FPTP family theory, REDLOG shadowing

[Markus Schulze]

Dear Craig,

you wrote (13 Dec 1999):
> Mr Schulze, are satisfied with my comments on geometry?

Unfortunately, there are still many questions waiting to
be answered.

******

You wrote (10 Dec 1999):
> My aim is to avoid numbers (and single election examples) because the aim
> is to have rules hold, and rules impose constraints over an infinity of
> winner sets. An obvious problem is: what happens when the problem is
> found to be quite implicit in that it can't be solved until it is already
> solved?. A method of 'guessing a method and testing it' is an  option but
> that method can keep failing for a decades. If numbers are replaced with
> algebra (geometry) the implictness can vanish maybe.

The reason why so many paradoxa stay undiscovered (e.g.: The Dodgson method
has been proposed in 1873; but the first person who discovered that the
Dodgson method violates monotonicity was Fishburn in 1977.) is the fact
that many paradoxa occur only when the number of candidates is large
(e.g.: The Dodgson method violates monotonicity only if there are at least
five candidates.). [Another example: Ossipoff's subcycle rules violate
Pareto only if there are at least six candidates.]

It seems to me that a problem of your geometrical interpretation of election
methods is the fact that -if you want to define an election method implicitely
by describing its simplex- you have a different simplex for every possible
number of candidates so that even if you have proven the absence of a given
paradox for up to 102 candidates (or whatever your favorite number of
candidates was) you still don't know whether this paradox occurs for 103
candidates unless you have also defined and investigated its simplex for 103
candidates. Do you agree?

******

You wrote (16 Sep 1999):
> What of Condorcet, a topic of this mailing list?:
> An example: In an election with 2,152,370 candidates and
> 430,927 winners, how can it be certain that pairwise
> comparing of two candidates is an idea that ever had some
> mathematical importance on the fist day?, in France is it?.

I don't understand your statement. Could you -please- explain it?

I don't remember that ever somebody said "that pairwise comparing
of two candidates had some mathematical importance." Those who support
Condorcet methods do this because their supported Condorcet method meets
monotonicity and is very difficult to manipulate by running additional
candidates. It never happened that anybody said that he prefers a
Condorcet method simply because it is a pairwise comparison method.

Do you want to say that pairwise methods are eo ipso worse than
non-pairwise methods? If the answer is "Yes!": Does your criticism
of Condorcet methods also include those methods that on the one side
meet the Condorcet criterion but that on the other side do not depend
only on the matrix of pairwise comparisons (e.g. the Dodgson method)?

******

You wrote (10 Dec 1999):
> Perhaps you would tell me where you got your "participation axiom"
> definition from. I recall that Mr Ron Holzman, in Israel, published a
> participation axiom definition. I can't recall anything about the
> definition particularly. (He has a website currently. It has no
> mention of voting. There was a bit on citations of graph theory
> publications by Mr Holzman.)

The participation criterion has been introduced by Moulin
[Herve Moulin, "Condorcet's Principle Implies the No Show Paradox," 
Journal of Economic Theory, vol. 45, p. 53-64, 1988].

A weaker version of the participation criterion has been
introduced by Fishburn and Brams [Peter C. Fishburn, Steven J. Brams,
"Paradoxes of Preferential Voting," Mathematics Magazine, vol. 56,
p. 207-214, 1983].

You wrote (10 Dec 1999):
> I do not know if the AV is passed or failed by the participation
> axiom. It will be passed in the 3 candidates case. An interesting
> question could be: assuming (P1) and/or the "participation axiom",
> deduce the values of quotas, for a AV method that was modified so
> that it used quotas (for losers). Some easy inductive argument
> maybe, or a computer simulation done in a day if it takes more
> than a few hours to solve the problem. Lord Alexander wasn't
> impressed with the method, and nor was Winston Churchill.

It has already been proven by Fishburn and Brams that Alternative
Voting violates the participation criterion even in the 3 candidate
case.

Example:

  7 voters vote A > B > C.
  6 voters vote B > A > C.
  8 voters vote C > B > A.
  
  If Alternative Voting is used, then candidate A is elected.
  But if three of the eight CBA voters didn't go to the polls,
  then candidate B would be elected.

Markus Schulze