[EM] Kevin, 8 April, '05, 0533 GMT

[MIKE OSSIPOFF]

James had said:

>	I just wanted to make sure that the criterion was failed by plurality,
>approval, and other methods that don't allow full rankings. Maybe there is
>a better way to do this.

Kevin says:

I see. I forgot who you were talking to.

I comment:

Ok, so only if someone is talking to me is it desirable for criteria to 
apply to
all methods in a way consistent with the criterion's intent and what people
reasonably expect from it, which often means distinguishing their favorite
method from Plurality?

Sure, there are some criteria that one would expect Plurality to meet. 
Consistency, Majority Favorite, etc. The mere fact that Plurality meets 
those suggests, to people who consider Plurality the worst, that much more 
is needed. Copeland and Margins meet the Smith Criterion (mine or Blake's). 
Smith is easy to meet. We can ask for more than that.

That doesn't mean that Consistency, Majority Favorite, and Smith aren't 
important or useful at all--only that much more is needed. If Plurality 
meets Consistency and Majority Favorite, then let's look for what it is that 
makes the best methods different from Plurality...because it isn't 
Consistency or Majority Favorite. And it isn't Blake's Condorcet's 
Criterion, Blake's Smith Criterion, or Kevin's Minimal Defense, because 
Plurality meets those too.

And he accomplished that goal...by posting my Smith Criterion. Maybe James 
knew that, for
universal applicability, it would be good to use my Smith Criterion instead 
of
Blake's.

James had said:

>the question
>is what is the most appropriate criterion to be identified as "majority
>rule". When we say that a given method is a majority rule method, what
>should we mean by this?

Kevin says:

I think the most intuitive is Steve Eppley's criterion that Markus quoted.
When v(i,j)>50% and there is no beatpath of strength >50% from J to I, then
J mustn't win. Basically, when more than half prefer I to J, in the "normal"
case you mustn't elect J.

I comment:

Very good. Rank methods that meet Steve's Beatpath Criterion (BC) meet all 
four
of my majority defensive strategy criteria.

Here's another way to state BC:

If a majority vote X over Y, and that defeat isn't in a cycle of majority 
pairwise
defeats, then Y shouldn't win.

[end of alternative BC definition]

But consider this example:

AB66, BC51, CA51

The Beatpath Criterion that you quoted doesn't say that B shouldn't win. It
nullifies the AB66 defeat because there's a majority beatpath from B to A.

But why should just any majority beatpath nullify a majority defeat? 
Shouldn't
it have to be a beatpath consisting of defeats at least as large?

I suggest that a defeat isn't nullified unless it's in a cycle with defeats 
that
are all at least as large as it is.

And, what do you know, it happens that I've recently posted a definition of
majority rule that takes that into account!

You may have missed it, and so I'll repeat it:

Majority rule definition:

A majority pairwise vote (MPV) is an instance of a majority of the voters 
voting
one candidate over another.

An MPV is nullified if it's in a cycle of MPVs,  and all of the other MPVs 
in that cycle are at
least as strong as it is.

Majority rule is violated if we elect someone who has an un-nullified MPV
against him.

[end of majority rule definition]

I haven't been calling that a criterion, and have only been using it in a
definition of offensive strategy, a definition that isn't important to me, 
and
which I'm not really satisified with.

But, just for completeness, let me define a criterion named after BC:

Strong BC (SBC):

Never elect a candidate in violation of majority rule (as I define it).

[end of SBC definition]

Because it's now more difficult to nullify an MPV, then more candidates can 
be
barred from winning, making an increased demand on the method, and making 
SBC
stronger than BC.

At least at first glance, BeatpathWinner, SD, and SSD meet SBC.

Because BC compliance implies SFC, GSFC, WDSC, & SDSC compliance, I use BC 
to
show that certain rank methods pass those 4 majority defensive strategy
criteria. I don't use BC for directly evaluating methods. And so if BC were
applicable only to rank methods, that would be ok. But BC is applicable to 
all
methods. Yes Plurality passes BC & SBC, but that's ok, because Plurality 
does
well by voted majority. For instance, Plurality passes Majority Favorite. , 
also
known as Majority, or Majority Winner.

I don't use SBC for anything. It's a conversation-piece.

BC or SBC versions defined in terms of preference would say something that 
the
votes-only versions don't say. For completeness I'll define such criteria 
here:

Preference Beatpath Criterion (PBC):

A majority pairwise preference (MPP) is an instance of a majority preferring 
one
candidate to another.

If no one falsifies a preference, and if a majority prefer X to Y and vote
sincerely, and if X's MPP against Y isn't in a cycle of MPPs, then Y 
shouldn't
win.

[end of PBC definition]

Sure, it could have stipulated that everyone vote sincerely, but there's no 
need
to require the voters outside that majority to vote preferences that 
sincerity
requires them to vote.

Preference Strong Beatpath Criterion (PSBC):

If no one falsifies a preference, and if a majority prefer X to Y, and vote
sincerely, and if X's MPP against Y isn't in a cycle of MPPs  in which all 
the
other MPPs are at least as strong as it is, then Y shouldn't win.

[end of PSBC definition]

Rank methods that meet BC or SBC meet PBC or PSBC respectively.

SFC and GSFC are really PBC, except that, where PBC is more general, SFC & 
GSFC
are about situations that make PBC's premise possible. l feel that that 
makes
SFC & GSFC more useful than PBC or PSBC.

I've here defined some criteria that I don't use.

By the way, I probably will change my offensive strategy definition to 
strategy
which, either in fact, or given the assumptions under which it is used,  
makes
defensive strategy necessary for someone else, and which would succeed if
defensive strategy isn't employed..

That's largely due to application problems of the current offensive strategy
definition that I've been using. But the definition of offensive strategy 
isn't
at all important to my criteria and their system of supporting definitions.

Anyway, that's consistent with the fact that the importance of what I call
offensive strategy is that it makes defensive strategy necessary. So why not
define it in that way.

Kevin continued:

If this criterion is too strong (although I doubt you think so), then I'd
suggest Minimal Defense: When a majority rank X>Y and Y over no one, then
Y mustn't win.

I comment:

Is that Steve's Minimal Defense, or is it Kevin's Minimal Defense?

If that's Minimal Defense, then Plurality meets Minimal Defense. You might 
want
to try SDSC instead, if you want a criterion that applies to Plurality in 
the
way that you would expect, a criterion that Plurality fails for the reason 
that
one would expect Plurality to fail. A criterion that tells a reason why we'd 
like something other than Plurality, and looks for that quality in methods.

There's some vagueness about what Kevin means by "rank". Say "rank X over Y" 
means "vote X over Y on a ranked ballot". In that case Kevin's Minimal 
Defense can't test Plurality, because there's no Plurality example that 
meets that criterion's premise.

Then, in addition to that definition of the verb "rank", there should be an 
explicit statement that the criterion applies only to rank methods. 
Otherwise Plurality passes because no failure example can be foiund. Paul's 
"A pork-chop passes" loophole.

Or say "rank Xover Y" means "vote X over Y" (but, if so, then why not say 
it?). Then Plurality meets the criterion for a different reason. For a 
majority to vote X over Y requires that a majority give their Plurailty vote 
to X. X wins. That means that Y doeesn't win, and Kevin's criterion's 
requirement is complied with. As required by the premise, those people are 
also voting Y equal last.

Kevin, you've held up your modification of Steve's Minimal Defense as an 
example of how my criteria should have been written, votes-only. But I don't 
want the faults that your Minimal Defense has.

Kevin says:

In my mind the problem is IRV's failure of Minimal Defense slash SDSC:

I comment:

Kevin is implying that Kevin's version of Minimal Defense is the same as 
SDSC. It is not.Try them both on Plurality.

James had said:

>	I suggest that narrower definitions, such as the one that Mike has
>formulated, are too narrow, in that it is necessary to choose one of
>several viable defeat strength definitions.

Kevin says:

I also think Mike's definition is too narrow

I comment:

I should be flattered, because James and Kevin want to attribute to me the 
nearly universally-accepted definition of majority, in the context of 
elections.

Kevin continues:

..., but because it doesn't seem
to allow other methods (i.e. non-pairwise) to be used.

I comment:

What? How does my definition of majority prevent non-pairwise methods from 
being used?

I assume that Kevin is referring to my definition of majorilty rule, based 
on majority pairwise votes.

It isn't quite clear why Kevin thinks that that definition of majority rule 
prevents Plurality or Approval from being used. Does Plurality violate 
majorilty rule, by my definition? No. If a majority vote X over Y, in 
Plurality, then there is an MPV for X against Y, it certainly isn't in a 
cycle of MPVs, since no majority is voting Y over anyone, and no majority is 
voting anyone over X. That majority must be voting for X. X wins. Y loses. 
Majority rule is not violated by Plurality. That isn't surprising, since 
Pluralitly has long been known to meet the well-known Majority Favorite 
Criterion (also called the Majority Criterion or Majority Winner Criterion).

Likewise all of my criteria and definitions apply to the non-rank methods.

For instance, does Plurality meet PSBC, the Preference Strong Beatpath 
Criterion? No. Here's a failure example:

A majority prefer X to Y. Half have Z as their favorite.  Half have X as 
their favorite. To vote sincerely, by my definition, they must not falsify a 
preference, which means, among other things, that they must not vote anyone 
over their favorite. That means that they must not vote for anyone other 
than their favorite. Being voters, however, they vote, and so they vote for 
their favorite.

In compliance with PSBC's premise, they prefer X to Y and have voted 
sincerely, for their favorites. Say all of the remaining voters like Y best, 
and that the majority who prefer X to Y is a 60%. 40% like Y best.  Y gets 
40% of the votes. X gets 30%, and Z gets 30%. Y wins by Plurrality. That, 
then, is a failure example for Plurality, with  PBSC. Plurality fails PBSC. 
PBSC applies to all methods, including non-rank methods, as does my majority 
rule definition, and as do all my criteria.

A very similar demonstration shows that Pluraltiy likewise fails SFC and 
GSFC. And WDSC and SDSC.

So, my majority rule definition, and my criterion definitions, including 
that of PSBC, SFC, GSFC, WDSC, and SDSC, apply to all methods, including 
non-ranked methods.

So my majority rule definition applies to non-ranked methods, contrary to 
what Kevin said. Or, if Kevin was referring to my other criteria, they too 
apply to non-rank methods. In the case of SFC and GSFC, the demonstration is 
very similar to the demonstration for PSBC.

Kevin says:

I don't think it
would be a crime against majority rule for a method to rule out the
candidates who must not win, and then pick the winner by e.g. the first
preferences.

I comment:

I have no idea what that means. What Kevin says above wouldn't be a crime 
against majority rule, but it could be a violation of majority rule, 
depending on what Kevin's method is. More details would be needed before one 
could say whether that method violates majorilty rule.

Kevin says:

An advantage of using Minimal Defense rather than Smith is that you don't
have to use terms like "innermost," "minimal," "pairwise," or even "beats."

I comment:

I can't speak for Kevin's definition of the Smith set, but my definition of 
the Smith set doesn't use the word "innermost" or "minimal". And though it 
uses "beats", I define "beats".
X beats Y if more voters vote X over Y than vote Y over X.

But Kevin is comparing "Smiith" to Minimal Defense, which suggests that he's 
referring to the Smith Criterion. Again, I can't speak for Kevin's Smith 
Criterion definition, but mine doesn't use the words "innermost", "minimal", 
or "beats", or probably even "pairwise".

It says that X is publicly preferred to Y if more voters prefer X to Y than 
prefer Y to X.

The sincere Smith set is the smallest set of candidates such that every 
candidate in that set is publicly preferred to every candidate outside that 
set.

The Smith Criterion says that if everyone votes sincerely then the  winner 
must come from the sincere Smith set.

The word "innermost" is used in my definition of the Schwartz set. In that 
definition, I clearly state what an innermost unbeaten set is. Keven also 
forgot to tell us why it's a disadvantage for a Schwartz set definition to 
use "innermost".

Though Kevin doesn't say it there, Kevin also thinks that it's an advantage 
for a criterion to not use the word "prefer". That's one reason why Kevin 
pointed out Steve's criteria that are counterparts to my majority defensive 
strategy criteria. What Kevin perhaps doesn't know is that, of those 3 
criteria of Steve's that are counterparts to the majority defensive strategy 
criteria, 2 of those 3 use the word "prefer" in their defintion, just as do 
the majorilty defensive strategy criteria.

Steve's Minimal Defense may be on of those. Someone might want to look up 
Steve's definition of Minimal Defense, to find out if Kevin has written his 
own Minimal Defense.

Kevin says:

I begin to think that Minimal Defense slash SDSC would be more popular if
they had been named without reference to defensive strategy. Those criteria
are very useful even if no offensive strategy is possible.

I comment:

"Minimal Defense slash SDSC" implies that Kevin's Minimal Defense is SDSC. 
It most certainly is not.

But Kevin is right when he says that the majority defensive strategy 
criteria are very useful even if no offensive strategy is used. That's 
because they're about defensive strategy, not offensive strategy. Plurality 
and IRV, for instance, create a drastsic defensive strategy need without any 
offensive strategy being used. So much so that Plurality and IRV fail all 
four of the majority defensive strategy criteria.

But I don't agree that it would be good to name the majorilty defensive 
strategy criteria without reference to defensive strategy--because defensive 
strategy is what they are about.

Mike Ossipoff


Kevin Venzke

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