Multiple districts paradox

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 majority-rule methods[2] and instant-runoff (or ranked-choice) voting.

[3][2] As shown below for Kemeny-Young and majority judgment, these three variants do not always agree with each other (which contrasts with most other voting criteria).

[2] This example shows that Copeland's method violates the consistency criterion.

Finally, the Copeland winner of the complete set of voters is determined.

This example shows that Instant-runoff voting violates the consistency criterion.

Finally, the instant runoff winner of the complete set of voters is determined.

C has the fewest first preferences and so is eliminated first, its votes are split: 4 are transferred to B and 3 to A.

Thus, A wins ahead of B and C. Now, the Kemeny-Young winner for the second group of voters is determined.

Finally, the Kemeny-Young winner of the complete set of voters is determined.

In fact, it is the only Condorcet method that satisfies ranking consistency.

is computed by summing up the number of pairwise comparisons on each ballot that match the ranking

), computing the Kemeny-Young scores for these subsets and adding it up: Now, consider an election with electorate

The premise of reinforcement is to divide the electorate arbitrarily into two parts

in the entire electorate is bigger than the Kemeny-Young score of every other ranking

: Thus, the Kemeny-Young method is consistent with respect to complete rankings.

Finally, the majority judgment winner of the complete set of voters is determined.

However, both groups combined elect B as the Majority Judgment winner.

This example shows that the ranked pairs method violates the consistency criterion.

The results would be tabulated as follows: The sorted list of victories would be: Result: B > C and A > B are locked in first (and C > A can't be locked in after that), so the full ranking is A > B > C. Thus, A is elected ranked pairs winner by the first group of voters.

The results would be tabulated as follows: The sorted list of victories would be: Result: Taking only the votes of the second group in account, A > C and C > B are locked in first (and B > A can't be locked in after that), so the full ranking is A > C > B.

Thus, A is elected ranked pairs winner by the second group of voters.

Finally, the ranked pairs winner of the complete set of voters is determined.

However, both groups combined elect B as the ranked pairs winner.

Thus, the ranked pairs method fails the consistency criterion.