Some thoughts on Condorcet and Burial
Kristofer,
I'm sorry for being a bit tardy in replying.
What is your definition of UMDT?
My definition of MDT is that if a set S of candidates are voted together
above all outside-S candidates on more than one third
of the ballots, and all the members of S pairwise-beat all the
outside-S candidates, then the winner must be a member of S.
My definition of UMDT is that if the winner T is a member of S, then it
must not be possible to make some outside-S candidate X
the winner just by altering some ballots that already vote X above S.
Perhaps to be a bit more strict we can replace "altering" with ' further
down-ranking S on'.
1: A>C>B
1: B>C>A
1: C>B>A
C is the CW, then
1: A>C>B
1: B>A>C <- burying C under A
1: C>B>A
is a perfect tie and every candidate has equal chance of winning.
This example clarified for me that MDT (and UMDT) refers to more than
a third (rather than exactly a third).
It is clear to me that MDT was meant to be analogous with Mutual
Majority (rather than "Mutual Half").
(A problem is that in English there is no word that means "more than a
third".)
So in neither case can this lead to nonmonotonicity if the base method
is monotone.
Sounds about right?
Yes.
Chris B.
Kristofer Munsterhjelmkm_elmet at t-online.de
mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Some%20thoughts%20on%20Condorcet%20and%20Burial&In-Reply-To=%3C59c996f1-4936-5cd3-7026-3b04198d192e%40t-online.de%3E
/Wed Nov 8 05:00:27 PST 2023/
On 11/8/23 05:07, C.Benham wrote:
/In my last EM post I included Smith//DAC in a list of Condorcet methods />/"that meets mono-raise". />//>/A knowledgeable correspondent has cast doubt on this claim, and I admit />/that I can't prove that it does. />//>/But I am fairly sure that any failure example needs there to me more />/than three candidates in the top cycle, and if I'm right about that />/then I'm not concerned enough to scratch it (at least) as a quite />/burial-resistant curiosity that is far less absurd than "elect the />/member of the Smith set that is voted the least desirable". /
I suspect that you're right, and this holds for monotone methods in
general. Here's the reasoning:
Suppose we have an ABCA cycle in Smith//X, where X is some monotone
method. Our strategy to show nonmonotonicity is to shrink the Smith set
to make the winner change, since raising A can never grow the Smith set.
(Note that there may be more candidates outside the Smith set, but
they're all irrelevant for our purposes.)
Suppose that we try to kick B off the Smith set. But this is impossible
since raising A can never alter B>C or C>B, and it can only increase
A>B. Since we already have A>B, raising A can't kick B off the set.
Okay, so suppose that we try to kick C off the Smith set since C>A. But
if we reverse this, then we have both A>B and A>C, which would make A
the Condorcet winner. Hence kicking C off the Smith set won't work.
So in neither case can this lead to nonmonotonicity if the base method
is monotone.
Sounds about right?
/So this method meets both Double Defeat and Unburiable Mutual Dominant />/Third. I doubt that an acceptable Condorcet method />/can get more Burial resistant than that. /
What is your definition of UMDT? I had some trouble trying to generalize
DMTCBR to an actual DMT set criterion, so it would be interesting to know.
The problem I encountered was that, if the method also passes DMT
(without which DMT burial resistance would be kind of strange), then the
pre-burial winner W is part of the innermost DMT set. And then when
voters who prefer X to W downrank W, they often change the relative
order of candidates within the DMT set on their ballots.
E.g. if A, B, C are also part of the DMT set, then X>W>A>B>C>D voters
changing their votes to X>A>B>C>D>W change their pairwise preference
between, for instance, A and W. Even though they prefer a non-DMT
candidate to the DMT set, their burial of the winner changes the
relative order within the set.
If we require that such alterations - burials within the DMT set -
should not matter, then the Smith-IRV hybrids fail since they're not
absolutely unburiable, e.g.:
1: A>C>B
1: B>C>A
1: C>B>A
C is the CW, then
1: A>C>B
1: B>A>C <- burying C under A
1: C>B>A
is a perfect tie and every candidate has equal chance of winning.
Perhaps something like "the new winner should not be preferred by the
buriers to the old winner". Or "should not be someone who was outside
the DMT set before the burial started"? But I'm not sure what properly
captures the strong type of resistance that the Smith-IRV hybrids pass
and that leads to generally low strategic vulnerability.
-km
Kristofer,
I'm sorry for being a bit tardy in replying.
What is your definition of UMDT?
My definition of MDT is that if a set S of candidates are voted together
above all outside-S candidates on more than one third
of the ballots, and all the members of S pairwise-beat all the
outside-S candidates, then the winner must be a member of S.
My definition of UMDT is that if the winner T is a member of S, then it
must not be possible to make some outside-S candidate X
the winner just by altering some ballots that already vote X above T.
Perhaps to be a bit more strict we can replace "altering" with ' further
down-ranking T on'.
1: A>C>B
1: B>C>A
1: C>B>A
C is the CW, then
1: A>C>B
1: B>A>C <- burying C under A
1: C>B>A
is a perfect tie and every candidate has equal chance of winning.
This example clarified for me that MDT (and UMDT) refers to more than
a third (rather than exactly a third).
It is clear to me that MDT was meant to be analogous with Mutual
Majority (rather than "Mutual Half").
(A problem is that in English there is no word that means "more than a
third".)
So in neither case can this lead to nonmonotonicity if the base method
is monotone.
Sounds about right?
Yes.
Chris B.
Kristofer Munsterhjelmkm_elmet at t-online.de
mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Some%20thoughts%20on%20Condorcet%20and%20Burial&In-Reply-To=%3C59c996f1-4936-5cd3-7026-3b04198d192e%40t-online.de%3E
/Wed Nov 8 05:00:27 PST 2023/
On 11/8/23 05:07, C.Benham wrote:
/In my last EM post I included Smith//DAC in a list of Condorcet methods />/"that meets mono-raise". />//>/A knowledgeable correspondent has cast doubt on this claim, and I admit />/that I can't prove that it does. />//>/But I am fairly sure that any failure example needs there to me more />/than three candidates in the top cycle, and if I'm right about that />/then I'm not concerned enough to scratch it (at least) as a quite />/burial-resistant curiosity that is far less absurd than "elect the />/member of the Smith set that is voted the least desirable". /
I suspect that you're right, and this holds for monotone methods in
general. Here's the reasoning:
Suppose we have an ABCA cycle in Smith//X, where X is some monotone
method. Our strategy to show nonmonotonicity is to shrink the Smith set
to make the winner change, since raising A can never grow the Smith set.
(Note that there may be more candidates outside the Smith set, but
they're all irrelevant for our purposes.)
Suppose that we try to kick B off the Smith set. But this is impossible
since raising A can never alter B>C or C>B, and it can only increase
A>B. Since we already have A>B, raising A can't kick B off the set.
Okay, so suppose that we try to kick C off the Smith set since C>A. But
if we reverse this, then we have both A>B and A>C, which would make A
the Condorcet winner. Hence kicking C off the Smith set won't work.
So in neither case can this lead to nonmonotonicity if the base method
is monotone.
Sounds about right?
/So this method meets both Double Defeat and Unburiable Mutual Dominant />/Third. I doubt that an acceptable Condorcet method />/can get more Burial resistant than that. /
What is your definition of UMDT? I had some trouble trying to generalize
DMTCBR to an actual DMT set criterion, so it would be interesting to know.
The problem I encountered was that, if the method also passes DMT
(without which DMT burial resistance would be kind of strange), then the
pre-burial winner W is part of the innermost DMT set. And then when
voters who prefer X to W downrank W, they often change the relative
order of candidates within the DMT set on their ballots.
E.g. if A, B, C are also part of the DMT set, then X>W>A>B>C>D voters
changing their votes to X>A>B>C>D>W change their pairwise preference
between, for instance, A and W. Even though they prefer a non-DMT
candidate to the DMT set, their burial of the winner changes the
relative order within the set.
If we require that such alterations - burials within the DMT set -
should not matter, then the Smith-IRV hybrids fail since they're not
absolutely unburiable, e.g.:
1: A>C>B
1: B>C>A
1: C>B>A
C is the CW, then
1: A>C>B
1: B>A>C <- burying C under A
1: C>B>A
is a perfect tie and every candidate has equal chance of winning.
Perhaps something like "the new winner should not be preferred by the
buriers to the old winner". Or "should not be someone who was outside
the DMT set before the burial started"? But I'm not sure what properly
captures the strong type of resistance that the Smith-IRV hybrids pass
and that leads to generally low strategic vulnerability.
-km
On 11/12/23 01:40, C.Benham wrote:
Kristofer,
I'm sorry for being a bit tardy in replying.
No problem; I haven't been the quickest lately either.
What is your definition of UMDT?
My definition of MDT is that if a set S of candidates are voted together
above all outside-S candidates on more than one third
of the ballots, and all the members of S pairwise-beat all the
outside-S candidates, then the winner must be a member of S.
My definition of UMDT is that if the winner T is a member of S, then it
must not be possible to make some outside-S candidate X
the winner just by altering some ballots that already vote X above T.
Perhaps to be a bit more strict we can replace "altering" with ' further
down-ranking T on'.
1: A>C>B
1: B>C>A
1: C>B>A
C is the CW, then
1: A>C>B
1: B>A>C <- burying C under A
1: C>B>A
is a perfect tie and every candidate has equal chance of winning.
This example clarified for me that MDT (and UMDT) refers to more than
a third (rather than exactly a third).
It is clear to me that MDT was meant to be analogous with Mutual
Majority (rather than "Mutual Half").
(A problem is that in English there is no word that means "more than a
third".)
Yes. What I had in mind here was {B, C} as the DMT set since it has 2/3
support. C wins. Then a voter who prefers B to C buries C under A and
B's chance of winning increases from zero to 1/3.
So clearly we can't protect against burial changing the winner from
someone in the DMT set to someone else. My thought was that there may
exist more subtle versions (with more voters) where the honest vote is
something like X>B>C>A, and then when burying to X>B>A>C, that changes
the winner from C to B even though that voter wanted X to win.
Your criterion handles that pretty well because it doesn't fail the
method just because the buriers got someone they preferred. The winner
has to change to someone not initially in the DMT set.
Do IRV and the Smith-IRV hybrids pass the stronger criterion you
suggested, where it's impossible to make X the winner by any kind of
modification, not just burial, by voters who prefer X?
-km