Monotonicity: Difference between revisions

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The monotonicity criterion renders the intuition that there should be neither need to worry about harming a candidate by (nothing else than) up-ranking nor it should be possible to support a candidate by (nothing else than) counter-intuitively down-ranking.
The monotonicity criterion renders the intuition that there should be neither need to worry about harming a candidate by (nothing else than) up-ranking nor it should be possible to support a candidate by (nothing else than) counter-intuitively down-ranking.

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There are several variations of that criterion; e.g., what Douglas R. Woodall called ''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. Agreement with such rather special properties is the best any ranked voting system may fulfill: The [[Gibbard–Satterthwaite theorem]] shows, that any meaningful ranked voting system is susceptible to some kind of [[tactical voting]], and [[Arrow's impossibility theorem]] shows that individual rankings can't be meaningfully translated into a community-wide ranking where the order of candidates {{math|''x''}} and {{math|''y''}} is always [[Independence of irrelevant alternatives|independent of irrelevant alternatives]] {{math|''z''}}.<!--


The result of David Austen-Smith and Jeffrey Banks that monotonicity in individual preferences is impossible is a nonissue: For given voter preferences v=v_1...v_n and a winner x under voting scheme alpha, they investigate changes in v, where e.g. altering v_i from a,b,c,d,x to d,c,x,b,a is allowed, which can't be seriously named a monotonicity property. That allows random permutations even ''ahead'' of x, and is therefore even more rigid than Woodall's mono-raise-random, which is already incompatible with [majority AND later-no-help AND later-no-harm].
The result of David Austen-Smith and Jeffrey Banks that monotonicity in individual preferences is impossible is a nonissue: For given voter preferences v=v_1...v_n and a winner x under voting scheme alpha, they investigate changes in v, where e.g. altering v_i from a,b,c,d,x to d,c,x,b,a is allowed, which can't be seriously named a monotonicity property. That allows random permutations even ''ahead'' of x, and is therefore even more rigid than Woodall's mono-raise-random, which is already incompatible with [majority AND later-no-help AND later-no-harm].
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==Other forms of Monotonicity ==
==Other forms of Monotonicity ==
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There are several variations of the "monotonicity criterion". For example, there's what Douglas R. Woodall called "mono-add-plump". These are described in the following section. Agreement with such rather special properties is the best any ranked voting system may fulfill: The [[Gibbard–Satterthwaite theorem]] shows, that any meaningful ranked voting system is susceptible to some kind of [[tactical voting]], and [[Arrow's impossibility theorem]] shows that individual rankings can't be meaningfully translated into a community-wide ranking where the order of candidates {{math|''x''}} and {{math|''y''}} is always [[Independence of irrelevant alternatives|independent of irrelevant alternatives]] {{math|''z''}}.


=== Single-winner monotonicity ===
=== Woodall ===


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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:

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


* l ([[Mono-raise criterion|'''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;
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* l (MONO--SUB-PLUMP) some ballots that do not have x top are replaced by ballots that have x top with no second choice;
* l (MONO--SUB-PLUMP) some ballots that do not have x top are replaced by ballots that have x top with no second choice;
* 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-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-PLUMP): A candidate {{math|''x''}} should not be harmed if further ballots are added that have {{math|''x''}} top with no second choice.
* l ([[Mono-add-top criterion|'''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>
* l (MONO-REMOVE-BOTTOM) some ballots are removed, all of which have x bottom, below all other candidates.

</blockquote>


=== Multi-winner monotonicity ===
=== Multi-winner monotonicity ===
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