The incompatibility of the Condorcet winner and loser criteria with positive or negative involvement and resolvability

Wesley H. Holliday

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Abstract: We prove that there is no preferential voting method satisfying the Condorcet winner and loser criteria, positive involvement (if a candidate $x$ wins in an initial preference profile, then adding a voter who ranks $x$ uniquely first cannot cause $x$ to lose), and $n$-voter resolvability (if $x$ initially ties for winning, then $x$ can be made the unique winner by adding some set of up to $n$ voters). This impossibility theorem holds for any positive integer $n$. In addition, positive involvement can be replaced by negative involvement (if a candidate $x$ loses in an initial preference profile, then adding a voter who ranks $x$ uniquely last cannot cause $x$ to win). In a previous note, we proved an analogous result assuming an additional axiom of ordinal margin invariance, which we now show is unnecessary for an impossibility theorem, at least if the desired voting method is defined for five-candidate elections.

Date: 2026-01, Revised 2026-01
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