- This article is about a voting system criterion. See monotonic function for a mathematical notion.
The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting systems. A voting system is monotonic if it satisfies one of the definitions of the monotonicity criterion, given below.
Douglas R. Woodall, calling the criterion mono-raise, defines it as:
- A candidate x should not be harmed [i.e., change from being a winner to a loser] if x is raised on some ballots without changing the orders of the other candidates.[1]
Note that the references to orders and relative positions concern the rankings of candidates other than X, on the set of ballots where X has been raised. So, if changing a set of ballots voting "A > B > C" to "B > C > A" causes B to lose, this does not constitute failure of Monotonicity, because in addition to raising B, we changed the relative positions of A and C.
This criterion may be intuitively justified by reasoning that in any fair voting system, no vote for a candidate, or increase in the candidate's ranking, should instead hurt the candidate. It is a property considered in Arrow's impossibility theorem. Some political scientists, however, doubt the value of monotonicity as an evaluative measure of voting systems. David Austen-Smith and Jeffrey Banks, for example, published an article in The American Political Science Review in which they argue that "monotonicity in electoral systems is a nonissue: depending on the behavioral model governing individual decision making, either everything is monotonic or nothing is monotonic."[2]
Although all voting systems are vulnerable to tactical voting, systems which fail the monotonicity criterion suffer an unusual form, where voters with enough information about other voter strategies could theoretically try to elect their candidate by counter-intuitively voting against that candidate. Tactical voting in this way presents an obvious risk if a voter's information about other ballots is wrong, however, and there is no evidence that voters actually pursue such counter-intuitive strategies in non-monotonic voting systems in real-world elections.
Of the single-winner voting systems, plurality voting (first past the post), Borda count, Schulze method, and Ranked Pairs (Maximize Affirmed Majorities) are monotonic, while Coombs' method and instant-runoff voting are not. The single-winner methods of range voting and approval voting are also monotonic as one can never help a candidate by reducing or removing support for them, but these require a slightly different definition of monotonicity as they are not preferential systems.
Of the multiple-winner voting systems, all plurality voting methods are monotonic, such as plurality-at-large voting (bloc voting), cumulative voting, and the single non-transferable vote. Those versions of the single transferable vote, which simplify to instant runoff when there is only one winner, are not monotonic.
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Example
Instant-runoff voting (Not Monotonic)
Suppose a president were being elected by instant runoff. Also suppose there are three candidates, and 100 votes cast. The number of votes required to win is therefore 51.
Suppose the votes are cast as follows:
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Number of votes 1st Preference 2nd Preference 39 Andrea Belinda 35 Belinda Cynthia 26 Cynthia Andrea
Cynthia is eliminated, thus transferring votes to Andrea, who is elected with a majority. She then serves a full term, and does such a good job that she persuades ten of Belinda's supporters to change their votes to her at the next election.
This election looks thus:
-
Number of votes 1st Preference 2nd Preference 49 Andrea Belinda 25 Belinda Cynthia 26 Cynthia Andrea
Because of the votes Belinda loses, she is eliminated first this time, and her second preferences are transferred to Cynthia, who now wins 51 to 49. In this case Andrea's preferential ranking increased between elections - more electors put her first - but this increase in support appears to have caused her to lose. Counterintuitively, it was the increase in support for Andrea (along with the properties of IRV) that hurt her.
In a real election, however, such problems may be more difficult to detect because there would be other movements of votes, and it may not be easily determined whether the same people cast the same votes.
Crispin Allard argues that the circumstances under which this could occur would be extremely rare, fewer than once per century under normal political conditions. [3] Nicholas Miller disputes this conclusion and provides a different mathematical model. [4] The 2009 Burlington, Vermont mayor election featured non-monotonicity.[5]
References
- ^ D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6, 1996
- ^ Monotonicity in Electoral Systems
- ^ Estimating the Probability of Monotonicity Failure in a UK General Election
- ^ Monotonicity Failure Under STV and Related Voting Systems
- ^ Burlington Vermont 2009 IRV mayor election
Some parts of this article are derived from text at http://condorcet.org/emr/criteria.shtml
See also
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