[EM] Consistency, Truncation, etc. (was CR ballots, etc.)
Forest Simmons
fsimmons at pcc.edu
Thu Sep 27 16:33:12 PDT 2001
On 26 Sep 2001, Buddha Buck wrote:
> Forest wrote:
<snip>
> >
> > If candidate A wins DECISIVELY in all the subsets in some partition of the
> > electorate (by restricting the election to the ballots from each of the
> > subsets of the partition in turn) then candidate A wins the entire
> > election.
>
> By defining "DECISIVELY" for Condorcet Methods as "has a Condorcet
> Winner", then Condorcet Methods pass HCC.
> >
> > IRV fails even this humble version of the CC.
>
> How do you define "Decisively" for IRV?
Of course "Decisively" is decisively vague. The point is that IRVies
believe that IRV is decisive unless in the final round of the instant
runoff there is a tie or near tie.
In the Greek tragedy example referred to below, the wins are definitely
decisive by IRVie standards, and arguably decisive even by the tougher
standards of EM list readers. The eliminations and wins are all by large
margins and significant percentages.
That's enough to show that IRV fails the Humble Consistency Criterion :-)
>
> > See my Greek Tragedy example in the archives at ...
> >
> > http://www.mail-archive.com/election-methods-list@eskimo.com/msg06168.html
> >
> > Forest
>
Here are some (I hope thought provoking) remarks on the use of vague words
like "large" and "decisively" for those whose neural pathways are not yet
set in stone :-)
For the first 200 years of Calculus, "Calculus" was short for "The
Calculus of Infinitesimals."
In that period great progress was made in differential equations and their
applications to physics. That was the era of the Bernoulli brothers and
the great Euler.
In the 19th century mathematicians like Karl Weierstrass who were
concerned with logical rigor (and embarrassed because they couldn't define
infinitesimals to the satisfaction of Bishop Berkeley and other
philosophers) banished infinitesimals from mathematics in favor of a
theory of limits as a foundation for the calculus of derivatives and
integrals.
The physicists and other applied mathematicians continued to use
infinitesimals because, although not "well defined," they were more
intuitive and less cumbersome to use than limits. [See the example
marked with "**" below.]
Then, after infinitesimals had been in exile for more than a century,
Abraham Robinson, who had a unique grounding in both applied math and
mathematical logic, vindicated the users (and continued use) of
infinitesimals by codifying the ('til then) informal rules of dealing with
them, and then proving that those rules were just as consistent as the
limit rules.
His 1966 book, "Nonstandard Analysis," is one of the great classics of
all time.
Now to bring this closer to home.
One of the things that infinitesimal users noted early on was that when
applying them naturally, you never have to worry about the precise
location of the boundary between infinitesimals and non-infinitesimals.
Notice the same feature of our application of "decisively".
In the Greek Tragedy example, the IRV decisions were obviously decisive by
IRV standards. We didn't have to worry about the precise definition of
"decisive".
There might be some other method for which it is impossible to determine
whether it fails the CC decisively or not. [If there were that much
ambiguity, it would seem that the failure could not be decisive :-]
In modern set theory (considered by orthodox mathematicians to be the
rigorous foundation of all mathematics) there are classes of objects
called sets, and other classes of objects which are not sets.
The class of all sets is an example of a class not considered to be a set.
Admitting this particular class as an element of itself would give rise to
various inconsistencies (like Russell's paradox) in set theory.
Some expanded formulations of set theory give special status to certain
classes called (in those formulations) semi-sets. They obey some of the
properties of sets (more than a class chosen at random would) but not all.
In some of these theories the class of all standard integers is admitted
as a semiset. [An integer is standard iff its reciprocal is not
infinitesimal.]
The class of standard whole numbers cannot itself be a standard set
because although it is bounded above (by each and every non-standard whole
number) it has no least upper bound. That is, it violates one of the
properties of numerical sets, the Least Upper Bound Axiom.
In other words, the class of non-standard integers is not crisp; it has no
definite boundary. A non-standard integer is an integer whose reciprocal
is infinitesimal, so the non-crispness of the class of non-standard
integers is inherited from the semiset of infinitesimals.
In mathematics both "tiny" and "large" are considered to be difficult
concepts to define, even though small children routinely use them fairly
consistently after being held to ridicule by older siblings for their
initial blunders.
**Example of the intuitive value of infinitesimals compared with limits:
Definition of uniform continuity using infinitesimals:
A function f is uniformly continuous if and only if
whenever x-y is infinitesimal, so is f(x)-f(y).
Same definition using limits:
A function f is uniformly continuous if and only if
for each positive number epsilon
there exists another positive number delta
such that
whenever the inequality
|x-y| < delta
is satisfied,
then so is the inequality
|f(x)-f(y)| < epsilon
which is a nice poem, but harder to get the gist of than the nonstandard
version.
The nonstandard version not only reaches our intuition more directly, it
is also easier to apply.
For example, let's use it to prove that the functions given by
F(t)=3*t and f(t)=t^2 are respectively, uniformly continuous and not
uniformly continuous:
F(x) - F(y) = 3*x -3*y = 3*(x-y),
and three times an infinitesimal is still infinitesimal.
Therefore, if x-y is infinitesimal, so is F(x)-F(y).
Now to show that squaring function f is not uniformly continuous:
Let t be any nonzero infinitesimal.
Let x=(1/t)+t and y=(1/t)-t.
Then x-y=2*t, which is infinitesimal, but
f(x)-f(y) = x^2 - y^2
= (x-y)*(x+y)
= (2*t)*(2/t)
= 4,
which is not infinitesimal, QED.
The standard proofs of these same two results are very cumbersome in
comparison. More importantly, they are harder to derive.
In this last proof, for example, it was easy to figure out appropriate
values of x and y based on the obvious requirement that they be large
numbers (out where the graph of f is steep) but close to each other.
The derivation of the epsilon/delta proof is much more difficult.
Forest
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