[EM] Wiki says IRV is monotonic--not fully democratic?

[steve bosworth]

To everyone:

Below, Steve is considering the following section of the following June 21, 2016 Wikipedia article:  ‘Monotonicity criterion’.  He will refer to the author of that section as ‘Wiki’:

Instant-runoff voting and the two-round system are not monotonic[edit<https://en.wikipedia.org/w/index.php?title=Monotonicity_criterion&action=edit&section=1>

Using an example that applies to instant-runoff voting<https://en.wikipedia.org/wiki/Instant-runoff_voting> (IRV) and to the two-round system<https://en.wikipedia.org/wiki/Two-round_system>, it is shown that these voting systems violate the mono-raise criterion. Suppose a president<https://en.wikipedia.org/wiki/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.

[STEVE’S Additions:

S: Given these preferences, the Center candidate rather than the Right candidate should get eliminated because he receives fewer 1st preference votes.  Still, the principle of one-citizen-one-vote requires that the preference of each of these supporter be counted until a majority winner is discovered.  Since 16 of them preferred the Left candidate next and 16 preferred the Right candidate next, the Left candidate receive a total of 51 votes and the Right candidate 49.  Given both that only one candidate can win in this election and the principle of one-citizen-one-vote, no citizen would be able to sustain an objection to this result.

Author’s

1st IRV

COUNT


Left


Center


Right


1


35


32


33


2


51





49


Wiki: 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 Center. 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.

1-[STEVE’S IRV exploration:

Wiki’s changed preferences (2nd Set of Ballots)


Number of votes


1st preference


2nd preference


28


Right


Center


3


Right


Left


2


Left


Right


30


Left


Center


5


Left


Right


16


Center


Left


16


Center


Right




2nd IRV

COUNT


Left


Center


Right


1


37


32


31


2


40


60


0


3





60





S: These 28 make the Center candidate’s total of 60 (and the winner) while the 2 make the total of the Left candidate 40.  While it is true that this change in preferences changes the win for the Left candidate to a defeat, they still helped the Left candidate both by increasing the number of first preferences he received and the average intensity of preference given to him.  At the same time, he could not expect to win because he had received only a total of 40 to the total of 60 votes received by the Center candidate.

Consequently, currently I do not yet see this example as containing any anti-democratic element. As I see it, the principle of ‘one-citizen-one vote’ only requires that each citizen’s vote (preferences) be counted equally as long as technically possible until the single-winner is discovered, i.e. the candidate who has received the highest intensity possible of majority support.  Accordingly, before the 2 preferences were changed by Wiki, the Left candidate won with 51 votes with an average intensity of 9.76 on a scale of 10.  After the change, the Center candidate won with 60 votes with an average intensity of 9.53.

Intensity of support calculations:

1st Set of Ballots

35X10 +16X9=350+144=494

494 divided by 51=9.76

2nd Set of Ballots

32X10 +28X9=320+252=572

572 divided by 60=9.53

In contrast, the intensity of support for a forced win by the Center candidate from the 1st Set of Ballots would only be 9.36:

32 X 1st preferences and 58 X 2nd preferences for the Center candidate:

32X10+58X9=320+522=842

842 divided by 90= 9.36

S: Again, using IRV, the originally winning Left candidate was defeated after the author changed the two voters’ preferences.  This happened even though the Left candidate was now given two 1st, rather than two 2nd, preferences.  Wiki sees this as a violation of the ‘mono-raise criterion’: ‘giving higher preferences to a candidate should never harm him’.  However, at least in some senses, these two higher preferences did help the Left candidate, i.e. they gave him two more 1st preferences. They also helped him by reducing the intensity of 2 preferences given to one of his competitors, i.e. by giving the Right candidate two 2nd rather than two 1st preferences.  By themselves, these changes would make it more likely that the Left candidate would win.  It is only because more other citizens gave their 1st preference to the Center candidate over the Right candidate that the Right candidate correctly had to be eliminated in the second count.  In turn, this appropriately required candidate Right’s 2nd preferences to be transferred:  28 to the Center candidate and 3 to the Center candidate.  Given these two change made by these two citizens, the particular preferences given by the other voters required that the Left candidate not be elected.  Consequently, it was not the isolated giving of two higher preferences to the Left candidate that ‘harmed’ him.  It was how these preferences had to be combined with the preferences of all the other citizens that required the Left candidate to be defeated.  This follows from the principle of one-citizen-one-vote.  Again, given both this principle and the fact that only one candidate can win in this election, no one would seem to be able to sustain an objection to this result.

S: Nevertheless, Wiki claims that IRV is not monotonic (i.e. violates the mono-raise criterion).  Does Wiki have Woodall’s definition of a mono-raise random criterion (see below) in mind [Douglas R. Woodall Discrete Applied Mathematics 77 (1997) 81-98]?

In the light of the above discussion and the following definitions, I would very much appreciate it if anyone could explain why they might still want to criticize IRV for being nonmonotonic.  Is IRV fully democratic in the sense defined above?

Section 1.3 in Woodall’s article defines how a nonmotonic set of rules might ‘harm’ or ‘help’ candidate x:  ‘We shall say a candidate x is either helped or harmed by a change in the profile if the result, respectively, is to increase or decrease [the probability of electing x, i.e] PE(x). The following two properties are well known to hold for STV [which reduces to IRV in a single-winner raced].

  *   Later-no-help:  Adding a later preference to a ballot should not help any candidate already listed.

  *   Latter-no-harm:  Adding a later preference to a ballot should not harm any candidate already listed.’

Next, Woodall goes on to define nine different ‘versions of monotonicity.  The basic theme is that a candidate x should not be harmed by a change in the profile that appears to give more support to candidate x.

Monotonicity:

1-(mono-raise) x is raised on some ballots without changing the orders of other candidates;

2-(mono-raise delete) x is raised on some ballots and all the candidates now below x on those ballots are deleted from them;

3-(mono-raise random) x is raised on some ballots and the positions now below x are filled (or left empty) in any way that results in a valid ballot;

4-(mono-append) x is added to the end of some ballots that did not previously contain x;

5-(mono-sub-plump) some ballots without x are replace by ballots with x placed top and with no second choice;

6-(mono-sub-top) some ballots that do not have x placed top are replaced with ballots that do place x top (and are otherwise arbitrary);

7-(mono-add-plump) further ballots are added that place x top and with no second choices;

8-(mono-sub-top) further ballots are added that place x top (and are otherwise arbitrary);

9-(mono-remove-bottom) some ballots are removed, all of which have x at bottom, below all other candidate.’

I look forward to any of your replies.

Steve



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