[EM] thoughts on the pairwise matrix

Abd ul-Rahman Lomax abd at lomaxdesign.com
Wed Nov 30 20:31:57 PST 2005

At 04:07 PM 11/29/2005, Paul Kislanko wrote:

> > Providing that overvoting is allowed, and also truncation (which is
> > generally a question of counting rules, not of ballot design, but
> > some machines may have interlocks which prevent overvoting), every
> > Condorcet ballot I've seen allows this. It is intrinsic to ranked
> > ballots. The only issue is the number of ranks; for practical
> > reasons, it may be limited.
>This is confusing.

Perhaps. However, if we want to describe the situation objectively, 
we'd have to specify *who* is confused. "Confusing" conceals this. Doesn't it?

>  If it is "counting rules" but "some machines may have
>interlocks" is relevant, it is "collection rules". "Overvoting" (whatever
>that is) should not be a theoretical consideration.

If you don't know what overvoting is, for sure, you will be confused. 
Overvoting is voting for more than one person for an office when only 
one can be elected, or marking the same rank for more than one 
candidate, which is tantamount to the same thing. (On a ranked 
ballot, we vote for "first choice," "second choice," etc. Standard 
plurality ballot is a ranked ballot with ranks truncated to two.)

Overvoting in the U.S. generally results in the ballot being 
considered spoiled, for no good reason. I think that the same is true 
with IRV here. The recent failed Washington State referendum (was it 
Washington or Oregon?) included specific language that considered an 
IRV ballot void at and beyond any level where more than one candidate 
was marked for that level of preference.

"Interlocks" prevent overvoting, hence they are collection rules 
(implemented to avoid a problem with a counting rule.

If overvoting is allowed, with a single-position ballot, you have 
Approval Voting. The situation is similar with ranked ballots, 
overvoting is marking more than one candidate with the same rank. 
Punch-card ballots would generally allow this, for example. But in 
the U.S., the ballot would be discarded (or, perhaps, if IRV has been 
implemented as in the State of Washington initiative, every vote at 
the overvoted rank or below is ignored).

>Re: "The only issue is the number of ranks; for practical reasons, it may be
>limited." that is a collecting-side question, but has significant impact if
>a "counting method" depends upon a "collection method" that has this
>limitation it should be accounted for some way.

Counting methods generally presume that the information necessary for 
counting is being collected! Unless the law requires discarding, say, 
undervoted ballots (less than full marking of ranks), the counting 
method must be able to handle truncated ballots. It has been assumed 
that if a ballot is truncated, it is as if the voter voted against 
that candidate in every pairwise contest where the opposing candidate 
was ranked, and as if the voter abstained from every contest 
involving two unranked candidates.

>  In the theoretical analysis
>of an EM such "practical" considerations do not apply. I have done some work
>on this, though, for a lesser cause. There are some significant differences
>in the outcome of the election if the collection method truncates ranked
>ballots vs a collection method that allows all rankings to be counted.

Obviously, if the truncation is sharp. Ballots get complex if full 
ranking is allowed; the utility of full ranking must be balanced 
against the expense of full ranking. My opinion generally is that 
gathering full information is desirable and that the cost of doing so 
is not really significant compared to the importance of elections....

> >
> > The Australian ballot has the voter write in a rank number. Ballots
> > with a limited number of ranks allow determining "A, B, Either,
> > Neither," but not for every pair, just for the top pairs.
>What is the definition of a "top pair"?

With a truncated ballot, the top pairs would be, in this case, those 
pairs with both candidates marked. With respect to a pair not marked 
at all, the vote is identical to Either or Neither. I.e., it has no 
effect on outcome, but only on vote counts, which can matter with 
some Condorcet resolution methods.

> > Any pair
> > with a member with an expressed rank and the other with no rank is
> > explicit for the expressed rank. Any pair with equal ranks is Either.
> > And any pair with no rank for either candidate is Neither.
> >
> > What is *not* shown in a Condorcet ballot is the strength of
> > preference. For that one needs Range. A Range ballot can be used to
> > infer pairwise preferences, and, again, Either is shown by rating
> > both candidates the same, and neither by rating both zero or
> > abstaining (depending on the rules).
>Range is one of those methods I dislike because it explicitly requires a
>specific ballot configuration in order for its counting mechanism to work.

Huh? Range is essentially a ranked method. Approval is Range, and 
approval uses a standard ballot. In this case there are two ranks.....

Range is often proposed to be ten ranks, which can be expressed as 
the digits from 0 to 9.

Range, however, is generally *counted* by averaging the ranks, which 
are considered as scores. Range allows overvoting, i.e., more than 
one candidate may be given the same rank or score.

>If you assign A a value of 69 and I assign A a value of 70, does that mean
>anything at all? You may like A more than I do, but you chose a different
>scale than I did, and without knowledge of the scales we both used the data
>is at best indeterminate.

For this reason I and others have proposed normalizing range ballots. 
Those numbers you gave are relevant if we know where they stand with 
respect to the maximum and minimum ratings on the ballot. 
Normalization would mean that the values would be multiplied by A/B 
where A is the highest possible rank on the ballot and B is the 
highest rank expressed by the voter.

Others consider that a voter might *want* to dilute his vote for any 
of various reasons, and would either simply inform voters that to 
exercise a full vote requires giving at least one candidate the 
maximum rank or would provide a means for voters to request that 
their ballots not be normalized.

Note that your objection to Range is not about collection but about 
analysis. And it is clear that a Range ballot does express 
preference: one voter's 69 and another voter's 70 bear little 
absolute relationship, but what can be compared is relative rank. 
This is why I think normalization, which would make averaging the 
scores meaningful, would be useful. On the other hand, Olympic 
scores, which are Range votes, are considered to have an absolute 
meaning, with 10 being perfect. So one judges 6.9 and another's 7.0 
would, in fact, bear some relationship, they would essentially be the 
same vote.

> > Another additional feature not intrinsic to Condorcet is Approval
> > cutoff, which may be implemented on any ranked ballot by allowing a
> > dummy candidate representing the Approval cutoff rank. If this is
> > done, all candidate pairs with both members below that cutoff are
> > Neither. But Neither is really not informative or active in this
> > case. Neither is quite equivalent to ranking both candidates in last
> > expressed rank in a system requiring no truncation but
> > allowing overvoting.
>Again, that's mixing the counting with the collection. An artificial thing
>nobody will understand and can't be used properly in any mathematical sense.

I've seen ballot designs with Approval cutoff that would be easy to 
understand. We are talking about both collection and analysis 
(counting). Hmmph.

(There are Condorcet methods which use the approval information to 
resolve Condorcet cycles. It is perfectly usable....)

> > Condorcet methods use the matrix, essentially, to determine the
> > winner, if there is a Condorcet winner. If not, the details of the
> > particular method determine the winner.
>If all such methods came up with the same winner, we wouldn't be having this

If they are Condorcet winners, they all come up with the same winner 
if there *is* a Condorcet winner.

>  But when there are cycles, different "Condorcet methods" come up
>with different winners, by evaluating the contents of the pairwise-matrix in
>different ways. I would prefer that they evaluated ballots to resolve

Yes. And it appears that you are proposing a method that requires 
individual ballot information for analysis. There are, I think, some 
proposals along this line. It's more complicated, that's all.

> > Showing the election by overlaying the pairwise matrix with a win
> > marker (as cell color, for example), and sorting the rows and columns
> > in a manner relevant to the determination of the winner -- which
> > varies with the specific Condorcet method), can show the maximum
> > information possible without going into detailed ballot analysis
> > (which could require a *huge* amount of data, though truncated
> > versions of it might be manageable).
>Yes. Just put all methods' winners the same color. That just shows that
>there is more than one "winner".


> > The raw ballot data should be available, *except* that some aspects
> > of it might be necessarily concealed, if secrecy of ballots is
> > important. This is because a ballot ranking *could* identify a
> > ballot. For example, in a 10-candidate election, one coercing a vote
> > could require the voter to rank the lower 9 candidates in a sequence
> > such as to make it extremely unlikely that a voter would
> > spontaneously rank them that way. This becomes even easier if
> > write-ins are allowed and are tabulated and reported.
>In a 10-candidate race there are 9,865,123 possible ballot configurations
>(with equal rankings and truncation allowed).  Of those a significant
>fraction (work it out) would have the "coerced" configuration just by

You are assuming that votes are random. They are not. Some vote 
configurations would be quite common, some extremely rare or 
non-existent. Start to think a little more deeply!

As a simple example, on a ballot with 10 candidates the coercer might 
require that the ballot have 9 of the candidates ranked equally, but 
below one other candidate. Yes, there might be other such ballots, 
not coerced. But there would be a reasonable possibility that there 
would be none, and thus the coercion could be effective.

>  I am not concerned about the chance that my anonymous ballot might
>be determined from the election results.

You may not be. Many are.

If write-in votes are allowed, identifying ballots becomes trivial.

> > However, a judicious choice of what data would be suppr9865123essed (by
> > being summarized in a way that conceals less-significant ballot data)
> > could leave the remainder of the data reasonable to open for public
> > access. In that 10-candidate election, perhaps only full data would
> > be available for the top N candidates, perhaps four or five. Or only
> > the data from ballots where there are N identical ballots, in the
> > topmost ranks.
> >
> > This is the data that would be of significant interest.
>To whom? No reason to suppress counts or summaries, and if a method requires
>that, it will NOT be adopted.

Would all write-in votes be reported with this exact ballot 
information? If so, you have completely defeated one major purpose of 
secret ballot. I'm saying that it is possible to reveal *most* ballot 
information, but revealing all of it could have undesired consequences.

But I prefer Asset Voting, which finesses the whole problem.

> > Of course, with Asset Voting, the whole exercise becomes unnecessary.
> > Asset approaches the voting problem in an entirely different way. As
> > one way to vote Asset, pick the person who you would prefer for the
> > office, or, alternatively, whom you would prefer to choose who wins
> > the election. The latter is the most important, but this might
> > usually also be the former.
>Frame that in a way that it can be analyzed and I might buy it. But it's
>just Borda in disguise.

Hog Wash. You don't know what Asset Voting is, apparently. It *looks* 
like Borda, perhaps, but the way the votes are used has a completely 
different effect.

> > The skill for governing and the skill for choosing reliable governors
> > is essentially the same skill, because one who does not have the
> > latter would be unsuited for the former, since any governor must
> > essentially delegate a great deal of authority, or be overwhelmed and
> > dysfunctional.
>And, judging by the constituents of our Executive and Legislative branches,
>US citizens are incompetent in this regard. So what else is new?

The citizens are competent about some things and not competent about 
others. They are competent, in my opinion, to choose someone they 
trust *under certain conditions,* which, unfortunately, do not exist 
in most jurisdictions. They do often exist in small towns.

This is another example of the problem of scale in democracy, and is 
one reason why I'm working for delegable proxy, though not 
immediately in formal governmental bodies, but in NGOs. I have a 
theory and a plan.

The point made was simply that delegating election power is quite 
reasonable: this is what Asset voting does. One objection made to 
Asset has been "I might want A to be elected, but I might not trust A 
to recast my vote." I'd say that this objection, examined closely, is 
essentially an objection to representative democracy....

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