Monotonicity: Difference between revisions

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==Other forms of Monotonicity ==

=== Single-winner monotonicity ===

There are several variations or types of monotonicity: <blockquote>MONOTONICITY. A candidate x should not be harmed if:

* l (MONO-RAISE) x is raised on some ballots without changing the orders of the other candidates;

* l ([[Mono-raise criterion|'''MONO-RAISE''']]) x is raised on some ballots without changing the orders of the other candidates;

* l (MONO-RAISE-DELETE) x is raised on some ballots and all candidates now below x on those ballots are deleted from them;

* l (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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* l (MONO-SUB-TOP) some ballots that do not have x top are replaced by ballots that have x lop (and are otherwise arbitrary);

* l (MONO-ADD-PLUMP) further ballots are added that have x top with no second choice;

* l (MONO-ADD-TOP) further ballots are added that have x top (and are otherwise arbitrary);

* l ([[Mono-add-top criterion|'''MONO-ADD-TOP''']]) further ballots are added that have x top (and are otherwise arbitrary);

* l (MONO-REMOVE-BOTTOM) some ballots are removed, all of which have x bottom, below all other candidates. <ref>https://www.rangevoting.org/Woodall97.pdf</ref>

</blockquote>

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~~</blockquote>~~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]].

=== Multi-winner 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]].

== See also ==