Multiple districts paradox

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A voting system satisfies join-consistency (also called the reinforcement criterion) if combining two sets of votes, both electing A over B, always results in a combined electorate that ranks A over B.^[1] It is a stronger form of the participation criterion. Systems that fail the consistency criterion (such as instant-runoff voting or Condorcet methods) are susceptible to the multiple-district paradox, a pathological behavior where a candidate can win an election without carrying even a single precinct.^[1] Conversely, it can be seen as allowing for a particularly egregious kind of gerrymander: it is possible to draw boundaries in such a way that a candidate who wins the overall election fails to carry even a single electoral district.^[1]

Rules susceptible to the multiple-districts paradox include all Condorcet methods^[2] and instant-runoff (or ranked-choice) voting. Rules that are not susceptible to it include all positional voting rules (such as first-preference plurality and the Borda count) as well as score voting and approval voting.