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Encyclopedia > Monotonicity criterion

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. A voting system criterion is a formally defined pass/fail criterion by which a voting system may be assessed. ... A voting system is a means of choosing between a number of options, based on the input of a number of voters. ...

Douglas 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.

Mike Ossipoff defines the monotonicity criterion as:

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.

The definitions are logically equivalent. 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. In logic, statements p and q are logically equivalent if they have the same logical content. ...

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." [1] In voting systems, Arrowâ€™s impossibility theorem, or Arrowâ€™s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ...

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. In voting systems, tactical voting (or strategic voting) occurs when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome. ...

Of the single-winner voting systems, First Past the Post, Borda count, Schulze method, and 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. The plurality voting system, also known as first past the post, is a voting system used to elect a single winner in a given election. ... The Borda count is a single winner election method in which voters rank candidates in order of preference. ... The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. ... Maximize Affirmed Majorities (MAM) is a voting method developed by Stephen Eppley that selects a single winner using votes that express preferences. ... The Coombs method, created by Clyde Coombs, is a voting system used for single-winner elections in which each voter rank-orders the candidates. ... Example Instant-runoff voting ballot Instant-runoff voting (IRV) is a voting system most commonly used for single member elections in which voters have one vote, but can rank candidates in order of preference. ... Range voting (also called ratings summation, average voting, cardinal ratings, 0â€“99 voting, or the score system or point system) is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins. ... On an approval ballot, the voter can vote for any number of candidates. ... Preferential voting (or preference voting) is a type of ballot structure used in several electoral systems in which voters rank a list or group of candidates in order of preference. ...

Of the multiple-winner voting systems, all plurality voting methods are monotonic, such as 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. The first-past-the-post electoral system is a voting system for single-member districts, variously called first-past-the-post (FPTP or FPP), winner-take-all, plurality voting, or relative majority. ... Bloc voting (or block voting) refers to a class of voting systems which can be used to elect several representatives from a single multimember constituency. ... A points method ballot design like this one is the most common for governmental elections using cumulative voting. ... The Single Non-Transferable Vote or SNTV is an electoral system used in multi-member constituency elections. ... This STV ballot for the Australian Senate illustrates group voting tickets. ...

### Approval voting

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### Borda count

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### Instant-runoff voting

Suppose a president were being elected by instant runoff. Also suppose there are 3 candidates, and 100 votes cast. The number of votes required to win is therefore 51. For other uses, see President (disambiguation). ... Example Instant-runoff voting ballot Instant-runoff voting (IRV) is a voting system most commonly used for single member elections in which voters have one vote, but can rank candidates in order of preference. ...

Suppose the votes are cast as follows:

 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. In fact, of course, it was not the increase in support for Andrea that hurt her.

Non-monotionic scenarios for IRV are frequently miss-presented along the lines of; "Having more voters support candidate A can cause A to switch from being a winner to being a loser." Note that it is not the fact that A gets more votes that causes A to lose. In fact that, by itself, can never cause a candidate to lose with IRV. The actual cause is the shift of support among other candidates, (in the example above, the decline in support for Belinda) which changes which candidate A faces in the final match-up.

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. [2] Nicholas Miller disputes this conclusion and provides a different mathematical model. [3]

### Kemeny-Young method

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### Minimax Condorcet

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### Plurality voting system

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### Range voting

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### Ranked Pairs

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### Two-round system (traditional runoff voting)

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### Schulze method

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## References

• Woodall, Douglas R. "Monotonicity and Single-Seat Election Rules" Voting Matters, Issue 6, 1996.

A voting system is a means of choosing between a number of options, based on the input of a number of voters. ... A voting system criterion is a formally defined pass/fail criterion by which a voting system may be assessed. ...

Results from FactBites:

 Monotonicity criterion - Wikipedia, the free encyclopedia (527 words) In mathematics, monotonicity usually refers to the different concept of a monotonic function. Furthermore, although all voting systems are vulnerable to tactical voting, systems which fail the monotonicity criterion suffer an unusual form, where voters might try to elect their candidate by voting against that candidate. Range voting and approval voting are monotonic as you never help a candidate by reducing or removing support for them, but require a slightly different definition as they are not preferential systems.
More results at FactBites »

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