Monotonicity criterion
Encyclopedia
The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting system
s. 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:
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."
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
, runoff voting
and instant-runoff voting
are not. The single-winner methods of range voting
, majority judgment
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
. Most versions of the single transferable vote
, including all variants currently in use for public elections (which simplify to instant runoff when there is only one winner) are not monotonic.
(IRV) and to the Two-round system
, it is shown, that these voting systems violate the mono-raise criterion.
Suppose a president
were being elected among three candidates, a left, a right, and a center candidate, and 100 votes cast. The number of votes for an absolute majority is therefore 51.
Suppose the votes are cast as follows:
According to the 1st preferences, Left finishes first with 35 votes, Right gets 33 votes, and Center 32 votes, thus all candidates lack an absolute majority of first preferences.
In an actual runoff between the top two candidates, Left would win against Right with 30+5+16=51 votes. The same happens under IRV, Center gets eliminated, and Left wins against Right with 51 to 49 votes.
But if at least two of the five voters who ranked Right first, and Left second, would raise Left, and vote 1st Left, 2nd Right, then Left would be defeated by these votes in favor of Left.
Let's assume that two voters change their preferences in that way, which changes two rows of the table:
Now Left gets 37 first preferences, Right only 31 first preferences, and Center still 32 first preferences, and there is again no candidate with an absolute majority of first preferences.
But now Right gets eliminated, and Center remains in round 2 of IRV (or the actual runoff in the Two-round system). And Center beats its opponent Left with a remarkable majority of 60 to 40 votes.
would be 1 in 4000; however, Lepelley et al.
found a probability of 397/6912=5.74% for 3-candidate elections.
Another probability model, the "impartial culture," yields about 15% probability. In elections with more than 3 candidates, these probabilities tend to increase eventually toward 100% (in some models this limit has been proven, in others it is only conjectured). Estimates of 5-15% order are easily confirmed in any probability model with "monte carlo experiments" and the aid of the "was it monotonic?" tests stated in the Lepelley paper. Nicholas Miller also disputed Allard's conclusion and provided a different mathematical model.
Both events are real-life monotonicity violations.
The ballots (or information allowing them to be reconstructed) are rarely released for instant runoff elections, which means there are few recorded monotonicity violations for real IRV elections.
under instant-runoff voting, where the necessary information is available. In this election, the winner Bob Kiss could have been defeated by raising him on some of the ballots. E.g. if all voters who ranked Kurt Wright over Bob Kiss over Andy Montroll, would have ranked Kiss over Wright over Montroll, and additionally some people who ranked Wright but not Kiss or Montroll, would have ranked Kiss over Wright, then these votes in favor of Kiss would have defeated him! The winner in this scenario would have been Andy Montroll, who was also the Condorcet winner according to the original ballots, i.e. for any other running candidate, a majority ranked Montroll above the competitor.
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. (The policy of Australia's election authorities not to release this data
is justifiable on privacy grounds. If rank-order ballots in an election with, say, 13 candidates, were released, even in a highly "anonymized" form, that would still provide enough information for a coercer to use to verify or deny that some voter had cast a pre-specified vote-pattern he'd demanded.)
However, for the Australian federal election, 2010, one article was aware of the non-monotonicity possibility: 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 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.
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
s. 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.
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
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
. 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."
Although all voting systems are vulnerable to tactical voting
Tactical voting
In voting systems, tactical voting occurs, in elections with more than two viable candidates, when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.It has been shown by the Gibbard-Satterthwaite theorem that any voting method which is...
, 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
Plurality voting system
The plurality voting system is a single-winner voting system often used to elect executive officers or to elect members of a legislative assembly which is based on single-member constituencies...
(first past the post), Borda count
Borda count
The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...
, Schulze method
Schulze method
The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners...
, and Ranked Pairs
Ranked Pairs
Ranked pairs or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners....
(Maximize Affirmed Majorities) are monotonic, while Coombs' method
Coombs' method
The Coombs' method is a voting system created by Clyde Coombs used for single-winner elections in which each voter rank the candidates in order of preference. It is very similar to instant-runoff voting , a more common preferential voting system.-Procedures:Each voter rank-orders all of the...
, runoff voting
Runoff voting
Runoff voting can refer to:* Two-round system, a voting system used to elect a single winner, whereby only two candidates from the first round continue to the second round...
and instant-runoff voting
Instant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...
are not. The single-winner methods of range voting
Range voting
Range voting 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.A form of range voting was apparently used in...
, majority judgment
Majority Judgment
Majority Judgment is a single-winner voting system proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner. If more than one candidate has the...
and approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...
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
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...
systems.
Of the multiple-winner voting systems, all plurality voting methods are monotonic, such as plurality-at-large voting
Plurality-at-large voting
Plurality-at-large voting is a non-proportional voting system for electing several representatives from a single multimember electoral district using a series of check boxes and tallying votes similar to a plurality election...
(bloc voting), cumulative voting
Cumulative voting
Cumulative voting is a multiple-winner voting system intended to promote more proportional representation than winner-take-all elections.- History :...
, and the single non-transferable vote
Single non-transferable vote
The single non-transferable vote or SNTV is an electoral system used in multi-member constituency elections.- Voting :In any election, each voter casts one vote for one candidate in a multi-candidate race for multiple offices. Posts are filled by the candidates with the most votes...
. Most versions of the single transferable vote
Single transferable vote
The single transferable vote is a voting system designed to achieve proportional representation through preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate, and then, after candidates have been either elected or eliminated, any surplus or...
, including all variants currently in use for public elections (which simplify to instant runoff when there is only one winner) are not monotonic.
Instant-runoff voting and the Two-round system are not monotonic
Using an example that applies to instant-runoff votingInstant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...
(IRV) and to the Two-round system
Two-round system
The two-round system is a voting system used to elect a single winner where the voter casts a single vote for their chosen candidate...
, it is shown, that these voting systems violate the mono-raise criterion.
Suppose a president
President
A president is a leader of an organization, company, trade union, university, or country.Etymologically, a president is one who presides, who sits in leadership...
were being elected among three candidates, a left, a right, and a center candidate, and 100 votes cast. The number of votes for an absolute majority is therefore 51.
Suppose the votes are cast as follows:
| Number of votes | 1st preference | 2nd preference |
| 28 | Right | Center |
| 5 | Right | Left |
| 30 | Left | Center |
| 5 | Left | Right |
| 16 | Center | Left |
| 16 | Center | Right |
According to the 1st preferences, Left finishes first with 35 votes, Right gets 33 votes, and Center 32 votes, thus all candidates lack an absolute majority of first preferences.
In an actual runoff between the top two candidates, Left would win against Right with 30+5+16=51 votes. The same happens under IRV, Center gets eliminated, and Left wins against Right with 51 to 49 votes.
But if at least two of the five voters who ranked Right first, and Left second, would raise Left, and vote 1st Left, 2nd Right, then Left would be defeated by these votes in favor of Left.
Let's assume that two voters change their preferences in that way, which changes two rows of the table:
| Number of votes | 1st preference | 2nd preference |
| 3 | Right | Left |
| 7 | Left | Right |
Now Left gets 37 first preferences, Right only 31 first preferences, and Center still 32 first preferences, and there is again no candidate with an absolute majority of first preferences.
But now Right gets eliminated, and Center remains in round 2 of IRV (or the actual runoff in the Two-round system). And Center beats its opponent Left with a remarkable majority of 60 to 40 votes.
Estimated likelihood of IRV lacking monotonicity
Crispin Allard argued, based on a mathematical model that the probability of monotonicity failure actually changing the result of an election for any given constituencyEuropean Parliament constituency
Members of the European Parliament are elected by the population of the member states of the European Union , divided into constituencies....
would be 1 in 4000; however, Lepelley et al.
found a probability of 397/6912=5.74% for 3-candidate elections.
Another probability model, the "impartial culture," yields about 15% probability. In elections with more than 3 candidates, these probabilities tend to increase eventually toward 100% (in some models this limit has been proven, in others it is only conjectured). Estimates of 5-15% order are easily confirmed in any probability model with "monte carlo experiments" and the aid of the "was it monotonic?" tests stated in the Lepelley paper. Nicholas Miller also disputed Allard's conclusion and provided a different mathematical model.
Real-life monotonicity violations
When the ballots of a real election are released, it is fairly easy to prove if it was possible- to defeat the winner by raising him on some of the ballots (without changing the orders of the other candidates)
- to push a loser at top by lowering him on some of the ballots (without changing the orders of the other candidates)
Both events are real-life monotonicity violations.
The ballots (or information allowing them to be reconstructed) are rarely released for instant runoff elections, which means there are few recorded monotonicity violations for real IRV elections.
2009 Burlington, Vermont mayoral election
A real-life monotonicity violation was detected in the 2009 Burlington, Vermont mayor electionBurlington, Vermont mayoral election, 2009
The Burlington mayoral election of 2009 occurred on Tuesday, March 3, 2009. Burlington's mayoral race is partisan election that occurs every three years, and there are no term limits. The current mayor, Bob Kiss, has served since 2006. This marks the second mayoral election in Burlington using...
under instant-runoff voting, where the necessary information is available. In this election, the winner Bob Kiss could have been defeated by raising him on some of the ballots. E.g. if all voters who ranked Kurt Wright over Bob Kiss over Andy Montroll, would have ranked Kiss over Wright over Montroll, and additionally some people who ranked Wright but not Kiss or Montroll, would have ranked Kiss over Wright, then these votes in favor of Kiss would have defeated him! The winner in this scenario would have been Andy Montroll, who was also the Condorcet winner according to the original ballots, i.e. for any other running candidate, a majority ranked Montroll above the competitor.
Australian elections and by-elections
On the other hand, since every or almost every IRV election in Australiahas 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. (The policy of Australia's election authorities not to release this data
is justifiable on privacy grounds. If rank-order ballots in an election with, say, 13 candidates, were released, even in a highly "anonymized" form, that would still provide enough information for a coercer to use to verify or deny that some voter had cast a pre-specified vote-pattern he'd demanded.)
However, for the Australian federal election, 2010, one article was aware of the non-monotonicity possibility: 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
By-election
A by-election is an election held to fill a political office that has become vacant between regularly scheduled elections....
in the South Australia
South Australia
South Australia is a state of Australia in the southern central part of the country. It covers some of the most arid parts of the continent; with a total land area of , it is the fourth largest of Australia's six states and two territories.South Australia shares borders with all of the mainland...
n seat of Frome
Electoral district of Frome
Frome is an electoral district of the House of Assembly in the Australian state of South Australia. It is named after Edward Charles Frome, the third surveyor-general of South Australia. The electorate is based around the industrial city of Port Pirie and the agriculture areas of Clare and Gilbert...
. 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
National Party of Australia
The National Party of Australia is an Australian political party.Traditionally representing graziers, farmers and rural voters generally, it began as the The Country Party, but adopted the name The National Country Party in 1975, changed to The National Party of Australia in 1982. The party is...
scored 4th place, their preferences were distributed beforehand, allowing the Independent to overtake the Australian Labor Party
Australian Labor Party
The Australian Labor Party is an Australian political party. It has been the governing party of the Commonwealth of Australia since the 2007 federal election. Julia Gillard is the party's federal parliamentary leader and Prime Minister of Australia...
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.
See also
- Voting systemVoting systemA voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
- Voting system criterion