Truncated ballots

[Steve Eppley]

David Marsay wrote:
> There are currently two main options for ranking ballots.
> 1) give first choice (FPP)
> 2) rank all choices.

I wouldn't categorize #1 as ranking.

> I am considering the case where voters are asked to rank, say, their 
> top 4 out of 10.
> 
> Does anyone know of any references to practical experience with such 
> truncated rankings?

Sorry, I can't reference any empirical data about elections with
limited rank positions.  (But I do have a number of comments 
below.)  We might look to Australia, which provides a truncating 
shortcut used by most voters in Australia's STV proportional 
representation elections.  The shortcut allows the voter 
to select one party instead of ranking all parties' candidates.  
(In other words, contrary to conventional belief, Australia 
doesn't really use STV PR; they use party list.)  But this is 
much more severe than the 4/10 that David suggested, so the data 
may not shed much light on his question.

Don't forget, it will be important to consider how the rankings
will be tallied when analyzing the effects of limiting the
rankings.  If the tally is a method which deters candidates
from competing because competing may backfire and elect "greater
evils", the way single-winner STV (Instant Runoff) can backfire 
when it's used to elect one choice, then the empirical data from
Australia is that 4 will unfortunately be *more* than enough:
in Australia's single-winner STV elections it's extremely rare 
for more than 2 candidates to compete.

The data from Ireland's presidential elections (also Instant 
Runoff) is useless since in Ireland that office is a powerless 
figurehead position.  

Gary Cox (professor of political science at UC San Diego, and
director of the Lipjhart Archive of worldwide data on election
systems) argued in his 1997 book _Making Votes Count_ that
Instant Runoff is a two-choice system just like FPP.

> The idea is that:
> 
> 1) FPP discourages voters from expressing their first
> preference when it is unlikely to win. With ranking methods
> they can express their preference and thus help it to build a
> constituency over repeated ballots.  

Not if the ranking method is STV (Instant Runoff).  Failure to 
rank the compromise choice first risks electing one's greater 
evil.

   Example:
   35%:  ABC
   16%:  BAC
   16%:  BCA
   33%:  CBA

   I presume we're all agreed that B should be elected here 
   given these voter preferences.  B is the best compromise, 
   being the first or second choice of every voter.  The 
   method used in Robert's Rules of Order also elects B.

   By ranking C ahead of B, the 33% whose greater evil is A 
   elect A if the ballots are tallied by Instant Runoff.  The 
   method never tallies their preference for B over A.  To 
   defeat A, some of them need to vote BCA instead of CBA.  

With the Instant Runoff method, it's only safe to rank
*sure-losers* ahead of the compromise choice, so rather than
build up a constituency the method places an artificially low
cap on the apparent size of that constituency and backfires when 
that constituency grows too large.

If the rankings of the example are tallied by a pairwise method, 
however,  it becomes clear that the compromise choice B trounces 
both A (by 65%) and C (by 67%).  There's no cap on the size of 
non-centrist constituencies, yet the centrist is easier to 
elect.  And polarization is reduced due to the improved 
measurement of popularity: B leads the "social preference 
order", A is second, and C is third.

> It also helps analysts to decide if the result is
> reasonable if the votes record preferences more accurately.

Truncated preferences will not be recorded accurately, since 
they won't be recorded at all.

Whether the method encourages or discourages the voters from 
expressing their preferences orders accurately depends on how 
the preference orders will be tallied.  I wouldn't assume that 
ranking only 4/10 would lead voters to rank their true top 4.  
For instance, if their "greater evil" (the least preferred of 
the candidates who have a significant chance to win) is one of 
their top 4, they have an incentive to misrepresent by 
truncating their greater evil.

> 2) Full ranking is tedious. Most methods are pair-wise, and thus a 
> candidate who is ranked 9th ahead of the 10th gets equal relative 
> benefit as if it had been ranked first ahead of 2nd. Is this 
> reasonable? 

It seems more reasonable to me than the alternative of forcing 
voters not to rank them at all.

Of more interest (to me at least) is that the 2nd choice, for
example, gets no more relative benefit over the 10th choice
than the 9th choice did: pairwise methods don't attempt to
tally preference "intensities."  Fans of Borda's method (which
isn't pairwise and which does attempt to tally inferred
intensities) find this counter-intuitive, but I see no reason
to assume specific extra intensities since that information
isn't contained in the rankings and therefore cannot be
accurately inferred.  There are an infinite number of Borda
scoring functions which are "reasonable" (monotonically
decreasing) and it's arbitrary to use one of them.

For example, two voters V1 and V2 may evaluate two viable 
candidates A and B identically:
    A:  -$20 billion
    B:  +$50 billion
It seems reasonable that the method should ideally tally their 
votes on B relative to A identically.  But if the voters 
evaluate a third candidate C differently, one voter believing 
that C is better than B (+$70B) and the other believing C is 
worse than B (+$30B), the Borda method will hiccup:
   V1:  CBA
   V2:  BCA
Given these two votes, Borda's method fallaciously assumes that 
V1 and V2 have different preference intensities on B relative to 
A.  Candidate A might get elected even though a vast majority of 
voters believe B is much better than A, since C can fragment B's 
votes.

In addition, by attempting to tally preference intensities 
Borda's method creates strong incentives for voters to 
misrepresent their preference orders in order to strategically 
increase the perceived relative intensity for the lesser evil 
over the greater evil.  As Gary Cox pointed out in his book 
(see above), Borda's method leads to "turkey raising" voting 
behavior: the voter has an incentive to rank rotten ("turkey") 
candidates ahead of viable candidates in order to help defeat 
the worst of the viable candidates.

Allowing the voter to rank only 4 choices forces a similar 
"turkey raising" behavior and perception in which turkey 
candidates appear no worse, and maybe better, than some more 
reasonable candidates.  (More on this below.)

> Will voters take as much care with the lower rankings? 
> Might they not be encouraged to vote tactically? ...

The question about care is of more concern than the question
about tactics, I think.  No one can be sure whether the voters
will receive enough information to be able to evaluate many
candidates sensibly so their rankings will truly reflect their
interests.  In today's U.S., most people tune out political
information if they can, so the news media sees their job as to
provide information mainly about candidates who have a
significant chance of winning.  If the method is reformed so 
voters can rank candidates and the rankings will be tallied 
pairwise, the media will presumably continue to offer little 
information about sure-loser fringe candidates, but can be 
expected to offer more about the centrists--even if the 
centrists are nominees of "small" parties since the centrists 
are best positioned to win.  (I put quotes around the word 
"small" because it's fallacious to measure a party's popularity 
by the number of voters who rank it first.  This is a major flaw 
in the primitive PR systems used around the world; they should 
be improved to allow the voters to rank the parties so that 
relative popularity can be better measured.  This will better 
limit the constituency-building of fringe parties, since those 
parties will be exposed as least popular even if they are 
"larger" than more-centrist parties.)  

It's unclear how many of the voters will gather enough solid
information to be able to intelligently rank 9th and 10th
choices.  Nevertheless, we can hope that even with those low
rank positions muddled, the results will be better than in the
"vote for only one" two-party system since candidates who agree
with majorities on more issues (in other words, candidates who
are more centrist) should win more readily.

Some months ago I posted a technique for mooting tactics like
order-reversal: allow voluntary withdrawal by candidates after 
the voting.  This option can be added to any ranked ballot 
method.  If David is suggesting limiting the rankings primarily 
because of a concern over voter misrepresentation strategies, 
then it's worth considering this "just in time" withdrawal 
feature as an alternative solution.

Mike Ossipoff agreed that the withdrawal feature would probably
be beneficial to any ranked ballot method.  The worse the
method, the more useful the withdrawal feature is at mitigating
the method's flaws.  It can mitigate the worst flaws of
inferior methods like STV (Instant Runoff), and even of
Plurality (FPP, "First Past the Post") if the voters can rank
the choices.  The withdrawal feature is such a simple technique
that the Withdrawal//Plurality ranked ballot method may be a 
good method for election reformers to advocate where voters may 
be put off by the complexity of better (pairwise) methods.  
"Plurality With Withdrawal" appears to be much better than 
Instant Runoff on all the important criteria, including the
Simplicity criterion which Instant Runoff advocates claim is 
where Instant Runoff is better than pairwise methods.

I've also been looking at "automatic involuntary withdrawal" 
possibilities to see whether it's possible for advanced 
pairwise methods to counter attempts at manipulation.  The 
Gibbard-Satterthwaite "manipulability" theorem may not apply if 
withdrawal is an option.  I suspect that there will still be 
some scenarios where manipulation can succeed, but that an 
advanced method can be devised which makes it more difficult 
for manipulators to succeed and more likely that reversal would 
backfire.

> For a truncated ranking, one would disregard the lower rankings, 
> treating them as if they were equal. It seems to me that this avoids 
> the worst faults of other methods, but may introduce some new ones.

Yes, such as the false perception that turkey candidates are 
no worse and maybe better than others, as in Borda's method. 
Clever voters will leave their "greater evils" (the least
preferred of the viable candidates) unranked, possibly ranking
turkeys ahead of them.

I don't think 4 will be enough, in general.  I consider it 
important for the rankings to expose the fact that rotten 
parties or candidates are considered least preferred by a lot of 
people.  If the truncations routinely make them appear no 
worse than average then this important info is lost, as it is in 
the primitive obsoleted PR methods currently used around the 
world.  (Modern PR: let each voter rank the parties, elect 
candidates proportional to the number of ballots which ranked 
them best, and tally the "social preference order" using an 
iterative pairwise method.)

Limiting the voter to ranking only 4 choices is going to 
create dilemmas for the voters.  It's important that the voter 
have room in the ballot to rank the best compromise choice(s) 
ahead of worse choices.  If there are so many choices that the 
voter prefers more than the compromise(s) that there isn't 
enough room to rank them all ahead of the compromise(s), the 
voter will have the classic "lesser of evils" dilemma which will 
distort the candidates' votes.

* *

Another option which David may want to consider is to offer a 
finite number of discrete rank positions, yet allow the voter to 
rank as many candidates as s/he wants.  (The voter may need to 
rank as equals some candidates which s/he actually considers 
unequally preferred.)  Here's an illustration, using a ballot 
format which is practical because it is machine-readable:

         <--more preferred   less preferred-->
    A         ( )   ( )   ( )   (o)   ( )
    B         ( )   (o)   ( )   ( )   ( )
    C         (o)   ( )   ( )   ( )   ( )
    D         ( )   ( )   ( )   ( )   ( )
    E         ( )   (o)   ( )   ( )   ( )
    F         ( )   ( )   (o)   ( )   ( )
    G         ( )   (o)   ( )   ( )   ( )
    H         ( )   ( )   ( )   (o)   ( )
    I         ( )   ( )   ( )   ( )   ( )
    J         ( )   ( )   ( )   ( )   (o)

Only 5 explicit rank positions were provided, and an implicit 
6th position is tallied for unranked choices.

Candidates D & I were unranked and will be treated as if the
voter ranked them last=6th.  

A compact syntax to express this vote is: 
    C > B=E=G > F > A=H > J > D=I

* *

If the method limits rankings, that's another reason to evaluate 
methods based on their Truncation Resistance.  (This property 
has been discussed at length already in this maillist.  To find 
those messages in the archive, try searching the web for the 
keywords "election-methods AND truncat AND resist".)  The two 
pairwise methods which led the 1996 poll in this maillist both 
perform better at resisting truncation than others.  (We 
referred to the two methods by the names Condorcet and 
Smith//Condorcet.  Loosely speaking, when the votes are 
circular, these two methods minimize the number of voters who 
would prefer a different winner.)

Here's an example to illustrate the truncation resistance of 
the two methods which led our poll, and how some other methods 
perform:

   1. The non-truncated preferences
      46: ABC
      10: BAC
      10: BCA
      34: CBA

   The tally:
                      po>A   po>B   po>C
      A vs B:          54L    46W
      A vs C:          44W           56L
      B vs C:                 34W    66L

      (The quantity "po>X" is the number of voters who ranked 
      X's pair-opponent "po" ahead of X.  The suffices 'W' and 
      'L' indicate that the candidate listed atop the column won 
      or lost the pairing listed at the left of the row.)

   B wins all its pairings: 54 voters ranked B ahead of A and 
   66 voters ranked B ahead of C.  Any pairwise method will
   elect B given these votes.

   2. Truncated preferences
      46: A
      10: BAC
      10: BCA
      34: CBA

   With this truncation, no candidate wins all its pairings.  
   (See below.)  Different pairwise methods may elect different 
   candidates.  To resist truncation by the "A" voters, the 
   method should still elect B.

   The tally:
                      po>A   po>B   po>C
      A vs B:          54L    46W
      A vs C:          44W           56L
      B vs C:                 34L    20W
                      ----   ----   ----
                wins:   1      1      1
              losses:   1      1      1
         largest MoD:   8     14     12
          largest OP:  54     46     56
          largest OL:  54     34     56

Some methods (such as Copeland and Regular Champion) elect the 
choice with the best "won lost" record.  Those methods are prone 
to ties, as in this example where all three candidates have the 
same score (1 win and 1 loss).  Usually those methods use 
Plurality as the tie-breaker when two or more choices have the 
same "won loss" record, but that elects A here since 46 voters 
ranked A first.  (To defend against their "greater evil", some 
of the "C" voters would need to rank B ahead of C, or C would 
need to avoid being on the ballot, or C would need to withdraw 
before the result is finalized.)

The acronym "MoD" in the tally table above stands for "margin of 
defeat."  Some methods elect the candidate whose largest margin 
of defeat is the smallest.  In this example, those methods will 
elect A since A's margin of defeat was only 8.  And B's margin 
of defeat is the largest, making it appear that B is the least 
popular choice.  Those methods don't protect B from truncation.  
(To defend against their "greater evil", some of the "C" voters 
would need to rank B ahead of C, or C would need to avoid being 
on the ballot, or C would need to withdraw before the result 
is finalized.)

The acronym "OP" stands for "relative Opposition in a Pairing."
The Simpson-Kramer method elects the choice whose largest OP
is the smallest.  In this example, B would be elected since B's 
largest OP is only 46.  None of the voters have an incentive to 
misrepresent preferences, and none of the candidates has an 
incentive to not compete nor a need to withdraw.)

The acronym "OL" stands for "relative Opposition in a
pairLoss."  The two methods which led our poll both use this 
measure to tie-break when the voted preferences are circular.  
Its definition is slightly more complicated than Simpson-Kramer, 
but these methods appear to deter manipulation more easily than 
Simpson-Kramer.  Both these methods elect B, since the largest 
"pairwise opposition" to B in B's pairlosses is only 34.

(I hope a better term than "opposition in a pairing" can be
devised, since opposition connotes an absolute feeling and we're
really talking about *relative* preferences.  Can anyone
suggest a short way to express the quantity "number of voters
who ranked the candidate's pair-opponent ahead of the
candidate" which doesn't have any misleading absolutist 
connotations?)

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)