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Majority voting can go round in a circle, with every option losing to another, a flaw Condorcet found in the 1780s
Condorcet's voting paradox shows that majority rule can produce contradictory results even when every voter's preferences are consistent. Imagine three options, A, B and C, and three groups of voters whose rankings are arranged so that a majority prefers A to B, a majority prefers B to C, and a majority prefers C to A. Each voter is perfectly rational, yet the group as a whole has no winner that beats all others. The Marquis de Condorcet described this in the 18th century. Such loops are called Condorcet cycles, and they show that the choice of voting method can decide the outcome.
Majority preference is decided by pairwise contests, so it is not always transitive. If A beats B and B beats C, a majority can still favour C over A. When every option is beaten by some other option, the cycle has no winner. Methods like Condorcet voting avoid this by comparing all options head to head, but any such method has its own trade-offs.
The paradox means that no voting rule can be perfect for every case. Kenneth Arrow later generalised the result, showing that any system choosing among three or more options must break some reasonable fairness condition or be controlled by a single person. It is a basic reason political scientists study how elections are designed.
Arrow's impossibility theorem in 1951 extended the idea, showing that no ranked voting system can meet a short list of basic fairness rules when there are three or more options. Elections in many countries have shown spoiler effects, where a third candidate changes the winner. The same tension between individual rationality and group outcomes appears in economics and public policy.
background
In social choice theory, Condorcet's voting paradox (also called Condorcet's paradox or the Condorcet paradox) is a fundamental discovery by the Marquis de Condorcet that majority rule can be self-contradictory as majority preferences can become intransitive with three or more options. For example, there can be rock-paper-scissors scenarios where a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called Condorcet cycles or cyclic ties.
In such a cycle, every possible choice is rejected by the electorate in favor of another alternative, who is preferred by more than half of all voters. Thus, any attempt to ground social decision-making in majoritarianism must accept such self-contradictions (commonly called spoiler effects). Systems that attempt to do so, while minimizing the rate of such self-contradictions, are called Condorcet methods.
From Wikipedia: Condorcet paradox (CC BY-SA 4.0) ↗
From Wikipedia, the free encyclopedia · text CC BY-SA 4.0
Illustrative photo · Tomas McIntee, Public domain, via Wikimedia Commons
