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

TL;DR

This paper proves that no voting method can simultaneously satisfy the Condorcet criteria, positive involvement, and resolvability, highlighting fundamental conflicts in voting rule design.

Contribution

It establishes an impossibility theorem showing the incompatibility of key voting criteria with positive involvement and resolvability.

Findings

No voting method satisfies all three criteria simultaneously.

The theorem holds for any number of voters.

Incompatibility persists under various criterion replacements.

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 $n$. It also holds if either the Condorcet loser criterion is replaced by independence of clones or positive involvement is replaced by negative involvement.