The Condorcet winner paradox

A Condorcet winner beats every other candidate head-to-head. The Condorcet winner paradox is a voting method failing to elect that candidate — the electorate's proven pairwise favorite loses. (Felsenthal classifies this as a simple paradox: the ballots as cast already produce the surprising result.)

→ Glossary: Condorcet winner · Felsenthal's taxonomy: README

The 7-voter demonstration

In BV2144 — Felsenthal Example 1, Bo beats Ana 4–3 and Cal 5–2 head-to-head — Bo is the Condorcet winner. Choose-One counts only first choices (Ana 3, Bo 2, Cal 2) and elects Ana, a candidate Bo defeats outright. Seven voters are enough.

The same election's STAR race elects Bo: the runoff is itself a head-to-head, so the pairwise favorite that reaches the finalist pair wins it.

Which methods are vulnerable

Method	Vulnerable?	Why / where demonstrated
Choose-One (Plurality)	Yes	First-choice counts ignore every pairwise fact — BV2144
RCV-IRV	Yes	Center squeeze eliminates the broad favorite early — BV2137 center squeeze, center_squeeze_irv, BV2158 (Ossipoff)
STAR	Rarely	Usually elects the Condorcet winner, but can miss one — honestly shown in BV2156 (STAR's miss)
Ranked Robin	No	Elects the Condorcet winner by construction whenever one exists — ranked_robin.md

Note the IRV-specific wording: center squeeze is a property of IRV's elimination order, not of ranked ballots — Ranked Robin counts the same ballots and finds the Condorcet winner.

Why it matters

"Beats everyone head-to-head" is the strongest simple claim a candidate can make on the ballots. A method that can reject that candidate is rejecting the electorate's own pairwise verdicts — which is exactly what voters point to when they feel a result was wrong.