[EM] Why Margins isn't as democratic or ethical as
Norman Petry
npetry at cableregina.com
Tue Feb 8 07:32:19 PST 2000
Hi Everyone,
Over a year ago, when the issue of margins vs. votes-against was being hotly
debated by Blake, Mike, and Markus, I prepared a lengthy response to Blake
setting out my position on this issue. I had planned to revise the article
somewhat before posting it, but unfortunately new time commitments forced me
to curtail active participation in the activities of this list, and the
article was never sent (I've been lurking ever since!). Since the margins
vs. votes against issue has now resurfaced, I thought it would be a good
opportunity to post the message. I still don't have time to make the
changes I would like, but I think it raises a few issues that have not
received much attention in this debate, and hope that you might find it
interesting.
Please note when reading this that at the time it was written, I was
assuming that the Schulze method as intended by Markus used absolute votes,
rather than pairwise majority-only beat-paths (there *is* a difference, as
you will see in the examples below). Only in the last couple of days did it
become clear to me that Markus uses pairwise majorities, rather than
absolute votes, based on one of his examples from February 3rd ("Re: [EM]
Tideman and GMC"). In fact, looking back now on Markus' definition of a
"beat-path", it is clear that I was mistaken in my earlier assumptions.
Therefore, it now appears that there are at least three approaches that can
be taken when dealing with the problem of truncation in pairwise methods:
1) Margins (M) -- replace the pairwise wins with the majority-minority
differences, and the pairwise losses with 0 prior to applying the pairwise
method.
2) Votes-Against (VA) -- replace the pairwise losses with 0 prior to
applying the pairwise method.
3) Absolute Votes (AV) -- replace nothing, and apply the method directly to
the original pairwise matrix.
My current view is that the best and simplest approach is (3). In terms of
results, it is probably a compromise between (1) and (2), and I believe that
Schulze(AV) will satisfy GMC, which I consider important.
Anyway, here's the article. I welcome your comments.
Norm.
*************
Dear Blake,
I have been quietly following the discussion on this list between you,
Markus, and Mike for a number of months concerning the issue of how to
resolve the problems caused by truncated ballots in pairwise methods. I've
prepared a rather lengthy response which presents my opinions on this issue,
which I hope you will find interesting.
First -- A Rant about "Votes Against" terminology
One problem I have with the terminology that is used on this list is the use
of "Votes Against" (VA) to refer to methods which consider absolute vote
totals rather than margins when comparing candidates. Aside from sounding
negative (suggesting we are talking about people voting 'against'
candidates, rather than 'for' them), it's also inaccurate and a source of
confusion. For example, if we have a pairwise matrix containing the
proposition:
A>C 57:30
this is neither "57 votes against C", nor "57 votes for A". We cannot tell
from these votes whether voters are "voting against" candidates they
dislike, or "voting for" candidates they like. Rather, all we know is that
there are 57 people who prefer A to B, and 30 prefer B to A. Plurality (and
IRO) teach us to think of voting for/against candidates, since in these
methods the voter bestows an actual quantity (the vote) on the candidate of
their choice. Some people mistakenly believe that this is an inherently
positive act, even though we know that voters under plurality are often
voting against a "greater-evil" rather than for a candidate they actually
like. In contrast, pairwise methods use votes that are derived mechanically
from a ranked ballot, and are all "votes for" various "propositions", or
relative orderings of candidates.
I think the VA terminology was adopted because it makes some sense when
describing Condorcet(EM) (Plain Condorcet (PC), Minimax), since in that
particular method the winner is the candidate having the fewest number of
"votes against" them in their worst defeat. The terminology makes no sense
at all when applied to Schulze's method.
It would be better to use the term "absolute votes", or just "votes" when
comparing the "VA" methods with methods using margins. This is what I will
be doing in the following discussion.
Accuracy of the Schulze Method vs. Path Voting
I think one of the reasons you prefer Path Voting to Schulze is because you
have an intuition that it will produce more accurate results, on average,
when voters mark their ballots sincerely. On September 17, 1998 you wrote
(Re: Margins, majority, strategy):
"So, in conclusion, marginal methods give better results when the voters are
sincere and do not have the problems inherent in violating SEC."
This conjecture may be based on a mistaken belief that Schulze's method
ignores minority beatpaths, while Path Voting does not (see below for an
example which refutes this). If this *were* the case, one might expect Path
Voting to be more accurate, since it makes use of *all* the information in
the pairwise matrix (majorities and minorities) by first computing margins
(which simultaneously destroys that information, unfortunately). Schulze's
method does *not* ignore minority beatpaths however, so it wasn't obvious to
me that Path Voting would give better results under sincere voting
assumptions.
Still, it seemed like an interesting theory, and one which could be analysed
empirically. Therefore, I constructed a computer simulation to compare the
two methods on the basis of accuracy in the presence of varying degrees of
truncation in order to test your assertion.
Methodology:
For my model, I decided to use ratings as the standard by which to compare
each method's results. For each trial, I first created a set of voters
represented as points normally distributed in an n-dimensional
"issue-space". I then did the same thing for candidates, but used a uniform
distribution. Ballots were then constructed for each voter by computing the
"distance" between the voter and each candidate in the issue-space. Each
candidate's rating was represented by this distance, so that low ratings
corresponded to most-preferred candidates. Rankings could obviously
inferred by sorting on these ratings low-high. To compute the ratings
winner, I simply summed the ratings between every voter and each candidate
to determine a total score, and then declared the candidate with the lowest
score the ratings winner. Effectively this method can be seen as minimising
the total "error" involved in picking a given candidate, or choosing the
candidate with the minimum average distance to all the voters. A more
complex heuristic could be devised, of course, but this seemed a simple and
reasonable way of determining the "best" choice in a single-winner contest.
To simulate the effects of truncation, I constructed a random number
generator based on a binomial pdf for n candidates. A value from this
generator would be used to determine the number of rankings that would
actually be expressed by a given voter when they cast their ballot in the
pairwise election. Truncated candidates would simply be ranked equally
lowest (the usual assumption on this list). By varying the per-trial
probability in the binomial pdf, I could easily construct simulations having
as much or as little truncation in the ballots as I wanted. For example, a
binomial pdf of B(n,p) = B(6,0.5) would select 0 to 6 candidates, and on
average rank 3 of them. By changing the value of p, I could skew the
distribution to allow different types of electorates to be simulated. I
called this my "involvement" index. Lazy electorates would have a low value
of p (generally ranking 1 or 2 candidates), and "involved" electorates might
always rank them all (p=1.0).
Once the pairwise matrix was calculated, I simply applied Schulze's method,
then the Path Voting method to the matrix, and counted the number of times
each method produced the "right" answer (i.e.: ratings result). The ratings
winner was always determined by using *full* ballots (no truncation).
This simulation has many variables, of course. Since I wanted to see how
each method would respond to varying degrees of truncation, I made the
following assumptions for the simulation shown here:
candidates = 6
voters = 100
dimensions = 2 (dimensions of "issue-space")
trials = 10,000
voter distribution = normal(SND)
candidate distribution = uniform(-2..2)
involvement = 0.1 .. 1.0
Here are the results of my simulation run based on the above parameters
(best viewed in a mono-spaced font):
Involvement Schulze PathVoting Schulze% PathVoting%
0.1 6,647 6,651 66.5% 66.5%
0.2 7,510 7,500 75.1% 75.0%
0.3 8,121 8,118 81.2% 81.2%
0.4 8,566 8,564 85.7% 85.6%
0.5 8,868 8,863 88.7% 88.6%
0.6 9,078 9,077 90.8% 90.8%
0.7 9,214 9,226 92.1% 92.3%
0.8 9,249 9,255 92.5% 92.6%
0.9 9,305 9,314 93.1% 93.1%
1.0 9,262 9,265 92.6% 92.7%
Average: 8,582 8,583 85.8% 85.8%
Note that although the two methods do not always produce the same results,
they are almost identical in accuracy over the entire range of possible
values for involvement. It may be possible to observe slightly better
results in Schulze under low involvement (0.2-0.6) and slightly better
results in Path Voting when involvement is high (0.7-1.0), but the
differences are negligible (under 0.1% in all cases), and may only be a
limitation of the simulation (perhaps I need 100,000 elections to make sure,
but this one took about 8 hours and I'm not that patient!).
Of course, a simulation like this tells us little about how a method will
perform under actual use, since the degree to which voters expressed
preferences match their sincere preferences will depend on how manipulable
the method is. For example, this simulation also gives quite good results
for Borda (not shown), yet the ease with which voters can gain a better
result by nominating clones or "turkey raising", etc. ensure that
sophisticated electorates using Borda will generally not reveal their true
preferences, so the entire election result becomes suspect. The
strategy-resistance of Schulze appears to be better than that of Path
Voting, so we can expect the accuracy of Schulze to be considerably better
than Path Voting in practice, even though Path Voting appears equally good
(although no better) in theory.
I have repeated this simulation using different values for most of the above
variables, and have found only small differences in accuracy between Path
Voting and the Schulze method, regardless of the parameters chosen. Thus, I
conclude that the Schulze method equals Path Voting in accuracy under
sincere voting assumptions.
I will be making my simulation software available to members of the list in
a few days so you can run your own simulations to (hopefully!) confirm these
results. In addition to Schulze and Path Voting, I have also simulated a
number of other methods which have been of interest to members of this list,
and I'm hoping that peoples' experiments with this software will provoke
some interesting discussion here on EM (I'm cleaning up the code a bit
before distributing it, which is why you're not getting it today -- write to
me if you're impatient!).
Example 1 - Schulze Method and Minority Beat Paths
I think that perhaps one reason you prefer the use of margins to absolute
votes when comparing candidates is a result of a slight misunderstanding of
how Schulze's method (VA) actually works. I note that on your EM Resource
website you've written:
"Some methods assign pair-wise victories a strength so that it can be
determined which prevail when there is a conflict. There have been a number
of ways suggested, but mainly either margins or winning-votes are used.
Consider a contest of 15 to 5
Margins-- use the margin of victories, in this case 15-5=10
Winning-Votes-- use only the votes on the winning side, 15
Losing-Votes-- use only the votes on losing side, 5"
[...]
If you want to use winning-votes as mentioned above, you would want to
replace the losing side with 0, and vice versa. To make a marginal pair-wise
matrix, each cell should show the margin of victory for the row's candidate
over the column's candidate, or in the case where there is no victory, a 0."
This is not a correct description of how absolute votes are used in
Schulze's method. My understanding of his method is that it makes *no*
distinction between majority and minority beatpaths. Except in the case of
unanimous pairwise decisions, all propositions are contradicted by at least
some voters, so Schulze's method considers *all* possible paths between each
pair of candidates to determine the path strength. Sometimes a minority
path will provide the strongest link. While it may be true that considering
only majorities is equivalent to considering all the votes in simpler
methods (Condorcet(EM), Smith//Condorcet(EM)), this is not true for
Schulze's method.
Consider the following 4-candidate example (100 voters).
A>C 61:18
A>D 39:15
B>A 81:3
B>D 19:15
C>B 42:13
D>C 44:43
All candidates are members of the Smith set. If we only consider pairwise
majorities in constructing the beatpath matrix, we get:
A>>C 61:42
B>>A 81:42
B>>C 61:42
D>>A 42:39
D>>B 42:39
D>>C 44:39
This produces a final preference order of: D>B>A>C
However, if we consider *all* possible beatpaths (Schulze's method) we get:
A>>C 61:42
A>>D 43:42
B>>A 81:42
B>>C 61:42
B>>D 43:42
D>>C 44:43
for a final preference order of B>A>D>C.
The Path Voting result (margins beat-path matrix) for this example is as
follows:
A>>B 43:29
A>>D 24:1
B>>A 78:29
B>>C 43:29
B>>D 24:1
C>>D 24:1
which yields a final order of B>A>C>D. Note that both Schulze's method and
Path Voting produce the same winner (B) in this case, whereas we get a
different winner (D) if only pairwise majorities are considered.
>From this example it is clear that minority beat-paths are *not* ignored in
Schulze's method, and will have an effect on the outcome in some cases. In
this case, the weak victory of D over C left a strong minority beatpath
which prevented it from winning under Schulze. Therefore, it is not
necessary for a beatpath method to use margins to distinguish between
strongly established (i.e.: high margin) propositions and weak ones.
On October 6th, 1998 you wrote (Re: VA, Margins, & voter wishes):
"For example, I have two statements, and I know one is wrong, one is
supported 52 to 48, the other 47 to 3. Which should I guess is right? I find
I am unable to simply disregard the opinion of those on the minority side.
Each of those 48 people is contradictory evidence to the 52. [...]"
I hope this example has made clear how Schulze's method does *not* ignore
the opinions of minorities. It merely uses those opinions in a better way
than is possible in Path Voting. It is an unfortunate fact that pairwise
methods which employ only margins cannot possibly satisfy strong
majoritarian criteria such as GMC, since they destroy the information needed
to make compliance even possible. In itself, this is not sufficient reason
to discount the importance of such criteria (as you seem to have done),
given that a method *is* available which can simultaneously respect both
pairwise and overall majorities. I personally consider GMC to be a very
important criterion.
Example 2 - Path Voting violates GMC
Consider the following example (100 voters, 3 candidates):
Ballots:
A>B>C 7
A>B=C 4
B>A>C 32
B>A=C 8
B>C>A 3
C>A>B 40
C>B>A 6
This produces the following pairwise matrix:
A>B 51:49
B>C 50:46
C>A 49:43
All 3 candidates are members of the Smith set. Applying Schulze's method,
we get the following beatpaths:
A>>B 51:49
A>>C 50:49
B>>C 50:49
The winner using Schulze (absolute votes) is therefore A. If we instead use
Path Voting (margins), we get a margins matrix of:
A>B 2
B>C 4
C>A 6
which has the following beatpaths:
B>>A 4:2
B>>C 4:2
C>>A 6:2
The Path Voting winner is therefore B.
Which is the better choice -- A or B?
It seems to me that it would be an egregious error for a voting method to
choose candidate B as its winner rather than A, given that an absolute
majority of voters prefer A to B 51:49. I think it would be difficult to
defend such a result to the voting public as being at all reasonable. They
are unlikely to be persuaded that B should win on the grounds that an
irrelevant 3rd candidate (C) beat A by a wider margin than A beat B (by
irrelevant, I mean irrelevant to a choice between A and B -- more on this
later).
This example demonstrates how Path Voting violates GMC (Mike Ossipoff's
'Generalised Majority Criterion', although I think the name 'Overall
Majority Criterion' would be a better description). I think that respecting
the wishes of an absolute majority of voters *is* a worthwhile goal for any
method to aspire to, so GMC does provide voters with a useful guarantee
that's missing in Path Voting.
Example 3 - Path Voting and Irrelevant Alternatives
My 2nd example was contrived to demonstrate how Path Voting violates GMC.
In contrast, this example was drawn from the computer simulation I conducted
which compared the two methods, so there is no reason to believe this to be
a particularly 'unrealistic' result. A six candidate simulation reduced to
a 3-candidate Smith Set having the following pairwise wins:
A=B 50:50
B>C 51:48
C>A 50:49
The Schulze method yields:
A==B 50:50
A==C 50:50
B>>C 51:50
The Schwartz set consists of [A,B], and since A=B, Schulze's method cannot
reduce the result further, so the tied winners are: [A,B]
The Path Voting result looks as follows:
A=B 0
B>C 3
C>A 1
which gives
B>>A 1:0
B>>C 3:0
C>>A 1:0
so the winner is decisively B.
While it's nice to have a method that produces decisive results rather than
ties, this is only true if the winner chosen is reasonable. In this case, I
think it's fairer to decide randomly between A and B, since support for each
candidate is equally divided 50:50. Decisively choosing B simply because a
3rd candidate (C) was preferred to one of the two, does not seem justifiable
to me.
Both of the above examples show how computing margins appears to cause
preference information from an irrelevant 3rd candidate to have a harmful
effect on the outcome in Path Voting, while the Schulze method successfully
avoids the problem. While it's true that both Schulze and Path Voting
satisfy LIIAC (since they only consider alternatives in the Smith set), the
Schulze method seems to go beyond Path Voting in ignoring (to some extent)
the irrelevant alternatives *within* the Smith set in making its choices,
and thus produces better results. You may object to me referring to
candidate C in these examples as being "irrelevant", since it is involved in
a voting cycle and does beat the Schulze winner. I agree that I may be
using the term somewhat loosely. What's clear, however, is that given a
choice between the Path Voting winner and the Schulze winner, the public
would clearly prefer the latter, so I think the correct choice between the
two is obvious.
If you have any examples where you think the Schulze result is inferior to
the Path Voting result, I'd be interested in seeing them. I haven't
encountered any yet myself.
Conclusion
I hope that the arguments I have presented here will lead you to reconsider
your views on the suitability of Margins in Path Voting. Like Mike Ossipoff
and others, I find it discouraging that even among the few people who
frequent the EM list that are interested in pairwise methods, we cannot
reach agreement on important standards and criteria, and come to some
consensus as to which method(s) are best. Successful electoral reform
depends on developing "mindshare" for a particular method, and as long as
pairwise advocates remain divided, it will be the IRV and Approval advocates
who are likely to be successful. While their methods may (?) offer some
modest improvements over plurality, there's no good reason why the public
should settle for second-best, given that much better rank-ballot methods
are available.
I have devoted considerable time and effort in preparing these arguments in
order to try to convince you, in particular, to support the use of Schulze's
true method, rather than your margins variant. I have visited your EM
Resource website, and found it to be an excellent source of information and
analysis on voting methods. I regret being unable to also recommend your
Path Voting advocacy pages (since they are also very good generally), due to
disagreement with you over this small, but critical issue of margins vs.
absolute votes. I hope our disagreement can now be resolved.
Sincerely,
Norman Petry
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