4,194
edits
Monotonicity: Difference between revisions
Created a new #Woodall section, and moved much of the prose into this section
(Created new #Details section, moving most of the introduction into that section) |
(Created a new #Woodall section, and moved much of the prose into this section) |
||
Line 8:
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.
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].
Line 101 ⟶ 99:
==Other forms of Monotonicity ==
▲There are several variations of
===
▲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;
Line 113 ⟶ 111:
* 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-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-REMOVE-BOTTOM) some ballots are removed, all of which have x bottom, below all other candidates.
=== Multi-winner monotonicity ===
| |||