[EM] pairwise, fairness, and information content

Craig Carey research at ijs.co.nz
Fri Aug 16 17:47:46 PDT 2002

At 2002\08\16 23:06 +0100 Friday, James Gilmour wrote:
 >Maybe there's something I'm missing here, but I just cannot see "the
 >issue" with 'two candidates, one winner' elections. I'll freely admit
 >most of the maths, the complex equations and the formal logic statements
 >(to say nothing of the jargon) go straight over my head, but just what is
 >the problem?

I suppose the text is replying to Alex Small rather than to me, e.g. about

At 02\08\16 08:06 -0700 Friday, Alex Small wrote:
 >In other words, one mark of genius is the ability to take highly advanced
 >matters and explain them in simplified terms.  It shows an understanding
 >of what is essential and what is not essential.  There is nothing to be
 >disdained about a "layman's discussion."
 >Just some food for thought.  I look forward to some day reading a clear
 >and informal description of how you think we should select a single winner
 >from a set of two candidates.

To reply to Mr Small. The questions is not one that I am answering.
However if it is mathematical and a request for an definition then I have
already answered it: (in the following, q=1/2 [unless the method is
At 02\08\16 11:11 +1200 Friday, Craig Carey wrote:
 >At 02\08\14 19:12 -0700 Wednesday, Richard Moore wrote:
 > >A couple of weeks ago, Craig Carey made an assertion that (to me, at
 >(q < a0/(a0+b0)) implies (A wins)
 >(b0/(a0+b0) < q) implies (B wins)

I corrected the B-wins equation:

At 02\08\16 11:46 +1200 Friday, Craig Carey wrote:
 >|(b0/(a0+b0) < q) implies (B wins)
 >              > q

 >I also fail to see the relevance of this boundary condition, ie "two
 >candidates, one winner", to the resolution of the real issues that do
 >arise as soon as you move away from this extreme, eg  move to "three
 >candidates, one winner".  So why is "two candidates, one winner" being
 >discussed in the context of the more general (and more common) problem?

Fortunately Richard was ruling out consideration of checkboxes in the
recent messages to me.

I can give some answers:

* a seemingly proper aim is to get a single set of axioms for all
numbers of candidates and winners. In my theory I combine monotonicity
and truncation resistance into a single rule named P1. [Truncation
Resistance by itself but monotonicity doesn't seem to be and I replace
it with P1.] I just wrote on P1 and described how I avoided doing
algebra by using a method of telnetting to computer running REDUCE. The
maths of the checking the P1/monotonicity compliance of a 2 candidate
method is not that simple.

* A clear principle is that readers and experts (excluding social
decision theorists who make mistakes), can't guess correctly, about
what rule testing all methods is best, and what methods are best.
These people that guess are always at a risk of making a rule too harsh.
If the rules are renamed axioms then a rule that is too harsh can
easily produce a method that seems quite unacceptable. So far it seems
that I am the only one that does the algebra from rules to methods.
(A rule that is too harsh is likely one that fails STV.) The rules
have to be perfectly designed or there is no method that both tests
and rejects the rule before it can reject old known methods. Also,
the considerations should start from the simplest cases because of the
assumed correct requirement to avoid guessing wherever possible, as
if it were an to add a candidate at a time and keep the rules as
weak as possible,

 >"Three (or more) candidates, one winner" elections present a number of
 >different problems, but still the best you can do is to guarantee
 >representation to only half of those who voted.  Of course, your method
 >of voting and counting may do much worse than that.  And there will be
 >different outcomes, depending on the methods you use.  And there will be
 >debate about which outcome "best" reflects the "wishes" of the voters.

Certainly not. A competent method designer would model the needs of the
voters using a set of rules. There need not be any great room for debate
too. In a real world, what could happen is that STV reformers almost
feel guilty for not going with the latest and they can't quite explain
it all. Also voters can have representatives. E.g. persons without
"wishes" who can out-argue an STV advocate in public but also in private.

 >And there will be debate about how those "wishes" are to be assessed and
 >how "best" is to be defined.

Debate does not occur in appeal courts: when a person says that the
exhibited non-monotonicity of the Alternative Vote was unfair, then how
is it possible to have a debate?. I imagine that any debate of the
public would be over the question of whether to have the obvious and
correct decision, ignored, rather than to enter inside of the topic of

 >Leaving aside the maths, there seem to be two quite different approaches
 >to determining what kind of outcome "best" reflects the "wishes" of the
 >voters in multi-winner elections.  Those who believe in representative
 >democracy want to see all significant viewpoints represented, ie maximise
 >the diversity of those elected, to reflect all the significant
 >diversities among those who voted.  (Let's leave aside for now the
 >determination of "significant".)  But there are others in this debate who
 >appear to come from a "social choice" background, where the objective is
 >not to ensure the representation of all significant diversity, but to
 >achieve "consensus" representation in those who are elected.  These two
 >approaches are fundamentally incompatible because their desired outcomes
 >are diametrically opposed.

It is not clear that your view is any of those two. Neither seems to
require 'One Man One Vote'. Possibly the idea of "0.2" approaches instead
of "2" approaches, might be nearer the facts.

 >rarely admit their true intent.  The concepts of social choice MAY have
 >some relevance to "one winner" elections, but the situation seems quite
 >diffferent for multi-winner elections to councils and boards that are
 >being elected to represent the communities they serve.

You wrote "social choice". That is about the 2nd item of the two above,
an idea of summing correctly.

Presumably it would correspond to a rule requiring that f=1 in this
this 2 candidate multiwinner FPTP/SNTV election:

The papers and their weights are

A   a0
AB  ab
B   b0
BA  ba

(Subtotal for candidate A) = a0 + f*ab
(Subtotal for candidate B) = b0 + f*ba
    The 0 to 2 candidates with the largest subtotals, win.

I can get the result that f=1 by saying that P1 implies truncation
resistance, and truncation resistance implies:

(1) Candidate A's win-lose status is unchanged as (A) papers are
altered into (and/or from) (AB) papers. Also we need to use the
right number of winners rule to stop B's win-lose state from

Alternatively, P2 can get the same result more easily by requiring
that the full set of winners is unchanged when -(A)+(AB) papers are

But both of those rules have not been admitted to be respected, by
Moore and all the rest.

You wrote about voters but suppose there are none?. It is the same
formula. You added something artificial that has no real relevance.
Whether a mayor is elected or the election is a mere simulation,
the method is constant. We could get someone from France [is that
the country?] to explain the imprecision that a skilled use of
English can bring ?. A mayor could be a mayor in a doll house.

Rather than say that the topic of 2 winner elections is simple,
instead it could be said (if true) that progress with checking the
FPTP method against the axiomatic rules, is too slow.

Craig Carey

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