House-Monotonicity Problem in Ranked STV -- an Example

[Norman Petry]

On July 31, I posted a message (Re: Vacancy Rule - STV cycles rule)
clarifying the workings of Ranked STV[NP], and showed how this method
differs from Mike Ossipoff's method (Ranked STV[MO]) in the way in which it
deals with house-monotonicity violations.

In this posting, I'll work through an example which should make these
differences clear, as well as provide some insight into the potential
problems each of these methods may introduce as they try to deal with the
non-monotonicity problem.

***

The example:

Candidates = {A,B,C,D}
V = Total Voters = 100
R = Current Rank = 1,2,3,4 (the rank position being filled by successive
candidates)
Q = Droop Quota (Q = [V/(R+1)]+1), where [] indicate truncation)
F = Previous Caucus (Fractie) size

Candidate lettering generally indicates how "far apart" candidates are in
terms of voter preferences (A's supporters strongly dislike D, and
vice-versa).  I've used this assumption to make reasonable guesses about
vote transfers.  I've ignored the effects of truncation, assuming that all
voters mark the full ballot (this should be unimportant anyway in this
example).  Note that I've used spaces to create the tables showing the STV
stages; you will need to view this using a mono-spaced font for the example
to be legible.  Where an asterisk appears beside a candidate's name, this
indicates that the candidate has been previously ranked, and may not be
eliminated.  "|" indicates "or".

STAGE 1: R=1, Q = [100/(1+1)]+1 = 51

Ranked STV[NP,MO]

A   34 +10  44  +1  45 -45   0
B   19 -19   0       0       0
C   21  +8  29 +26  55 +45 100
D   26  +1  27 -27   0       0

STV Winners: {C}
Ranking:     C


STAGE 2: R=2, Q = [100/(2+1)]+1 = 34

Ranked STV[NP,MO]

A   34      34      34
B   19 -19   0       0
C*  21 +18  39  -5  34
D   26  +1  27  +5  32

STV Winners: {A,C}
Ranking:     C,A


STAGE 3: R=3, Q = [100/(3+1)]+1 = 26

Ranked STV[NP]                  Ranked STV[MO]

A*  34  -8  26      26          A   34  -8  26      26
B   19  +6  25 -25   0          B   19  +6  25 +23  48
C*  21  +2  23 +25  48          C   21  +2  23 -23   0
D   26      26      26          D   26      26      26

                                STV Winners: {A,B,D}
Ranking[NP]: C,A,D              Ranking[MO]: A,(B,D)|(D,B) for R<=F
                                             C,A,(B|D)     for R>F



Comment:

Stage 3 is the interesting case.  Here, house-monotonicity has been
violated, since the STV winners are {A,B,D}, but were {A,C} previously.

Ranked STV[NP]'s solution to this problem is to "lock" previous winners, so
C _cannot_ be eliminated.  This means that B must be eliminated, even though
B has more votes than C (25 to 23).  In this example, C would receive
sufficient transfer votes from B's defeat to reach the quota, but this will
not necessarily happen.  If, for example, all of B's votes had transferred
to A (or ballots were truncated), C's final vote total would have been 23 --
3 short of quota. Therefore, it's important when using this method to
eliminate all but one of the unranked candidates before declaring the final
winner for the round (D).  Normally, it's possible to pick any unranked
candidate which has reached the quota, but this anomaly might result in more
than one unranked candidate satisfying that requirement.

Ranked STV[MO]'s solution is different.  Since the STV election for the
round chooses {A,B,D}, one of two things can happen.  If R<=F, the new list
replaces the old one, and the new candidates are placed at the end of the
ranking, either in order of first-choice vote totals(26:19 gives A,D,B) or
according to overall voter preferences (a single-winner method is used to
order B and D, in which case B would probably appear first: (A,B,D)).  If
R>F, then C must remain in the rankings, so either B or D would be ranked
3rd.  If first-choice vote totals are used to choose between B and D
(26:19), the new ranking becomes (C,A,D).  If a better single-winner method
is used, the result would probably be (C,A,B).


Analysis:

Of the two methods that might be used in Ranked STV[MO] for either ordering
or choosing new candidates, I prefer using first-choice vote totals rather
than a better single-winner method that counts *all* the ballots.  Not only
is it simpler (which is desirable, especially considering how infrequently
this rule will need to be invoked), but as this example shows, the effect of
using, say, Condorcet's method to choose between B and D will introduce a
*very slight* bias away from strict proportionality towards overall
popularity in the final rankings.  In this case, B would be chosen over D
because A's supporters strongly prefer B to D, and C's supporters' opinion
would be divided.  However, if the single-winner method were applied _only_
to the first-choice ballots of the affected candidates, this _would_ be a
technical improvement (again, probably needlessly complicated in practice),
since it may happen that *more* than 2 new candidates would appear in the
latest STV count.  Good single-winner methods improve on plurality voting
when there are more than 2 candidates, and the same argument could be
applied here when we need to choose the next 1 of N candidates, when N>2.
For example, if candidates B,D,E were "new" winners in the latest STV round
(different election, obviously), use Condorcet's method applied to the
first-place ballots of B,D,E to pick the single winner.

STV[NP]'s workings are a little more "STV-like", since it chooses winners
only through elimination and surplus transfers.  House-nonmonotonicity
introduces some unfairness into the system, since we can't always eliminate
the lowest-scoring candidate (fewest votes).  However, supporters of the
(unfairly) eliminated candidate will be the ones who determine subsequent
choices, since their transfer votes will raise their second-favourites
closer to quota.  The result is that the system is probably
self-compensating in terms of faction-proportionality.  Of course, now that
these voters' compromise candidate has been elected, that "locks" their
compromise choice higher in the rankings, which gives them less power to
elect their true favourite in later stages.  Still, the progressively lower
quota should cancel out these effects in most cases.  One last thing to note
about this method is that if possible, "locked" candidates _do_ have to
capture and hold one quota's worth of votes (which aids proportionality),
since no previously ranked candidate is ever eliminated.  STV[MO] in
contrast, allows the full transfer of votes from an eliminated candidate to
influence the final choices for the round (although not the single-winner
choice/ordering of them).  If that eliminated candidate (such as C, above)
cannot be removed from the rankings (R>F), his election is essentially
"free" in terms of voter support, and this may give his supporters undue
influence over the final choices (here: B,D.  This is not evident in this
example, BTW, due to the limited number of candidates).


Conclusion:

As I indicated previously, either of these methods should produce similarly
excellent results in practice.  Due to the rarity of non-monotonicity
problems, it would probably be necessary to run computer simulations of both
to determine conclusively which method's solution to the non-monotonicity
problem creates more accurate proportional representation in a ranked party
list.



Norm Petry