A voting system is consistent if, whenever the electorate is divided (arbitrarily) into several parts and elections in those parts garner the same result, then an election of the entire electorate also garners that result. Smith calls this property separability and Woodall calls it convexity.
It has been proven a ranked voting system is "consistent if and only if it is a scoring function", i.e. a positional voting system. Borda count is an example of this.
The failure of the consistency criterion can be seen as an example of Simpson's paradox.
As shown below under Kemeny-Young, passing or failing the consistency criterion can depend on whether the election selects a single winner or a full ranking of the candidates (sometimes referred to as ranking consistency); in fact, the specific examples below rely on finding single winner inconsistency by choosing two different rankings with the same overall winner, which means they do not apply to ranking consistency.
Examples
Copeland
This example shows that Copeland's method violates the consistency criterion. Assume five candidates A, B, C, D and E with 27 voters with the following preferences:
Now, the set of all voters is divided into two groups at the bold line. The voters over the line are the first group of voters; the others are the second group of voters.
First group of voters
In the following the Copeland winner for the first group of voters is determined.
The results would be tabulated as follows:
[X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption [Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption
Result: With the votes of the first group of voters, A can defeat three of the four opponents, whereas no other candidate wins against more than two opponents. Thus, A is elected Copeland winner by the first group of voters.
Second group of voters
Now, the Copeland winner for the second group of voters is determined.
The results would be tabulated as follows:
Result: Taking only the votes of the second group in account, again, A can defeat three of the four opponents, whereas no other candidate wins against more than two opponents. Thus, A is elected Copeland winner by the second group of voters.
All voters
Finally, the Copeland winner of the complete set of voters is determined.
The results would be tabulated as follows:
Result: C is the Condorcet winner, thus Copeland chooses C as winner.
Conclusion
A is the Copeland winner within the first group of voters and also within the second group of voters. However, both groups combined elect C as the Copeland winner. Thus, Copeland fails the consistency criterion.
Instant-runoff voting
This example shows that Instant-runoff voting violates the consistency criterion. Assume three candidates A, B and C and 23 voters with the following preferences:
Now, the set of all voters is divided into two groups at the bold line. The voters over the line are the first group of voters; the others are the second group of voters.
First group of voters
In the following the instant-runoff winner for the first group of voters is determined.
B has only 2 votes and is eliminated first. Its votes are transferred to A. Now, A has 6 votes and wins against C with 4 votes.
Result: A wins against C, after B has been eliminated.
Second group of voters
Now, the instant-runoff winner for the second group of voters is determined.
C has the fewest votes, a count of 3, and is eliminated. A benefits from that, gathering all the votes from C. Now, with 7 votes A wins against B with 6 votes.
Result: A wins against B, after C has been eliminated.
All voters
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, B wins with 12 votes against 11 votes of A.
Result: B wins against A, after C is eliminated.
Conclusion
A is the instant-runoff winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the instant-runoff winner. Thus, instant-runoff voting fails the consistency criterion.
Kemeny-Young method
This example shows that the Kemeny–Young method violates the consistency criterion. Assume three candidates A, B and C and 38 voters with the following preferences: