Monotonicity: Difference between revisions
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In elections via the single-winner methods [[range voting]] and [[majority judgment]] nobody can help a candidate by reducing or removing support for them. The definition of the monotonicity criterion with regard to these methods is disputed. Some voting theorists argue that this means these methods pass the monotonicity criterion; others say that, as these are not ''ranked'' voting systems, they are out of the monotonicity criterion's scope.
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The general pattern of monotonicity criteria is:
If X is a winner under a voting rule, and one or more voters change their preferences in a way favourable to X, then X should still be a winner▼
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or for methods that employ some element of chance,
{{definition|If one or more voters change their preferences in a way favourable to X, then the chance that X is elected should never decrease.}}
Different definitions of "favourable to X" lead to different monotonicity criteria. The primary monotonicity criterion, mono-raise, is:
{{definition|If X is a winner under a voting rule, and one or more voters change their preferences by ranking or rating X higher without otherwise changing their ballots, then X should still be a winner.}}
==Instant-runoff voting and the two-round system are not monotonic==
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Suppose the votes are cast as follows:
{{ballots|
28: Right > Center
5: Right > Left
5: Left > Right
16: Center > Right
▲| Center
▲| Center
▲|}
According to
In an actual runoff between the top two candidates, Left would win against Right with 30+5+16=51 votes. The same happens (in this example) under IRV, Center gets eliminated, and Left wins against Right with 51 to 49 votes.
But if at least two of the five voters who ranked Right first, and Left second, would raise Left, and vote 1st Left, 2nd Right; then Left would be defeated by these votes in favor of Center.
Let's assume that two voters change their preferences in that way, which changes
{{ballots|
4: Right > Left
7: Left > Right
▲|}
Now Left gets 37 first preferences, Right only 31 first preferences, and Center still 32 first preferences, and there is again no candidate with an absolute majority of first preferences.
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