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 ranked-choice voting or Condorcet methods) are susceptible to the multiple-district paradox, which allows 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]

There are three variants of join-consistency:

1. Winner-consistency: if two districts elect the same winner A, A also wins in the combined district.
2. Ranking-consistency: if two districts rank a set of candidates exactly the same way, then the combined district returns the same ranking of all candidates.
3. Grading-consistency: if two different districts assign a candidate the same overall grade to a candidate, the overall grade for the candidate must still be the same.

A voting system is winner-consistent if and only if it is a point-summing method; in other words, it must be a positional voting system or score voting (including approval voting).^[2][3]

As shown below under Kemeny-Young, whether a system passes reinforcement can depend on whether the election selects a single winner or a full ranking of the candidates (sometimes referred to as ranking consistency): in some methods, two electorates with the same winner but different rankings may, when added together, lead to a different winner. Kemeny-Young is the only ranking-consistent Condorcet method, and no Condorcet method can be winner-consistent.^[3]