[EM] Why Not Condorcet?

[Abd ul-Rahman Lomax]

At 10:12 PM 5/16/2010, Dave Ketchum wrote:
>On May 16, 2010, at 6:11 PM, Abd ul-Rahman Lomax wrote:
>>At 02:16 PM 5/16/2010, Dave Ketchum wrote:
>>>On May 16, 2010, at 9:24 AM, Abd ul-Rahman Lomax wrote:
>>>>At 06:34 PM 5/15/2010, Dave Ketchum wrote:
>>>>>>Some objections to Condorcet could be:
>>>>>>1. It is not expressive enough (compared to ratings)
>>>>>Truly less expressive in some ways than ratings.
>>>>>    This is balanced by not demanding ratings details.
>>>>>    And more expressive by measuring differences between each pair
>>>>>of candidates.
>
>The base topic is Condorcet.  It would take a book to respond to all
>your extensions such as IRV.  Likewise I see no benefit in adding
>Borda - Range/score is an adequate source for ratings.

Dave, you apparently don't understand a good deal of what you read. 
That's okay, take your time.

My point was about your use of "demanding ratings details," which is 
not intrinsic to range methods. In particular, I've been pointing 
out, Borda is a ranked method that is a Range method, and it becomes 
full range if the method simply allows one to equal rank any two (or 
more) candidates without disturbing the points given to other candidates.

The most fully expressive ballot is a Range ballot, of course, and 
the higher the resolution, the higher the allowed expression.

The simplest way to understand this is through this progression:

Plurality. Vote for one, candidate with the most votes wins.
Approval. Vote for one or more, candidate with the most votes wins.
Range. Vote for one or more, fractional votes allowed, candidate with 
the most votes wins.

What distinguishes Range from Ranked methods is the allowance of 
fractional votes. However, once it is possible to vote fractions, and 
particularly if the resolution (fractional increment) is fine enough, 
a Range ballot can fully express ranking, which is a Range ballot 
interfaces with a ranked ballot. Borda is a range method that is a 
ranked method, and the connection is that the number of unique 
ratings is equal to the number of candidates, thus a Borda ballot has 
adequate resolution; however, typically, Borda ballots prohibit 
assigning the same rating to more than one candidate, and all ratings 
are assigned or the vote is diluted. If overvoting and empty ranks 
are allowed, Borda is simply Range (N-1), where N is the number of candidates.

>   I used care in
>mentioning ranking to avoid complications such as you add - and
>clearly included equal ratings and rankings.  Your extensions could be
>useful if they contributed value, but not if they just complicate.

You have not understood the "extensions," which may be because your 
lack of understanding causes them to seem complex.

All this stemmed from your complaint or comment that range methods 
"demand" ratings details. They don't. You can vote a Range ballot as 
Borda, generally. Just spread the votes across the range. It's 
trivial if the number of ratings allowed is the number of candidates.

But you seem to assume that "Range" involves some particular number 
of ratings; you cited Range 99, when, quite likely, Range will be 
implemented with much less than 100 ratings (0-99 in Range 99).



>>>>"Demanding" is an odd word to use for "allowing." "Condorcet"
>>>>doesn't really refer to ballot form, though it is often assumed to
>>>>use a full-ranking ballot. In any case, a ballot that allows full
>>>>ranking, if it allows equal ranking and this causes an empty space
>>>>to open up for each equal ranking, is a ratings ballot, in fact.
>>>>It's Borda count converted to Range by having fixed ranks that
>>>>assume equal preference strength. Then the voter assigns the
>>>>candidates to the ranks. It is simply set-wise ranking, but the
>>>>voter may simply rank any way the voter pleases, and full ranking is
>>>>a reasonable option, just as is bullet voting or intermediate
>>>>options, as fits the opinion of the voter.
>>>
>>>Assuming I LIKE A, B & C are almost as good, and I DISlike D:
>>>
>>>I can rate A=99, B=98, C=98, D=0 or rank A high, B&C each medium, and
>>>D low (A>B=C>D).
>>
>>Dave, you are assuming that the ratings ballot has more ratings than
>>candidates. That is precisely what I did not suggest. That's why I
>>mentioned "Borda." It seems you are thinking of Range 99 as "Range,"
>>when Range is a family of methods, with the range of ratings being,
>>normally, from 1-N for Range N. With 4 candidates, the equivalent
>>Borda ballot has four ranks (1st, 2nd, and "no vote" perhaps). If
>>the ballot allows equal ranking, then, you really have a Range 3
>>ballot. So your "simple ranking" would be A>B>C>D or A>C>B>D. With
>>no equal ranking allowed, you must choose one of these, but the
>>condition of the problem is that you have no basis for this. Is that
>>hard, or what?
>
>Since the topic is Condorcet equal ranking can be allowed, and I
>clearly indicate use of that.

Yes, you did. However, you also assumed a very high resolution range, 
thus creating an appearance of some difficulty. Your stated condition 
can be expressed with a Range 3 ballot: A 3, B 2, C, 2, D, 0. In real 
terms, if the difference between A and B=C is 99 to 98, and if D is a 
viable candidate, there really is no difference at all, it is a 
formal expression of preference without significant substance. 
However, if D is not a viable candidate and is "Satan," then the 
ratings of B=C have been disturbed by the presence of D. It's a 
complex subject, in fact.

>After describing B and C as equally ranked I used common symbology -
>(A>B=C>D) - and am not used to the symbology you use below.

Sure, you aren't. But the only difference is the existence of empty 
ranks. I'd have thought that obvious. I expressed an empty rank with a "."

I wrote that it was the "same ballot." That means that the ranks are 
laid out on the ballot for you to specify. You can leave a rank 
empty. This is actually how Bucklin was implemented, and it is this 
that made Bucklin a Range method. The empty ranks have significance. 
They indicate a preference strength. The problem with pure 
preferential ballots is that they show no preference strength. A>B>>C 
is just A>B>C.

>>Now allow equal ranking on the same ballot.

I.e., a ballot with fixed ranks. IRV ballots are sometimes set up 
this way, with facility for ranking all candidates (making the ballot 
the same, effectively, as a Borda ballot), or for ranking a fixed 
number. But preferential interpretation of that ballot means that an 
empty rank is meaningless, it is simply skipped, as if it did not 
exist. In Borda/Range, it has meaning.

>>  Yes, you have a choice,
>>with the simplest ballot rules: You can rank them A>B=C>.>D (D
>>perhaps not being on the ballot, but I'll show the bottom rank), or
>>as A>.>B=C>D.

Do you understand the notation now? The first example, A>B=C>.>D 
means just what you said. But you could also have ranked them 
A>.>B=C>D, which would mean something a little different. Instead of 
liking B and C almost as much as A, you, rather, dislike B and C 
almost as much as D.

>>  It's a trade-off, and which one you pick depends on
>>two factors: how strongly do you want to prefer A, and how strongly
>>do you want to act against D? Strongly preferring A indicates you
>>put both middle candidates in third rank, strongly acting against C
>>indicates you might put both middle candidates in second rank. In
>>addition, there are the probabilities to consider, which may
>>outweigh the preference strength issue. Is it possible for A to win?
>>If so, indication is that you should rate B and C lower. Is it
>>possible for D to win? If so, then you might want to rate B and C
>>higher.

And then I've introduced a real consideration: strategy. There is a 
serious problem with assuming that ratings on a range ballot should 
simply reflect relative "like" or "dislike." The fact is that we make 
choices based on real possibilities, not merely on absolute 
preferences. And it's necessary unless there is some way for a method 
to "amplify" our preferences once irrelevant alternatives have been 
removed. That is, in a way, what preferential methods do, though it 
isn't normally described that way, and doing it without 
discrimination means that trivial preferences are given the same 
weight as strong ones. The paper from Voting Matters, latest issue, 
that used a Range ballot allowed what might be called a "sincere 
absolute range expression," with other data that, if I understand the 
paper correctly, caused "expansion" of the voting power over a 
narrower set, thus allowing the kind of vote that could be:

A:100
B:51
C:49
D:0

Where A was the Messiah, and D was the Antichrist, and B and C are 
the real candidates. If I get it correctly, the voter's additional 
expression would cause the B and C votes to be expressed with full or 
appropriate strength. In Dhillon-Mertens Rational Utilitarianism -- 
which is Range voting -- the voter does this, and might vote A:100, 
B:99, C:1, D:0. The absolute utilities have been modified by real 
election probabilities.

Unless one, for example, thinks of D as a frontrunner, in which case 
one might well vote A:100, B:99, C:98, D:0.

Bucklin with runoff can handle this case reasonably well.

>In ranking all I can say is to rank B&C above D and below A..
>
>Go back to the example and see B and C each rated 98 because I DO NOT
>want them to lose to D.

Yes. And that's what I described. But you made the decision more 
complex by imagining a high-resolution Range method. If D is not a 
realistic outcome, you then may have over-rated B and C. Essentially, 
you rated A, B, and C all the same, for most practical purposes.


>>If the frontrunners are A and D, *it matters very little where you
>>rank B and C*
>
>True, but ranking them below A and above D gave what insurance was
>possible.

Depends on the canvassing method. Bucklin is cool because you can 
express any significant first preference with very little harm, yet 
still vote maximum strength for your second or third choices. 
Bucklin/Runoff even allows you to postpone the lower choice decisions 
to a runoff, unless you see a danger of your least favorite winning 
in the primary.

>>If you have trouble deciding to go for low ranking or high ranking,
>>there is an option that might be allowed in Bucklin or Range: half- 
>>ranking. The way that A low-res Range 3 ballot might be shown would
>>be a list of candidates, with three options for each candidate. If
>>you mark more than one option, your vote would be, with range, 
>>half- assigned to one rank and half to the other. (or a third, etc., if
>>you mark more than two, but with this particular ballot you could
>>just neglect the middle rank vote, it would end up the same). With
>>Bucklin analysis, same, except that in the counting rounds, a
>>"middle rank" would be counted after the higher rank and before the
>>lower.
>
>Huh?

Yes, you don't understand. It's been explained before, you either 
didn't read it (which is fine) or you didn't understand then, either.

Take a three-rank Bucklin ballot. Suppose a voter votes for a 
candidate in both second and third rank. When is the vote counted? 
The only law I've seen that considered this specified that the vote 
was counted in the higher rank. That's arbitrary, but better than 
tossing the vote. It could have been that it would be counted at the 
lower rank. The problem with Bucklin and voting machines was that 
both the votes would be counted, unless the ballot was arranged 
properly so that such overvotes would be impossible. This requires 
that the ballot be arranged so that each candidate is, as it were, an 
independent race, vote-for-one-rank-only.

But with hand counting or more sophisticated machine counting, the 
vote could be counted as if it were in rank 2.5. That would mean that 
it is not counted in the second round, but before the normal third 
round. This would allow finer ranking. Otherwise the possible 
information is discarded. Which is better? In any case, the voters 
should know how such a vote will be counted.

This interpretation of such overvotes could turn a three-rank Bucklin 
ballot into a five-rank one, with no fuss or extra ballot space taken 
up. Indeed, it would convert a two-rank ballot to a three-rank one. 
And I don't think it would be at all difficult to understand.

>>There are other reasons for defining what such "overvotes" mean,
>>basically to avoid discarding ballots that have an apparent meaning.)
>>
>>It is, in general, easier to rank candidates if the equal ranking
>>option exists. The issue, then, is how such equal ranking is to be
>>interpreted. IRV rules typically toss the vote. Not allowed. But, in
>>some small level of progress, in the U.S., the ballot simply is
>>considered exhausted at that point, the higher ranked candidate
>>still have their votes (which, if the lower ranked votes, where the
>>overvoting was, are being counted, the higher ranked candidates have
>>been eliminated. But at least the whole ballot hasn't been tossed.)
>
>Why say this?

Why not?

How overvotes are handled is an important topic. It's the "equal 
ranking" topic, really, which certainly is relevant to Condorcet 
counting methods.


>>>The example ratings of A, B,&C do the most I can to make any of them
>>>win over D; the example rankings do the most I can to make A win, D
>>>lose, and give B&C an equal chance.
>>>
>>>In Condorcet I ranked A over B and C over D but could not express the
>>>magnitude of these differences.  In Score I must rate with numeric
>>>values that include the differences.

You are showing, Dave, that you have completely missed the point. 
Again, you use "must." No, a Range ballot can simply be a list of 
ranks. On a real ballot, you would not enter numbers at all, but the 
voting positions might have numbers attached. With low-res Range, 
they might have names. I've described a Bucklin ballot that is a true 
and complete Range 4 ballot, it would have names like

Favorite(s)
Preferred
Acceptable
Less than Acceptable
Rejected

Sort the candidates into these five categories. The top three 
categories are approval votes if you place the candidate there. The 
lower two categories are disapproved categories and are used for 
Condorcet analysis and runoff determination, probably.

The magnitude of preference between two candidates is expressed on 
this ballot by the rank distance. On this ballot, the rating step 
between ranks can be expressed as 0.25.

Bucklin is a method which steps down what can easily be seen as a 
Range ballot, adding in approvals when the sliding down of Approval 
cutoff reaches candidates. Classic Bucklin had only three ranks, and 
was equivalent to a Range 4 ballot with the ratings of 0 and 1 
combined (into 0). Any approved candidate was rated at or above 
midrange (which sets midrange as the average expected election value, 
classic approval voting strategy).

The theory can get somewhat complex, but the actual voting was very 
simple. I'm suggesting tweaks for the use of Bucklin in runoff 
voting; these tweaks would make it Condorcet compliant, or so close 
to compliant that the difference would be merely theoretical and very 
unlikely to show up in actual elections. (Depends on the runoff 
rules, but I've suggested that any Condorcet winner should be 
included in a runoff, and the ballot contains good data for the 
determination of that.)

The above ballot, with an explicit Rejected rating (which would also 
be assumed, I'd prefer, for any non-rated candidate), can become even 
finer in expression of the "overvoting" scheme is followed. By using 
multiple rank expression, the five single ratings become nine 
possible ratings. My opinion is that this is more than adequate. But 
the voter does not need to pay attention to these complications 
unless the voter needs the flexibility.


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