Monotonicity: Difference between revisions
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[[Douglas Woodall]] defined several variations or types of monotonicity.<ref name="Woodall-Monotonicity" /> In Woodall's definitions, a candidate x should not be harmed if: |
[[Douglas Woodall]] defined several variations or types of monotonicity.<ref name="Woodall-Monotonicity" /> In Woodall's definitions, a candidate x should not be harmed if: |
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* ([[Mono-raise criterion|'''mono-raise''']]) x is raised on some ballots without changing the orders of the other candidates; |
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* ('''mono-raise-delete''') x is raised on some ballots and all candidates now below x on those ballots are deleted from them; |
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* ('''mono-raise-random''') x is raised on some ballots and the positions now below x on those ballots are filled (or left vacant) in any way that results in a valid ballot; |
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* ('''mono-append''') x is added at the end of some ballots that did not previously contain x; |
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* ('''mono-sub-plump''') some ballots that do not have x top are replaced by ballots that have x top with no second choice; |
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* ('''mono-sub-top''') some ballots that do not have x top are replaced by ballots that have x lop (and are otherwise arbitrary); |
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* ('''mono-add-plump'''): A candidate {{math|''x''}} should not be harmed if further ballots are added that have {{math|''x''}} top with no second choice. |
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* ([[Mono-add-top criterion|'''mono-add-top''']]) further ballots are added that have x top (and are otherwise arbitrary); |
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* ('''mono-remove-bottom''') some ballots are removed, all of which have x bottom, below all other candidates. |
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=== Multi-winner monotonicity === |
=== Multi-winner monotonicity === |
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Monotonicity would be more aptly named ''endorsement monotonicity'' since it is the preservation of monotonicity relative to endorsement. Since it is the most important form of monotonicity is bears the simple naming. There are however two other important forms of monotonicity for multi-winner voting systems, [[Population monotonicity]] and [[House monotonicity criterion |House monotonicity]]. |
Monotonicity would be more aptly named ''endorsement monotonicity'' since it is the preservation of monotonicity relative to endorsement. Since it is the most important form of monotonicity is bears the simple naming. There are however two other important forms of monotonicity for multi-winner voting systems, [[Population monotonicity]] and [[House monotonicity criterion |House monotonicity]]. |
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Multi-winner monotonicity could also be considered in a weaker and stronger sense: the weak form is satisfied whenever, if A is one of the winners, ranking A higher does not kick A out of the winning set; whereas the stronger form is satisfied whenever, if A is one of the winners, ranking A higher does not kick anyone out of the winning set. In a single winner election, these criteria are the same, but the stronger form is harder to satisfy for multi-winner. |
Multi-winner monotonicity could also be considered in a weaker and stronger sense: the weak form is satisfied whenever, if A is one of the winners, ranking A higher does not kick A out of the winning set; whereas the stronger form is satisfied whenever, if A is one of the winners, ranking A higher does not kick ''anyone'' out of the winning set. In a single winner election, these criteria are the same, but the stronger form is harder to satisfy for multi-winner. |
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==Footnotes== |
==Footnotes== |
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