Subcycles and the Rich Party Problem

[Mike Ossipoff]

Steve Eppley writes:
> 
> Mike O wrote:
> -snip-
> >Adding the subcycle rule to Copeland would be a patch that would
> >necessitate a generalization of ITC in order to still be able to
> >demark methods that have the Rich Parties problem from methods that
> >don't have it. 
> -snip-
> 
> What generalization of ITC do you have in mind that subcycle-patched 
> Copeland would fail?
> 
> ---Steve     (Steve Eppley    seppley at alumni.caltech.edu)
> .-


I've writtens some generalizations of ITC which are based on
"co-partisan sets" of alternatives. A co-partisan set is 
a set of alternatives such that everyone who ranks 1 member
of that set over some alternative X ranks every member of
that set over X. And everyone who doesn't rank X over some
particular member of the set doesn't rank X over any member
of the set. And everone who ranks some alternative Y over 1
member of the set ranks Y over every member of the set. And
everyone who doesn't rank Y over some particular member of
the set doesn't rank Y over any member of the set.

I've written several criteria about co-partisan sets. One of
them says:

A method meets the Co-Partisan Independence Criterion #1
(CPI-1) if:

Given any configuration of candidates & votes, there's always
a way of adding or subtracting members to or from any co-partisan
set without changing the election by giving or denying victory
to an alternative outside that co-partisan set, regardless of
how many are added or subtracted.

[For this criterion I couldn't say "adding or subtracting members
to or from a co-partisan set can't change the election result
by giving or denying victory to somethign outside the set", because
it would usually or always be possible, with any or almost any
method, to add to a co-partisan set an alternative that can
deny victory to a member of that set, without itself being able
to win].

Of course, for any initial configuration of candidates & voters,
adding a new candidate, for whom the voters vote, with respect
to the other candidates, however you choose to specify, is
permissible, and doesn't amount to changing that _initial_
configuration. And of course no other change is permissible
other than the removal of a candidate (in the co-partisan set)
from the election.

The point of this is to determine whether a method's
result can depend on the size of a co-partisan set.

In Copeland it's easy to have a situation where the co-partisan
set initially (say it just has 1 member) is in a tie with
the candidate that it beats. Adding another member to the
co-partisan set gives an additional defeat to that other
candidate, the one that the set beats, and is in a Copeland
tie with. This lowers that non-co-partisan candidate's
Copeland score by 1, and gives the election to the initial
member of the co-partisan set, if the new member is beaten
by him, or to the new member if he beats the initial member
of the co-partisan set.

Of course any 1 candidate is a co-partisan set.

***

I emphasize that this is a new criterion, and if it has
some bug in it, that shouldn't cast doubt on my
previous criteria: LO2E-1, LO2E-2, IME, Truncation
Resistance, & GMC. Those latter 5 criteria aren't new,
and have been thoroughly discussed, unlike this new
criterion defined in this letter.

***

I never meant to imply that precise criteria aren't important
& useful. Though standards can be used without criteria, 
criteria give precision, and make it possible to make
definite statements. 

***

I've written several other versions of CPI, which do say
that adding or subtracting an alternative from a co-partisan
set can't change the election by giving or denying victory to
anything outside the set, _under specified conditions_, such
as: 

A method meets CPI-2 iff adding a new member to a co-partisan set
can never change the election result by denying victory to anyone
outside that co-partisan set unless that new addition to the
set is the winner by that method's rules. 

[Obviously one could always deny victory to the previous winner
by adding a new candidate who's the winner by the method's rules.
We're trying to test for whether the mere addition of a member
to a co-partisan set can deny victory to someone outside the
set, merely because we've added a member & enlarged the set,
not because we've added a new candidate to the election who
is the winner by the method's rules].

***

I just want to add that Copeland's candidate-counting standard
just doesn't make any sense. A few examples, like the Red, Green
& Blue parties example, and the movie-genres story, _are_
sufficient to establish that. It doesn't take a mathematical
proof. Those examples are sufficient to show that Copeland
counts something that's irrelevant.

***

Mike








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