Monotonicity criterion

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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 can support their candidate by counter-intuitively voting against that candidate.

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 ranked voting 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.

Instant-runoff voting and the Two-round system are not monotonic[edit]

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. 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:

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 (in this example) 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[edit]

Crispin Allard argued, based on a mathematical model that the probability of monotonicity failure actually changing the result of an election for any given constituency would be 1 in 4000;[3] however, Lepelley et al.[4] 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.[citation needed] Nicholas Miller also disputed Allard's conclusion and provided a different mathematical model.[5]

Real-life monotonicity violations[edit]

If 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.

However, 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[edit]

A real-life monotonicity violation was detected in the 2009 Burlington, Vermont mayor election 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![6] 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[edit]

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. (The policy of Australia's election authorities not to release this data is justifiable on privacy grounds.[according to whom?] 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 Australian 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.[7]


  1. ^ D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6, 1996
  2. ^ Monotonicity in Electoral Systems
  3. ^ Estimating the Probability of Monotonicity Failure in a UK General Election
  4. ^ Dominique Lepelley, Frederic Chantreuil, Sven Berg: The likelihood of monotonicity paradoxes in run-off elections, Mathematical Social Sciences 31 (1996) 133-146
  5. ^ Monotonicity Failure Under STV and Related Voting Systems
  6. ^ Burlington Vermont 2009 IRV mayor election
  7. ^

See also[edit]