Monotonicity: Difference between revisions

{{wikipedia|Monotonicity criterion}}

 

 

 

{{Definition|If an alternative X loses, and the ballots are changed only by placing X in lower positions, without changing the relative position of other candidates, then X must still lose.}}

[[Plurality voting]], [[Majority Choice Approval]], [[Borda count]], [[Schulze method|Schulze]], [[Maximize Affirmed Majorities]], and [[Descending Solid Coalitions]] are monotonic, while [[Coombs' method]] and [[Instant-runoff voting]] are not. [[Approval voting]] is monotonic, using a slightly different definition, because it is not a preferential system: You can never help a candidate by not voting for them.<ref>Some parts of this article are derived from text at https://web.archive.org/web/20090610060543/http://condorcet.org/emr/criteria.shtml which was released to the [[project:public domain|public domain]].</ref><ref>Mono-raise test copied from https://electowiki.org/w/index.php?title=Mono-raise_criterion&oldid=15949</ref>

 

In deterministic single winner elections that is to say no winner is harmed by up-ranking and no loser can win by down-ranking. If the method relies on chance, then up-ranking a candidate can not decrease that candidate's chance of winning, nor can down-ranking the candidate increase it. Douglas R. Woodall called the criterion '''mono-raise'''.<ref name="Woodall-Monotonicity">{{cite journal| title = Monotonicity and single-seat election rules| last = Woodall| first = Douglas R.| journal = Voting matters| volume = 6| pages = 9–14| year = 1996|url=http://www.votingmatters.org.uk/ISSUE6/P4.HTM}}</ref>

 

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.

 

 

The general pattern of monotonicity criteria 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.}}

 

 

Using an example that applies to [[instant-runoff voting]] (IRV) and to the [[two-round system]], it is shown that these voting systems violate the mono-raise criterion.

 

If the ballots of a real election are released, it is fairly easy to prove if

 

would have been possible (nothing else is altered on any ballot). Both events can be considered as real-life monotonicity violations.

A party-list strategy exploiting something similar (down-ranking CDU '''and''' additionally up-ranking another party, e.g. FDP) happened in the German federal election of 2005, in which conservative voters in Dresden deliberately voted against the CDU, their party of choice, in order to maximize that party's number of seats in the federal parliament. This was possible due to Germany's voting system (mixed member proportional with overhang seats computed independently for each federal state) and the fact that the vote in Dresden took place a week after the rest of the country due to the death of a candidate, enabling voters in Dresden to vote tactically in full knowledge of the results already achieved elsewhere. As a result of this, the German Constitutional Court ruled on July 3 2008 that the German voting system must be reformed to eliminate its non-monotonicity.<ref> See e.g. [http://fruitsandvotes.com/blog/?p=117]</ref>

 

 

Since every or almost every IRV election in Australia has been conducted in the black (i.e. not releasing enough information to reconstruct the ballots), nonmonotonicity is difficult to detect in Australia, even though thanks to the Lepelley ''et al'' probability estimates it seems safe to say that it must have occurred in over 100 of their elections.<ref group="nb">If the Australian election authorities were to publish the full ranking of every voter, this could be used to facilitate vote buying or coercion. However, anonymization could close the hole, e.g. by truncating the reported ballots so that for every reported ranking, at least 0.1% of the voters began their ballots that way.</ref>

 

However, for the [[Australian federal election, 2010]], one article was aware of the non-monotonicity possibility: [http://andrewnorton.info/2010/08/16/why-labor-voters-in-melbourne-need-to-vote-liberal/ Why Labor Voters In Melbourne Need To Vote Liberal]. In 2009, the theoretical disadvantage of non-monotonicity worked out in practice in a state [[by-election]] in the [[South Australia]]n seat of [[Electoral district of Frome|Frome]]. The eventual winner, an Independent who was a town mayor, scored only third on the primaries with about 21% of the vote. But since the [[National Party of Australia]] scored 4th place, their preferences were distributed beforehand, allowing the Independent to overtake the [[Australian Labor Party]] Candidate by 31 votes. Thus Labor was pushed into third place, and their preference distribution favoured the Independent, who overtook the leading [[Australian Liberal Party]] candidate to win the election. However, had anywhere between 31 and 321 of the voters who preferred Liberal over Labor and Independent switched their support from Liberal to Labor, it would have allowed the Liberal to win the IRV election. This is classic monotonicity violation: the 321 who voted for the Liberals took part in hurting their own candidate.<ref>http://blogs.abc.net.au/antonygreen/2011/05/an-example-of-non-monotonicity-and-opportunites-for-tactical-voting-at-an-australian-election.html</ref>

 

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''}}.

 

 

 

 

=== 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]].

 

 

 

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[[Category:Voting system criteria]]