[EM] Other criteriion for reference method?

[Richard Moore]

MIKE OSSIPOFF wrote:

> Because I still don't fully understand the Monotonicity definition self 
> reference problem,
> and the extent to which it limits Monotonicity definitions, I'm printing 
> that letter out, to
> study the matter further. But it's occurred to me that the Participation 
> Criterion might
> be a better test of a reference method than the Consistency Criterion is.
> 
> Participation is so similar to Monotonicity that people have asked me 
> what the difference
> is between those 2 criteria.


Participation, consistency, and monotonicity seem to be very closely
related. I don't know which would be the better criterion for a reference
method. I was working from the assumption that consistency implies
monotonicity; that consistency is a stronger criterion than monotonicity.
I don't know if participation is stronger or weaker than monotonicity,
and stronger or weaker than consistency.

Of course, if we define monotonicity in terms of a reference method that
is consistent, then consistency implies monotonicity *by definition*.
This can be seen from the formal definition I posted last night: If a
method M passes the reference criterion, then you can choose M as the
reference method (C = M). Then simply letting T be the empty set,
condition 2 turns into:

(2) For every pair (S,S') of multisets of ballots, if M(S') != M(S), then
at least one of the following is true:

  (a) M(S') != M(S) and M(S') = M(S')
  or
  (b) M(S) = M(S) and M(S) != M(S')

which is a tautology. The same tautology would result whether the
requirement for C in condition 1 is consistency or participation (or
anything else).

So whatever criterion we choose for C, monotonicity will be a weaker
criterion than that criterion *by definition* (unless it works out
somehow that it is an equivalent criterion).

Now the question is, is participation stronger or weaker, or about
the same level of strength as consistency?


> If I understand Participation, here's a wording of its definition:
> 
> Participation:
> 
> Adding to the count some ballots that vote Smith over Jones should never 
> change
> the winner from Smith to Jones.
> 
> [end of definition]

> 
> Someone could say "identical ballots" instead of "ballots", but 
> Plurality, Approval, CR,
> & some other methods pass without that addition, and it seems to 
> unnecessarily weaken
> the criterion.


I actually see a couple of variations on the definition; changing
"ballots" to "identical ballots" is one. The other variation is to say
that the ballots that are added, if counted according to the method
under consideration, would elect Smith if only Smith and Jones are
running, but it isn't necessary for each of those ballots individually
to vote Smith over Jones. This isn't what I think of as the
Participation Criterion, but for the sake of comparing it to
Consistency I'll consider this definition. Let's call it the Strong
Participation Criterion (if the name isn't already taken), because,
since it allows more ways of adding ballots than the normal PC, the
method being tested has to tolerate those additional ways without
changing the winner from Smith to Jones.

I believe that Strong Participation is stronger than Consistency,
because Consistency only tests a method by adding sets of ballots
that would elect Smith without deleting any other candidates. Strong
Participation uses those tests, but it also tests the method with
sets of ballots that wouldn't elect Smith unless every candidate but
Smith and Jones were deleted. Personally, I think any criterion
stronger than Consistency isn't of much interest.

Now back to ordinary Participation. It tests the method only with
ballots that vote Smith over Jones. Some of the sets that are used
in Participation will also be used in Consistency, but because some
of the sets allow winners (within that set) other than Smith, those
sets won't be used in Consistency. Also, some of the sets used in
Consistency don't consist exclusively of ballots that vote Smith
over Jones. So I would say that Participation and Consistency
partially overlap each other, but that neither one is stronger
or weaker in an absolute sense.

And yes, adding the word "identical" to "ballots" does weaken the
criterion. I don't believe there's any need for that change.

None of this answers Mike's question of which criterion should be
used to select the reference method, however.


> Consistency too is defined in terms of actual votes, isn't it? Is this 
> Consistency?:
> 
> If an election region is divided into subregions, and a regionwide 
> election is held,
> but we also count in each subregion, to determine who'd win if only the 
> subregion's
> ballots were counted, then if a certain candidate wins in each 
> subregion, s/he should
> also win in the overall region, counting all the ballots.


That certainly matches my understanding of Consistency.

  -- Richard