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Concept
Consistency criterion
Summary
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.
Copeland's method
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.
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.
Now, the Copeland winner for the second group of voters is determined.
Official source
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