[EM] Deconstructing Craig

[Richard Moore]

Craig Carey wrote me back to say that the test failed by approval
voting is his P4, which apparently is a definition of "ballot power" 
that Craig likes.

Recall that the EM list discussion on this subject started last week
with a question about the "one person, one vote" rule, which approval
meets. "Ballot power" is something undefined and may be quite
different from "one person, one vote". It may be something irrelevant 
as the infamous "Natapoff power" demonstrates.

We could discuss approval's ballot power meaningfully if only we have
a consensual and useful definition of "ballot power". I am not aware 
of any consensus about the meaning of that phrase. We need to start 
with a look at what possible meanings are or are not useful.

Here is a description of P4 from Craig's site.

"P4: (description_2). For all papers, there exists a collection of
FPTP papers that the paper can be altered into, without the set of
winners [?or just those named on the paper], altering. That got
revised to saying that FPTP papers are able to produce a
not-less-desirable outcome even in the case when the winners can't be
reproduced.

"If adding a paper 1:(BCDE) leads to winners {B,D}, then the
satisfaction = 1/2^1 + 1/2^3 = 5/8. The P4 rule would require this
(and other things): A satisfaction of 5/8 (with respect to (BCDE)) has
to be bettered or reproduced for some x,y,z, when instead the added
papers are x:(BC) + y:(BD) + z:(BE), subject to
(0<=x)(0<=y)(0<=z)(x+y+z<=1).

"That statement has an umbral shadow of the tetrahedron be cast to a
point. The idea is that by splitting the vote out onto papers with
less preferences, it must be possible to get an outcome that is as
good. If it can't be done then the shorter preference papers have less
power to achieved the outcome desired by the given paper. Perhaps it
needs to be improved a bit, but that may be able to be deferred. The 3
candidate Alternative Vote fails this test."

The actual definition, if I understand it right, is in the first and 
second paragraphs and is awkwardly split by an example. On first 
reading I thought the definition was contained in the first paragraph 
alone. The second paragraph adds constraints on x, y, and z to the 
definition.

For his example, Craig calculates a "satisfaction" number, but the 
method of calculating this number (for multiple-winner elections) 
isn't justified. Which is more satisfactory for the B>C>D>E voter: if 
the winners are C and D, or if the winners are B ane E? Why? How do 
you calculate this number for a "B=C>D>E" ballot?

A question about the constraints: If a method can pass the test so 
long as every ballot in that method has a voting power between 0 and 
1, isn't there room for a huge disparity in voting power between 
ballots? Wouldn't it take 10000 ballots of power = 0.0001 to equal the 
strength of one ballot of power = 1.0?

The 3rd paragraph contains an admission that the definition may want 
some revision, yet "[maybe that can] be deferred". Statements about 
whether a method passes or fails a criterion can't be judged true or 
false if the criterion isn't fully developed; what if next year's 
version of the criterion changes the truth value?

Craig does not justify how splitting into FPTP papers relates to
"ballot power" (as we might understand it). One would think a ballot's
power is related to it's ability to achieve an intended result, which
may be the election of one specific candidate, or the prevention of
one specific candidate from being elected, or some in-between goal
such as confining the winner to be a member of a certain set. 
Asymmetrical FPTP, which allocates more power (in the sense I just 
described) to those who wish to elect one specific candidate than to 
those who primarily wish to defeat that candidate, seems a rather odd 
method to be used as a standard for measuring other methods, 
particularly symmetrical methods such as approval voting.

Without a consensus behind it, P4 is just an arbitrary criterion and 
may be irrelevant. Use of FPTP as a reference may be prevent any
consensus for P4 from forming. Suppose a ballot is cast in FPTP, with 
the intent of preventing W from winning (out of the set of candidates 
W, X, Y, and Z). The ballot could be cast for either X, Y, or Z. This 
ballot might have only one third the power of a ballot cast for W with 
the intent of making W win. That is, to cancel a certain number of W
ballots, as many as three times that number would have to be cast if
the ballots are spread equally across X, Y, and Z. An alliance could
be formed if voters can agree on which of the three alternatives to
put all their votes on; this changes the power structure of the
electorate and increases the power of ballots for those within the 
alliance, though not equally. If the alliance chooses X as its 
candidate, then those voters who like X the most will be more 
empowered by this alliance than those who prefer Y or Z. P4's reliance 
on FPTP's variable-power ballots is like using a rubber band to 
measure linear dimensions.

A few excerpts from Craig's message follow (CC = Craig, RM = me):

CC: When an "(All M,x, M a method, x, an election)Rule(x,M(x))"
becomes false at a single point, we quite reject the rule or the
method or both, but you instead wrote on an idea of me being
pedantic.

RM: I challenged both aspects of Craig's message. Unfortunately the
technical aspects of his post were so lacking information that I
could not address them in full, so I also challenged him to 
substantiate his statements. That's a fair request and I acknowledge 
his attempt to comply with it.

CC: I will ignore you said you did not want me to write to you

RM: I don't recall asking that. I did challenge Craig, hoping for a
response, but it would have been more appropriate to respond to the
list than to me directly, so that all can share in his wisdom.

CC: You could have followed the link I gave on Borda and found the
test that can prove Approval fails One Man One Vote.

RM: It is true that we were discussing "One Man One Vote" but Craig's 
post was about ballot power. They are not equivalent.

CC: A person would have to be as dumb as a dog to imagine that
Approval did not increase the power of papers when more sub-votes are
added

RM: That may depend on the definition of power (and P4 doesn't look 
like a good candidate). And if we adopt a reasonable definition for 
which that is true, we should find that adding more "sub-votes" 
decreases power after the midway point. In fact, by the above 
definition of P4, an approval ballot supporting 5 candidates has a 
power of 5, even though in a 6-candidate election it has no more power 
than a vote supporting a single candidate.

CC: (I actually use REDLOG so I am not part of the crowd).

RM: I have used many computer languages and tools. I find that they 
only do useful work with meaningful input. GIGO.

[from my post that prompted Craig's response:]
 >I would like Craig to present an argument, at least as coherent as the
 >one above, for why Approval fails if the number of candidates is
 >large, and to state what test it is failing.

RM: That challenge (other than the part about identifying the failed 
test) was unfortunately not answered. Some thought needs to be given 
to whether failing P4 is actually a bad thing.

 >Also, why is IFPP only formulated for small numbers of candidates, if
 >large fields of candidates are such an important consideration?

CC: IFPP is defined implicitly quite well. The P4 rule defining equal
suffrage and implying "One Man One Vote" is seemingly imperfect so the
axioms might be unable to get a 6 candidate method out.  By keeping
the axioms differential, I can have a system that does not fall over
and become eliminate when a critic arrives. I have limited time for
research into IFPP equations.

RM: As I've shown, P4 is more than "seemingly" imperfect, but at least 
we know Craig is aware of flaws.

CC: Here you seem to identify the acceptable meaning:

 >I suspect those using this phrase in such a context intend it to mean
 > "one person, one ballot mark per race".

RM: Actually, I identified that meaning as *unacceptable*.

CC: That agrees that A single FTP paper (or a positive sum of some 
with the sum being 1) is the baseline for the number power, and where 
power = 1.

RM: I've just shown that the FPTP ballot is an unstable baseline, and 
that it provides high power numbers for some low-power ballots, so 
Craig's statement can be rejected.

CC: But what you did not say is that you and anybody else are not free
to alter the definition. Directing the word "believe" towards others
and then appearing to investigate them isn't flip the balance between
you being wrong and the rest being right. The power of one man's paper
is its power to oppose an FPTP paper [actually a +ve sum of them].

RM: That statement isn't easy to parse. It might be translated as
follows:

De-obfuscated CC: Nobody is allowed to change the definition. Noticing 
that others "believe" something and then trying to understand their 
beliefs doesn't make you right or them wrong. A ballot's power is 
determined by the number of FPTP ballots it can cancel out.

RM: Now Craig cites a definition that is quite different from the one 
identified above as "description_2", so maybe the freedom to alter the 
definition does exist. Whether we have the freedom to alter a 
definition we don't like, we are free to reject it if it has no merit.

  -- Richard

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