[EM] thoughts on the pairwise matrix

Paul Kislanko kislanko at airmail.net
Tue Nov 29 13:07:13 PST 2005

> Abd ul-Rahman Lomax
> At 01:32 PM 11/29/2005, Paul Kislanko wrote:
> >I actually have no method. But "Condorcet ballots" is an 
> ambiguous term as
> >used in the reply to me. I actualy suggested that there BE a 
> well-defined CB
> >such that for each pair of choices I vote "A, B, Either, or 
> Neither". Then
> >the matrix can be guaranteed to reflect the voters' pairwise 
> preferences,
> >instead of having to infer them (under different rules depending upon
> >whether equal rankings and/or truncation is allowed).
> 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. 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. 

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. 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. 

> 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"?

> 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.
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.

> 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.

> 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
discussion. 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

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

> 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.

> 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.

> 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?

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