Local Copy of FairVote's page titled "Monotonicity and IRV – Why the Monotonicity Criterion is of Little Import" (downloaded 8 Aug 2010)

This page is copied from FairVote's web site unaltered aside from slight formatting changes. Since we were not its authors we do not necessarily endorse anything it says. Indeed, it contains many false or misleading statements, pointed out in added red-ink annotations. We mainly have posted this here to document the fact that Rob Richie, the head of FairVote, and other FairVoters such as "Senior Advisor" Terry Bouricius, have repeatedly contradicted their own page's definition of monotonicity in desperate but incorrect attempts to deny the non-monotonic nature of the Burlington's 3 March 2009 mayoral election, conducted via instant runoff voting (IRV). FairVote and Bouricius were instrumental in bringing IRV to Burlington, but after the 2009 election exhibited this and numerous other pathologies, Burlington repealed IRV.

Opponents of instant runoff voting (IRV) claim that it fails the “monotonicity criterion." What does this mean? And is this significant in real world elections? Before explaining why this failure is of little consequence – certainly of far less consequence than the majority criterion and "later-no-harm" criterion that IRV meets, but other proposed voting methods violate – a little background is useful.

What are voting method criteria?

Numerous formal criteria, that make good sense, have been proposed for evaluating voting methods. But no voting method satisfies all of them, as some are mutually exclusive. Since no voting method satisfies all criteria (those frequently cited and others yet to be devised), it is important to have a sense of how important a criterion is and how often any particular voting method is likely to exhibit a problem.

There can be different reasons for failing a criterion. First, selecting a set of criteria to examine can be arbitrary in the sense that there is no definitive set of criteria to use – any advocate of a particular voting method might believe the criteria that are met by that voting method are more important than the criteria that voting method fails. Second, the specific way in which a criterion is defined may cause some voting methods to flip from failing to meeting the criterion.

For example, one criterion that most people believe is of crucial importance is the "majority criterion." This can be defined as: "If more than 50% of voters consider a particular candidate to be the absolute best choice, then that candidate should win." Some proposed voting methods, such as range voting and approval voting fail this criterion. Advocates of these voting methods generally take two approaches in confronting this reality. Either they argue that the majority criterion is not really that important, or they attempt to modify the definition of the majority criterion so that their preferred method doesn't fail it. For example, the majority criterion could be re-defined so that approval voting passes by saying that as long as any of the candidates who are  "approved" by a majority of voters wins (under approval rules multiple candidates can exceed 50%) , then the re-defined criterion is met. Approval voting would meet this re-defined majority criterion even if the candidate that an absolute majority (more than 50%) thinks is the best choice is defeated by a candidate that nobody (0%) thinks is the best choice, as can happen with approval voting.

Another version of the "majority criterion" satisfied by both approval and range voting is: "if a majority of voters regards candidate X as the unique best choice, they can force X to win." This has been called by names such as "majority defense."

It is also important to be clear about the meaning of "meeting" or "failing" a criterion. Failing a criterion is not necessarily absolute. As used in election theory, a criterion is met only if a voting method satisfies it in 100% of conceivable elections. A voting method might comply with the criterion in 99.9999% of cases, but still be said to "fail."

What exactly is the monotonicity criterion?

Now we turn to the monotonicity criterion. Monotonicity can be defined as follows: 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 relative orders of the other candidates.

An error commonly made by FairVote is failing to understand that nonmonotonicity is about an election pair, not a single election. I.e. you start with election#1, raise X on some ballots resulting in a new election#2 – and if X loses in election#2 but wins in election#1, then 1↔2 is a non-monotonic election pair. One can also speak of monotonic election systems like Range Voting in which no such paradoxical pair can exist, as opposed to non-monotonic systems like Instant Runoff (IRV) in which an infinite number of such pairs exist. To speak of a single election as non-monotonic, one would have to mean "a second election exists, which together with this one, forms a non-monotonic pair."
    One amusing stunt that has several times been pulled by FairVoters in their effort to pretend IRV is flawless, is this. You show them some non-monotone election-pair 1↔2. Let us say election#1 in the pair was real, i.e. the other was the one arising from the hypothetical ballot alterations. Then the FairVoter says: "Aha! Election#2 was non-monotonic, but election#1 was real! So sorry, this was not a real example of IRV non-monotonicity!" If you then show them essentially the same pair 1↔2 but now with 2 being the real election, they pull the same stunt in reverse! Of course, that's just another confusion-scam FairVote uses on the unwitting. But anyhow, once I found the present FairVote page where they actually gave an example of an election they agreed was non-monotonic, they could no longer pull their 2-card-monte trick once I found an election (Burlington 2009) exactly paralleling their own example. Right? Wrong – they pulled it anyhow and disregarded their own definition! Watch this:
The Burlington election, as in fact it was not a nonmonotonic election – e.g, no candidate lost because of winning "too many" votes. What the rankings showed is that it _could_ have been a nonmonotonic election. For that to happen, a number of the backers of the Republican candidate who finished first would have needed to switch their first choice rankings to the Progressive party incumbent – the candidate who in fact most of them saw as a last choice among the three leading candidates. If enough had done so, then the Republican would have lost enough support to fall behind the Democrat, and the Democrat was a stronger runoff candidate against the Progressive. But that did not happen.
– Rob Richie, head of FairVote, posted on internet 24 July 2010.

Here is a standard explanation of IRV failing a monotonicity criterion paraphrased from Wikipedia. Suppose there are 3 candidates, and 100 votes cast. The number of votes required to win is therefore 51. Suppose the votes are cast as follows in an IRV election. 

         Number of ballots          1st Preference        2nd Preference
39 Andrea Belinda
35 Belinda Cynthia
26 Cynthia Andrea

No candidate has a majority of the vote. Last-place candidate Cynthia is eliminated, and in the instant runoff her votes count for Andrea, who wins in the second round with a majority of 65 to 35.

Now suppose 10 Belinda voters drop their support for her and rank Andrea first instead.

       Number of ballots         1st Preference          2nd Preference
49 Andrea Belinda
25 Belinda Cynthia
26 Cynthia Andrea

Andrea again is the plurality winner on the first count, but falls short of a majority. This time, however, Belinda is in last place. She is eliminated first this time, and in the second round all ballots cast for her are counted for Cynthia, who vaults to a victory 51 to 49. In this case Andrea's preferential ranking increased between elections - more voters put her first - but this increase in support appears to have caused her to lose because they led to Belinda being eliminated instead of Cynthia.

In order to emphasize the appearance of a paradox, criticism of IRV based on non-monotonicity is frequently presented in a misleading way, along the following lines: "Having more voters rank candidate Andrea first, can cause Andrea to switch from being a winner to being a loser." This is not correct, however. It is not the fact that Andrea gets more votes that causes her to lose. In fact getting more first preferences, by itself, can never cause a candidate to lose with IRV. With regards to additional voters casting votes that rank Andrea as the top choice, IRV is indeed monotonic.

The actual cause of a non-monotonic flip with IRV is the shift of support among other candidates (the decline in support for candidate Belinda in the Wikipedia example above), which changes which candidate Andrea faces in the final match-up. The fact that those ten voters shifted to Andrea was irrelevant, and did not cause Andrea to lose. The result would have been the same if those voters had shifted their votes to a fourth candidate or not been cast at all.

This difference is quite important. The rhetorical impact of this reality is less persuasive and certainly sounds a lot less paradoxical—i.e., now the failure becomes  "If support for other candidates shifts so that candidate Andrea faces a stronger opponent in the final runoff, Andrea could switch from being a winner to being a loser." Indeed it is this "paradox" that is often the basis for primary election campaigns in our system where a candidate makes the claim of "electability." Essentially that candidate is saying, "you might like my primary opponent better, but I am a stronger general election candidate."

FairVote is ducking and weaving to try to spin this. But the fact is simple. Before: Andrea wins. After a hypothetical raising of Andrea to top by 10 Belinda>Cynthia>Andrea voters (their votes become Andrea>Belinda>Cynthia), Andrea would stop winning.

Note how this example illustrates an important point about hypothetical voting examples concocted to demonstrate pathologies. They are often extremely unrealistic, which can be lost in a blizzard of A's, B's and C's. In this case, in order to switch from Belinda to Andrea, 10 voters have to skip over their original second choice, Cynthia, in favor of their original last choice. And this has to happen without any other changes taking place in the electorate. How often is this going to happen in real elections?  

(a) It happened in the Burlington 2009 Mayoral election essentially exactly as in the example here, renaming Andrea→Kiss, Belinda→Wright, Cynthia→Montroll; if 753 Wright-top voters had raised their rank for Kiss to top, that would have stopped Kiss winning.
(b) The question of how frequent non-monotone IRV elections are, is analysed here. It is found that it happens in 3-candidate IRV elections 5.4% to 14.5% of the time (depending which of three mathematical models is analysed) and in N-candidate IRV elections in the limit where N is large, it appears to happen 100% of the time in all 3 models (although this is currently only proven in two of them).

What does this mean, and is monotonicity significant in real world elections?

In terms of the frequency of non-monotonicity in real-world elections: there is no evidence that this has ever played a role in any IRV election -- not the IRV presidential elections in Ireland, nor the literally thousands of hotly contested IRV federal elections that have taken place for generations in Australia, nor in any of the IRV elections in the United States.

Not only was Burlington a counterexample (and, yes, Burlington is in the United States), the Ireland 1990 presidential election also exhibited a hybrid form of the non-monotonicity and no-show paradoxes:
  1. If 344362 new voters had magically appeared all ranking Currie top and Lenihan dead last...
  2. and 12% of the Lenihan-top voters had changed their vote to rank him dead last...
  3. That would have made Lenihan win.
[Note this was the opposite kind of non-monotonicity; Lenihan "lost because he had too many votes." In Burlington, by contrast, Kiss "won because he had too few votes."] There is no doubt there were many Australian non-monotone IRV elections too, but FairVote is correct that I presently have "no evidence" for that (nevertheless true) fact. The reason I have no evidence, is Australia has apparently never published the ballots (or enough information to allow them to be reconstructed) for any IRV election they ever held, and definitely have refused to provide them to me when I requested. Without the full ballot set, it is difficult (albeit sometimes possible, as in the Irish 1990 case) to prove there was non-monotonicity. In Burlington we had the full ballot set, so it was easy.

True, in theory, in a close election, if enough supporters of candidate A knew enough about the likely rankings of other voters they could, in some rare situations vote strategically as follows: Instead of ranking their true favorite as number one, they could give that first ranking to the weaker of the two likely opponents in the likely final match-up with A, in hopes of helping their favorite candidate win in the final runoff tally. Indeed you can see this happen in traditional runoff systems or in "open primary" systems – consider Rush Limbaugh's "operation chaos" strategy in the 2008 Democratic presidential nomination where he urged his conservative radio listeners to vote in the Democratic primary for Hillary Clinton, secure in his knowledge that John McCain was already assured of receiving the Republican nomination.

But this scenario is far-fetched in IRV elections for a number of reasons.

FairVote here is setting up a mythical and ludicrous "straw man," then attacking it, then declaring their attack successful. The main reason non-monotonicity is a bad thing, is not that it could be used to vote strategically. It is a bad thing because of this: suppose you voted maximally-against Nixon because you hated him. Suppose it was exactly your anti-Nixon vote that was the crucial swing vote that caused Nixon to win. And then you realize: "if only I had ranked Nixon top that would have made him lose!" Would you be a happy voter at that point? Would you think IRV was a great voting system? I suspect not. I suspect you'd feel ripped-off and defrauded. Suppose, at that point, FairVote came to you and said "Don't worry! Nothing was wrong with Nixon's election, because it is 'far-fetched' to imagine that many people intentionally voted against Nixon in order to try to make Nixon win." Would this restore your happiness? I suspect not.

Firstly, it is a tremendously risky venture, since if too many voters follow the strategy it could seriously backfire and cause the favored candidate to be eliminated before the final runoff is reached or lose in the final runoff. Unlike the Limbaugh strategy in the 2008 Democratic primary, one's true first choice isn't guaranteed a spot in the final pairing without real support – with IRV, voters don't get to switch their first choice between rounds, and so lack of monotonicity is less significant than with two-round runoff elections, which also fail the criterion. Second, the strategy would also require a substantial amount of reliable information about the likely first and alternate rankings of other voters – information that will not be easy to obtain, and certainly not in a way that would likely govern voting decisions. Combined with the fact that the strategy is counter-intuitive, these facts make its use extremely unlikely.

The "later-no-harm" (LNH) criterion is far more important than monotonicity because, unlike the monotonicity failure, it has direct strategic consequences. In a nutshell, a voting method fails the later-no-harm criterion if there is a risk that by indicating a second choice in any way (a ranking as in the Borda count or Bucklin system, another vote as in approval voting and points as in range voting), a voter might help defeat his or her first choice. This criterion has serious real-world implications, as there is substantial evidence that it leads some voters to honestly rank only their favorite choice under such methods as approval, Bucklin, Borda, Condorcet and Range Voting.

The actual evidence is that voters rank "only their favorite choice" more often with IRV, than they do with either approval or range voting, despite the fact IRV obeys LNH while approval & range voting do not. Thus FairVote is exactly wrong and LNH is entirely unimportant in its real-world consequences, indeed acting if anything in the opposite direction to the way FairVote thinks it acts.

Even worse, perhaps, would be if many voters grasped the strategic value of such "bullet voting" and many others didn't, thereby giving insincere tactical voters a big advantage over sincere voters casting ballots as the instructions suggest they should. Thus all of the mathematical niceties of these other methods go out the window by voters' refusal to play along and risk hurting their first choice.

Monotonicity has little if any real world impact, and voting methods that satisfy that criterion tend to fail the majority and later-no-harm criteria, which can dramatically affect voting behavior and produce what are considered by most to be undemocratic outcomes.

Actually the "later no harm" criterion is exceedingly unimportant as is also revealed here where, e.g. 387 papers in the poly-sci literature were found about the "Condorcet criterion" that FairVote "senior advisor" Bouricius there called "far less important" than the "later no harm" criterion; the corresponding search turned up zero papers on "later no harm." Here FairVote says Monotonicity is relatively unimportant, but for it the search now finds 107 papers...
    Obviously, nonmonotonicity had a serious "real-world impact" in Burlington 2009, and Ireland 1990, and indeed Burlington then repealed IRV... And FairVote has here produced no evidence either LNH or majority-top ever "dramatically affected" voting behavior. Their claims about the so-called rarity of non-monotone IRV elections have already been refuted above, and their straw-man argument non-monotonicity is unimportant, was shown above to be a deceptive red herring.

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