[EM] "Mutual Plurality" criterion suggestion

[Chris Benham]

The Majority for Solid Coalitions (aka Mutual Majority) criterion
reflects a strong standard of mine, but I'm not happy
that the concept is vulnerable to irrelevant ballots. In other words in
some election the criterion might insist that A must
win but then if we add a handful of ballots that vote for no-one the
criterion says that it's now ok for A to not win.

To address this I've come up with a somewhat stronger and more generally
useful criterion that implies compliance
with Majority for Solid Coalitions.

*If there exists one or more sets S of at least one candidate that is
voted above (together in any order) above all other
candidates on a greater number of ballots than any outside-S candidate
is voted above any member of S (in any positions)
then the winner must come from the smallest S.*

In other words if a candidate or set S of candidates need only the
ballots on which they are voted above all others to win
all their pairwise contests versus all the other (outside-S) candidates,
then that is good enough.

The brief and I hope adequate name I suggest is the "Mutual Plurality"
criterion.

As I earlier implied, everything that meets this also meets Majority for
Solid Coalitions but vice versa isn't the case.

49: A>B
41: B
10: C

Here my suggested criterion says A must win, but Majority for Solid
Coalitions says nothing but will agree if we remove
two or more of the C ballots.

Bucklin (and some similar methods) meet Majority for Solid Coalitions
but elect B.

Chris Benham



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[Greg Dennis]

Chris, I think this is an excellent improvement over mutual majority. My
only suggestions are around how it's phrased and named.

I'd probably drop the "smallest S" when describing it, since it's implied,
i.e. just "If there exists a set S ... then the winner must come from S."
Mutual majority has the same "smallest" implication but I think is usually
omitted from descriptions.

I'm concerned that the name "mutual plurality" makes it sound like a weaker
condition than "mutual majority." Maybe something like "Undefeated
coalition," not sure.

It's clear to me that the Smith set is always a subset of every "mutual
plurality" set, right?

The Majority for Solid Coalitions (aka Mutual Majority) criterion reflects
a strong standard of mine, but I'm not happy
that the concept is vulnerable to irrelevant ballots. In other words in
some election the criterion might insist that A must
win but then if we add a handful of ballots that vote for no-one the
criterion says that it's now ok for A to not win.
To address this I've come up with a somewhat stronger and more generally
useful criterion that implies compliance
with Majority for Solid Coalitions.
*If there exists one or more sets S of at least one candidate that is
voted above (together in any order) above all other
candidates on a greater number of ballots than any outside-S candidate is
voted above any member of S (in any positions)
then the winner must come from the smallest S.*
In other words if a candidate or set S of candidates need only the ballots
on which they are voted above all others to win
all their pairwise contests versus all the other (outside-S) candidates,
then that is good enough.
The brief and I hope adequate name I suggest is the "Mutual Plurality"
criterion.
As I earlier implied, everything that meets this also meets Majority for
Solid Coalitions but vice versa isn't the case.
49: A>B
41: B
10: C
Here my suggested criterion says A must win, but Majority for Solid
Coalitions says nothing but will agree if we remove
two or more of the C ballots.
Bucklin (and some similar methods) meet Majority for Solid Coalitions but
elect B.
Chris Benham
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[Chris Benham]

Greg,

I'm glad you like my idea.

I'm sure the definition could be polished and/or made more succinct. At
the moment I don't have a strong view on your suggestion
on how that should be done. In general I don't mind the odd redundancy
if it makes it more likely that more people will understand it.

I won't be dying in a ditch for the "Mutual Plurality" name, but I think
your "Undefeated coalition" suggestion is a bit misleading
and vague.

It was conceived as an irrelevant-ballot independent version of Mutual
Majority, so I suppose it could be called "Irrelevant-Ballot
Independent Mutual Majority".  Another possible clumsy name: "Mutual
Dominant Relative Majority"?

Post by Greg Dennis
It's clear to me that the Smith set is always a subset of every
"mutual plurality" set, right?

Yes, but of course there isn't always a "Mutual Plurality" set (or
subset) while there is always a Smith set.

Chris Benham

Post by Greg Dennis
Chris, I think this is an excellent improvement over mutual majority.
My only suggestions are around how it's phrased and named.
I'd probably drop the "smallest S"  when describing it, since it's
implied, i.e. just "If there exists a set S ... then the winner must
come from S." Mutual majority has the same "smallest" implication but
I think is usually omitted from descriptions.
I'm concerned that the name "mutual plurality" makes it sound like a
weaker condition than "mutual majority." Maybe something like
"Undefeated coalition," not sure.
It's clear to me that the Smith set is always a subset of every
"mutual plurality" set, right?
The Majority for Solid Coalitions (aka Mutual Majority) criterion
reflects a strong standard of mine, but I'm not happy
that the concept is vulnerable to irrelevant ballots. In other
words in some election the criterion might insist that A must
win but then if we add a handful of ballots that vote for no-one
the criterion says that it's now ok for A to not win.
To address this I've come up with a somewhat stronger and more
generally useful criterion that implies compliance
with Majority for Solid Coalitions.
*If there exists one or more sets S of at least one candidate that
is voted above (together in any order) above all other
candidates on a greater number of ballots than any outside-S
candidate is voted above any member of S (in any positions)
then the winner must come from the smallest S.*
In other words if a candidate or set S of candidates need only the
ballots on which they are voted above all others to win
all their pairwise contests versus all the other (outside-S)
candidates, then that is good enough.
The brief and I hope adequate name I suggest is the "Mutual
Plurality" criterion.
As I earlier implied, everything that meets this also meets
Majority for Solid Coalitions but vice versa isn't the case.
49: A>B
41: B
10: C
Here my suggested criterion says A must win, but Majority for
Solid Coalitions says nothing but will agree if we remove
two or more of the C ballots.
Bucklin (and some similar methods) meet Majority for Solid
Coalitions but elect B.
Chris Benham
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[Kristofer Munsterhjelm]

Post by Chris Benham
Greg,
I'm glad you like my idea.
I'm sure the definition could be polished and/or made more succinct. At
the moment I don't have a strong view on your suggestion
on how that should be done. In general I don't mind the odd redundancy
if it makes it more likely that more people will understand it.
I won't be dying in a ditch for the "Mutual Plurality" name, but I think
your "Undefeated coalition" suggestion is a bit misleading
and vague.
It was conceived as an irrelevant-ballot independent version of Mutual
Majority, so I suppose it could be called "Irrelevant-Ballot
Independent Mutual Majority".  Another possible clumsy name: "Mutual
Dominant Relative Majority"?

Post by Greg Dennis
It's clear to me that the Smith set is always a subset of every
"mutual plurality" set, right?

Yes, but of course there isn't always a "Mutual Plurality" set (or
subset) while there is always a Smith set.

Isn't the set of all candidates always a Mutual Plurality set, in a
vacuously true sense?
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[Chris Benham]

Post by Chris Benham
*If there exists one or more sets S of at least one candidate that is
voted above (together in any order) all other
candidates on a greater number of ballots than any outside-S candidate
is voted above any member of S (in any positions)
then the winner must come from the smallest S.*
Isn't the set of all candidates always a Mutual Plurality set, in a
vacuously true sense?

I meant to imply that if there aren't any "other candidates" then the
"set" doesn't exist.  Maybe:

*If there exists one or more subsets S of at least one candidate that is
voted above (together in any order)  all of the (one or more) outside-S
candidates on a greater number of ballots than any outside-S candidate
is voted above any member of S (in any positions) then the winner
must come from the smallest S.*

But as I initially defined it, then I suppose yes. But that doesn't much
matter. All methods might then elect from at least one Mutual Plurality
set, but only those who elect from the smallest one meet the criterion.

Chris Benham

Post by Chris Benham

Post by Chris Benham
Greg,
I'm glad you like my idea.
I'm sure the definition could be polished and/or made more succinct.
At the moment I don't have a strong view on your suggestion
on how that should be done. In general I don't mind the odd
redundancy if it makes it more likely that more people will
understand it.
I won't be dying in a ditch for the "Mutual Plurality" name, but I
think your "Undefeated coalition" suggestion is a bit misleading
and vague.
It was conceived as an irrelevant-ballot independent version of
Mutual Majority, so I suppose it could be called "Irrelevant-Ballot
Independent Mutual Majority".  Another possible clumsy name: "Mutual
Dominant Relative Majority"?

Post by Greg Dennis
It's clear to me that the Smith set is always a subset of every
"mutual plurality" set, right?

Yes, but of course there isn't always a "Mutual Plurality" set (or
subset) while there is always a Smith set.

Isn't the set of all candidates always a Mutual Plurality set, in a
vacuously true sense?

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[Kristofer Munsterhjelm]

Post by Chris Benham

Post by Chris Benham
*If there exists one or more sets S of at least one candidate that is
voted above (together in any order) all other
candidates on a greater number of ballots than any outside-S candidate
is voted above any member of S (in any positions)
then the winner must come from the smallest S.*
Isn't the set of all candidates always a Mutual Plurality set, in a
vacuously true sense?

I meant to imply that if there aren't any "other candidates" then the
*If there exists one or more subsets S of at least one candidate that is
voted above (together in any order)  all of the (one or more) outside-S
candidates on a greater number of ballots than any outside-S candidate
is voted above any member of S (in any positions) then the winner
must come from the smallest S.*
But as I initially defined it, then I suppose yes. But that doesn't much
matter. All methods might then elect from at least one Mutual Plurality
set, but only those who elect from the smallest one meet the criterion.

I think the original definition works, as the same thing happens for
mutual majority. Every method elects from some solid coalition that has
greater than majority support (namely, the coalition of all candidates),
but the method only passes the mutual majority criterion if it elects
from the smallest such set. In some situations, that smallest set *is*
the set of all candidates, which means there's no special case logic
needed for such a case; a method that passes mutual majority in the
"proper" cases is then free to choose any candidate to be elected
without violating the criterion.

I agree, though. It doesn't much matter, beyond in an elegance of
definition sense.
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[Greg Dennis]

Chris, do you have a precise definition of "irrelevant ballot"? Just a
ballot that expresses indifference between the smallest mutual majority set?

Post by Kristofer Munsterhjelm

Post by Chris Benham

Post by Chris Benham
*If there exists one or more sets S of at least one candidate that is
voted above (together in any order) all other
candidates on a greater number of ballots than any outside-S candidate
is voted above any member of S (in any positions)
then the winner must come from the smallest S.*
Isn't the set of all candidates always a Mutual Plurality set, in a
vacuously true sense?

I meant to imply that if there aren't any "other candidates" then the
*If there exists one or more subsets S of at least one candidate that is
voted above (together in any order) all of the (one or more) outside-S
candidates on a greater number of ballots than any outside-S candidate
is voted above any member of S (in any positions) then the winner
must come from the smallest S.*
But as I initially defined it, then I suppose yes. But that doesn't much
matter. All methods might then elect from at least one Mutual Plurality
set, but only those who elect from the smallest one meet the criterion.

I think the original definition works, as the same thing happens for
mutual majority. Every method elects from some solid coalition that has
greater than majority support (namely, the coalition of all candidates),
but the method only passes the mutual majority criterion if it elects
from the smallest such set. In some situations, that smallest set *is*
the set of all candidates, which means there's no special case logic
needed for such a case; a method that passes mutual majority in the
"proper" cases is then free to choose any candidate to be elected
without violating the criterion.
I agree, though. It doesn't much matter, beyond in an elegance of
definition sense.

[Chris Benham]

Greg,

I did have, but that wasn't it.  For the purpose of applying the test to
methods, I think I defined it thus:

*If there is some losing candidate X  with fewer above-bottom votes than
any other candidate, and all the ballots either
vote X below all other candidates (or ignore/truncate X) or vote X above
all other candidates and all the other candidates equal bottom
(or ignored/truncated), then removing any number of the X-supporting
ballots can't change the result.*

Maybe a better version is possible. My idea is that those ballots  
contain no information about any of the remotely competitive
candidates, but would normally (in jurisdictions that allow truncation
or voting candidates equal-bottom)) be counted as valid, which
might not be the case if the criterion just talked about "blank" ballots.

Chris Benham

Post by Greg Dennis
Chris, do you have a precise definition of "irrelevant ballot"? Just a
ballot that expresses indifference between the smallest mutual
majority set?

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[Greg Dennis]

I've been thinking about this property more recently, and I'd like to offer
what I believe is an equivalent formulation of it. I'm not saying this is
_the_ way it should be expressed, but this formulation helps me see the
importance of the property:

Consider a set S of candidates such that the following is true. For every
candidate C outside of S, exclude all the ballots that express indifference
between C and all the candidates in S (i.e. C+S all equally ranked, perhaps
left off the ballot altogether), more than half of the remaining ballots
(aka the "relevant ballots"), prefer every member of S to C. If there
exists an S, the winner must come from S.

Do you agree this is equivalent or have I missed something? If so, I like
how this formulation reveals the "majority" threshold lurking inside the
original formulation, and to me makes the name Mutual Relevant Majority
tempting.

Greg,
I did have, but that wasn't it. For the purpose of applying the test to
*If there is some losing candidate X with fewer above-bottom votes than
any other candidate, and all the ballots either
vote X below all other candidates (or ignore/truncate X) or vote X above
all other candidates and all the other candidates equal bottom
(or ignored/truncated), then removing any number of the X-supporting
ballots can't change the result.*
Maybe a better version is possible. My idea is that those ballots
contain no information about any of the remotely competitive
candidates, but would normally (in jurisdictions that allow truncation or
voting candidates equal-bottom)) be counted as valid, which
might not be the case if the criterion just talked about "blank" ballots.
Chris Benham

Post by Greg Dennis
Chris, do you have a precise definition of "irrelevant ballot"? Just a
ballot that expresses indifference between the smallest mutual majority set?

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[Chris Benham]

Greg,

Sorry to be so tardy in replying.

Your idea seems fine to me.

Chris Benham

Post by Greg Dennis
I've been thinking about this property more recently, and I'd like to
offer what I believe is an equivalent formulation of it. I'm not
saying this is _the_ way it should be expressed, but this formulation
Consider a set S of candidates such that the following is true. For
every candidate C outside of S, exclude all the ballots that express
indifference between C and all the candidates in S (i.e. C+S all
equally ranked, perhaps left off the ballot altogether), more than
half of the remaining ballots (aka the "relevant ballots"), prefer
every member of S to C. If there exists an S, the winner must come from S.
Do you agree this is equivalent or have I missed something? If so, I
like how this formulation reveals the "majority" threshold lurking
inside the original formulation, and to me makes the name Mutual
Relevant Majority tempting.
Greg,
I did have, but that wasn't it.  For the purpose of applying the
*If there is some losing candidate X  with fewer above-bottom
votes than any other candidate, and all the ballots either
vote X below all other candidates (or ignore/truncate X) or vote X
above all other candidates and all the other candidates equal bottom
(or ignored/truncated), then removing any number of the
X-supporting ballots can't change the result.*
Maybe a better version is possible. My idea is that those
ballots   contain no information about any of the remotely competitive
candidates, but would normally (in jurisdictions that allow
truncation or voting candidates equal-bottom)) be counted as
valid, which
might not be the case if the criterion just talked about "blank" ballots.
Chris Benham
Chris, do you have a precise definition of "irrelevant
ballot"? Just a ballot that expresses indifference between the
smallest mutual majority set?
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Voter Choice Massachusetts
p :: 617.863.0746 <tel:[phone]>
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