartial culture — each voter's ranking drawn uniformly and independently from the $|X|!$ possible strict orderings — the probability that no Condorcet winner exists, for $|X| = 3$:

Voters $n$	Probability of no Condorcet winner
3	$1/18 \approx 5.6%$
5	$5/72 \approx 6.9%$
7	$\approx 7.5%$
$n \to \infty$	$\approx 8.77%$

The limiting value for three alternatives is $\tfrac{3}{4} - \tfrac{3}{2\pi}\arccos!\left(\tfrac13\right) \approx 0.0877$.

Adding alternatives is far more damaging than adding voters. As $n \to \infty$ the limiting probability of no Condorcet winner is roughly 17.6% for four alternatives and 25.1% for five, and it tends to 1 as $|X| \to \infty$. The number of pairwise contests grows quadratically in $|X|$ while the constraint that they cohere does not weaken, so eventually a cycle is a near-certainty.

These figures are upper-bound-flavoured. Impartial culture maximises the disorder of the profile; any correlation between voters reduces the cycle probability, and real electorates are strongly correlated.

Black's escape

Fix a linear order $<$ on $X$ (the "left–right axis"). A preference $\succ_i$ is single-peaked with respect to $<$ if there is a peak $p_i \in X$ such that

$$x < y \le p_i \implies y \succ_i x \qquad\text{and}\qquad p_i \le y < x \implies y \succ_i x$$

Black's theorem (1948). If every voter's preference is single-peaked with respect to a common order $<$ and $n$ is odd, then $\succ_M$ is transitive and the peak of the median voter is the Condorcet winner.

The proof is short. Let $p_{(1)} \le \dots \le p_{(n)}$ be the sorted peaks and $m$ the median. For any $y > m$, every voter with $p_i \le m$ — at least $(n+1)/2$ of them — prefers $m$ to $y$, since $m$ lies weakly between $p_i$ and $y$. So $m \succ_M y$. The symmetric argument covers $y < m$.

The pizza profile is not single-peaked under any ordering of the three toppings. Check the three possible orders: in each, some voter ranks the middle option last, which violates the definition. That failure is not incidental — for three alternatives and three voters, "some voter ranks the middle alternative last under every ordering" is exactly the condition that permits a cycle.

The relationship to Arrow

Pairwise majority rule satisfies unrestricted domain, the Pareto condition, independence of irrelevant alternatives, non-dictatorship and anonymity. What it fails is the requirement that the social relation be transitive. Arrow's theorem says this trade is unavoidable: keep all of Arrow's conditions and no rule delivers a transitive social ranking. Condorcet's cycle is the single most important worked example of the theorem, discovered 166 years early.