Hi,
Guest
Re: [EM] Pairwise Median Rating
C
C.Benham
Tue, Jan 23, 2024 3:07 AM
Ted,
...you didn't comment on whether ballots with all Smith candidates
below top rating should have their ratings bumped up: i.e., D > E > A
blank > B (A and B in Smith) would be recounted as A > blank > B.
I don't think I left anything ambiguous.
Assuming in your example we are using say 5-slot ratings ballots then we
interpret it as a score ballot thus: D5, E4, A3, B0.
If A and B are in Smith then the average score of candidates in the
Smith set is 3+0/2 = 1.5. Only A is scored above 1.5 so only A is
approved.
I also noticed that there were cases where Smith//ASM(implicit) would
get different results (better, IMO) than Smith//Implicit-approval.
What does "ASM" stand for?
Chris
On 23/01/2024 10:56 am, Ted Stern wrote:
Chris:
Thanks for the clarifications, though you didn't comment on whether
ballots with all Smith candidates below top rating should have their
ratings bumped up: i.e., D > E > A > blank > B (A and B in Smith)
would be recounted as A > blank > B. I think this makes the most sense
as a voter whose favorites are eliminated would want to ensure that
their highest ranked Smith candidate is counted as approved.
In general I agree with your comments, though I think
Condorcet//Approval with all ranked ballots approved is probably not
optimal, and Approval Sorted Margins with explicit approval would be
too complex for a public proposal. I'd be happy with
Condorcet//Top-ratings as a public proposal.
Smith//Implicit-approval seems to perform well in a number of
situations, but not appreciably better enough to make it worth the
effort of trying to get people to accept something more complicated
than Condorcet/Top-ratings. I also noticed that there were cases where
Smith//ASM(implicit) would get different results (better, IMO) than
Smith//Implicit-approval.
On Fri, Jan 19, 2024 at 4:05 PM C.Benham cbenham@adam.com.au wrote:
How is the average calculated?
We interpret the ratings ballots as score ballots, giving zero
points for the bottom rating (which is default for unrated),
1 point for the next highest, 2 points for the next above that and
so on.
Then for any given ballot we add up the scores of the candidates
in the Smith set and divide that by the number of candidates
in the Smith set and interpret that ballot as approving those
Smith set candidates it scores higher than that average score.
That simulates the best approval strategy if the voters only know
which candidates are in the Smith set.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval?
Do you mean Approval Sorted Margins using ranking ballots with an
explicit approval cutoff?
Assuming yes, it uses a more expressive ballot, it is less
vulnerable to Defection strategy, and burial strategies are more
likely to have no effect rather than backfire.
In the method I proposed, omitting the ASM step and just electing
the candidate with the highest approval score (derived
as specified) would I concede make for a simpler method that is
nearly as good.
I worry a bit that with all methods that begin with eliminating or
disqualifying all candidates who aren't in the Smith set or
just "elect the CW if there is one", over time if there is never a
top cycle then the top-cycle resolution method could stop
being taken seriously.
An attractive feature of ASM is that it is a Condorcet method that
a lot of the time would work fine without anyone needing
to know if there is top cycle or not.
If the Approval order is A>B>C and A pairwise beats B and B
pairwise beats C no-one needs to enquire about the pairwise
result between A and C.
If we want something super simple to explain and sell, then
Condorcet//Top Ratings and Condorcet//Approval (voted above bottom)
are both not bad and much better than STAR.
Chris Benham
On 18/01/2024 10:13 am, Ted Stern wrote:
On Tue, Jan 16, 2024 at 7:27 AM C.Benham <cbenham@adam.com.au> wrote:
Ted,
3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
Why not simply drop all ballots that make no distinction
among members of the Smith set?
I believe it passes LNHelp.
Douglas Woodall showed some time ago that Condorcet and
LNHelp are incompatible. I can't find
his proof, but it says so here:
https://en.wikipedia.org/wiki/Later-no-help_criterion
TheCondorcet criterion
<https://en.wikipedia.org/wiki/Condorcet_criterion>is
incompatible with later-no-help.
From your post again:
It probably fails Participation ..
It has been known (for a longer time) that Condorcet and
Participation are incompatible.
So since we know for sure that your method meets Condorcet,
we also know that it doesn't meet
Later-no-Help or Participation.
Using a multi-slot ratings ballot for a Condorcet method of
similar complexity I like:
*Eliminate all candidates not in the Smith set.
Interpret each ballot as giving approval to those remaining
candidates they rate above average (mean
of the ratings given to Smith-set members).
Now, using these approvals, elect the Margins-Sorted Approval
winner.*
It seems to me that Smith//Implicit-Approval or
Smith//implicit-approval-sorted-margins would be affected by a
couple of factors:
* How is the average calculated? Do you normalize scores? In
other words, if a ballot has non-Smith candidates in the
first, say, three ranks, do you up-rank the Smith candidate
scores on that ballot by three? Also, if there are ranks
below the top that contain only non-Smith candidates, do you
collapse those ranks or leave the relative rank spacing on
the ballot between Smith candidates untouched?
* Approving Smith Candidates with scores above the mean has
similarities to Median Ratings. It would be more similar and
probably more stable to use the trimmed mean -- drop the top
and bottom 25% of scores. This would give you an average
score closer to the median.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval? I like ASM but fear it is probably too
complex for any advantage it gives you.
Has Smith//Median Rating been proposed before?
Not that I know of.
Chris Benham
*Ted Stern*dodecatheon at gmail.com
<mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
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------------------------------------------------------------------------
Continuing my search for a summable voting method that discourages burial
and defection, I've come across this hybrid of Condorcet and median ratings
that acts like Smith/Approval with an automatic approval cutoff. I'm
calling it Pairwise Median Rating (PMR), but it could also be described as
Smith//MR//Pairwise//MRScore:
1. Equal Ranking and ranking gap allowed (essentially a ratings method
with rank inferred). For purposes of this discussion, assume 6 slots (5
ranks above rejection).
2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
3. [Smith]
1. Compute the pairwise preference array
2. The winner is the candidate who defeats each other candidate
pairwise.
3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
4. [Median Rating]
1. Set the MR threshold to top rank.
2. While no Smith candidate has a majority of undropped ballots at or
above the threshold, set the threshold to the next lower rank,
until there
is no lower rank.
3. The winner is the single candidate that has a majority of
undropped ballots at or above the threshold.
5. [Pairwise]
1. Otherwise, if more than one candidate passes the threshold, look for
a pairwise beats-all candidate among candidates meeting the MR threshold.
(i.e. Condorcet on just the MR threshold set).
2. If there is one, you have a winner.
6. [MR Score]
1. Otherwise, the winner is the Smith set candidate with the largest
number of ballots at or above the Median Rating threshold (their MRscore).
This method is essentially Smith//Approval(explicit) with the approval
cutoff automatically inferred via median ratings
Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
considered optional, but by doing that, you ensure Immunity from Irrelevant
Ballots (IIB), aka the zero ballot problem that affects other Median Rating
/ Majority Judgment methods. In other words, the majority threshold is
unaffected by ballots that do not rank a viable candidate. It is possible
to do this summably if need be.
PMR either passes the Chicken Dilemma criterion without adjustment, or
there is a downranking strategy for defending against defection.
Consider the following examples from Chris Benham's post re MinLV(erw)
Sorted Margins (
http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
):
/* 46 A>B /> 44 B>C (sincere is B or B>A)
*>* 05 C>A
*>* 05 C>B
*>>* A>B 51-49, B>C 90-10, C>A 54-46.
*
With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
threshold is 2nd place, and A and B both pass the threshold. A defeats B,
so A is the winner and B's defection/burial fails.
/* 25 A>B /> 26 B>C
*>* 23 C>A
*>* 26 C
*>>* C>A 75-25, A>B 48-26, B>C 51-49*
C wins with PMR (MR threshold is first place). B would win with most
other Condorcet methods.
/* 35 A /> 10 A=B
*>* 30 B>C (sincere B > A)
*>* 25 C
*>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
*A wins with sincere voting. When B defects to try to win, which it
would do with most other Condorcet methods, B wins. With PMR, C wins,
an undesirable outcome for B.
Here is another example from Rob LeGrand
(https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
good example for chicken dilemma resistance, but it does demonstrate
differences from Schulze, MMPO, RP and Bucklin:
# example from method description page
98:Abby>Cora>Erin>Dave>Brad
64:Brad>Abby>Erin>Cora>Dave
12:Brad>Abby>Erin>Dave>Cora
98:Brad>Erin>Abby>Cora>Dave
13:Brad>Erin>Abby>Dave>Cora
125:Brad>Erin>Dave>Abby>Cora
124:Cora>Abby>Erin>Dave>Brad
76:Cora>Erin>Abby>Dave>Brad
21:Dave>Abby>Brad>Erin>Cora
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin
23:Dave>Cora>Brad>Abby>Erin
The pairwise matrix:
against
Abby Brad Cora Dave Erin
for Abby 458 461 485 511
Brad 463 461 312 623
Cora 460 460 460 460
Dave 436 609 461 311
Erin 410 298 461 610
There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
wins with Bucklin.
In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
score of 484 is only slightly over the 50% mark of 460.5, so the Dave
voters hold the balance of power. Dave defeats Brad pairwise, so Dave
voters might not be as happy with a Brad victory, and Abby might be
able to persuade Dave voters to downrank Brad but not Abby. If
successful, Brad drops 44 points in MRScore and is no longer in the MR
threshold set. Abby defeats Erin, so Abby wins.
* 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin
* 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
I believe it passes LNHelp. It probably fails Participation and IIA.
There are probably weird examples where changing one vote changes the
MR threshold. But overall, I think it has a good balance of incentive
to deter burial and deliberate cycles.
Has Smith//Median Rating been proposed before? It seems like a simple
modification to MR on its own.
----
Election-Methods mailing list - see https://electorama.com/em
for list info
Ted,
> ...you didn't comment on whether ballots with all Smith candidates
> below top rating should have their ratings bumped up: i.e., D > E > A
> > blank > B (A and B in Smith) would be recounted as A > blank > B.
I don't think I left anything ambiguous.
Assuming in your example we are using say 5-slot ratings ballots then we
interpret it as a score ballot thus: D5, E4, A3, B0.
If A and B are in Smith then the average score of candidates in the
Smith set is 3+0/2 = 1.5. Only A is scored above 1.5 so only A is
approved.
> I also noticed that there were cases where Smith//ASM(implicit) would
> get different results (better, IMO) than Smith//Implicit-approval.
>
What does "ASM" stand for?
Chris
On 23/01/2024 10:56 am, Ted Stern wrote:
> Chris:
>
> Thanks for the clarifications, though you didn't comment on whether
> ballots with all Smith candidates below top rating should have their
> ratings bumped up: i.e., D > E > A > blank > B (A and B in Smith)
> would be recounted as A > blank > B. I think this makes the most sense
> as a voter whose favorites are eliminated would want to ensure that
> their highest ranked Smith candidate is counted as approved.
>
> In general I agree with your comments, though I think
> Condorcet//Approval with all ranked ballots approved is probably not
> optimal, and Approval Sorted Margins with explicit approval would be
> too complex for a public proposal. I'd be happy with
> Condorcet//Top-ratings as a public proposal.
>
> Smith//Implicit-approval seems to perform well in a number of
> situations, but not appreciably better enough to make it worth the
> effort of trying to get people to accept something more complicated
> than Condorcet/Top-ratings. I also noticed that there were cases where
> Smith//ASM(implicit) would get different results (better, IMO) than
> Smith//Implicit-approval.
>
>
> On Fri, Jan 19, 2024 at 4:05 PM C.Benham <cbenham@adam.com.au> wrote:
>
>> How is the average calculated?
>
> We interpret the ratings ballots as score ballots, giving zero
> points for the bottom rating (which is default for unrated),
> 1 point for the next highest, 2 points for the next above that and
> so on.
>
> Then for any given ballot we add up the scores of the candidates
> in the Smith set and divide that by the number of candidates
> in the Smith set and interpret that ballot as approving those
> Smith set candidates it scores higher than that average score.
>
> That simulates the best approval strategy if the voters only know
> which candidates are in the Smith set.
>
>> What advantage does Approval Sorted Margins have over
>> Smith//Implicit-Approval?
>
> Do you mean Approval Sorted Margins using ranking ballots with an
> explicit approval cutoff?
>
> Assuming yes, it uses a more expressive ballot, it is less
> vulnerable to Defection strategy, and burial strategies are more
> likely to have no effect rather than backfire.
>
> In the method I proposed, omitting the ASM step and just electing
> the candidate with the highest approval score (derived
> as specified) would I concede make for a simpler method that is
> nearly as good.
>
> I worry a bit that with all methods that begin with eliminating or
> disqualifying all candidates who aren't in the Smith set or
> just "elect the CW if there is one", over time if there is never a
> top cycle then the top-cycle resolution method could stop
> being taken seriously.
>
> An attractive feature of ASM is that it is a Condorcet method that
> a lot of the time would work fine without anyone needing
> to know if there is top cycle or not.
>
> If the Approval order is A>B>C and A pairwise beats B and B
> pairwise beats C no-one needs to enquire about the pairwise
> result between A and C.
>
> If we want something super simple to explain and sell, then
> Condorcet//Top Ratings and Condorcet//Approval (voted above bottom)
> are both not bad and much better than STAR.
>
> Chris Benham
>
>
>
> On 18/01/2024 10:13 am, Ted Stern wrote:
>>
>>
>> On Tue, Jan 16, 2024 at 7:27 AM C.Benham <cbenham@adam.com.au> wrote:
>>
>> Ted,
>>
>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>> any member of the Smith Set.
>>
>> Why not simply drop all ballots that make no distinction
>> among members of the Smith set?
>>
>>> I believe it passes LNHelp.
>>
>> Douglas Woodall showed some time ago that Condorcet and
>> LNHelp are incompatible. I can't find
>> his proof, but it says so here:
>>
>> https://en.wikipedia.org/wiki/Later-no-help_criterion
>>
>>> TheCondorcet criterion
>>> <https://en.wikipedia.org/wiki/Condorcet_criterion>is
>>> incompatible with later-no-help.
>>
>> From your post again:
>>> It probably fails Participation ..
>>
>> It has been known (for a longer time) that Condorcet and
>> Participation are incompatible.
>>
>> So since we know for sure that your method meets Condorcet,
>> we also know that it doesn't meet
>> Later-no-Help or Participation.
>>
>> Using a multi-slot ratings ballot for a Condorcet method of
>> similar complexity I like:
>>
>> *Eliminate all candidates not in the Smith set.
>>
>> Interpret each ballot as giving approval to those remaining
>> candidates they rate above average (mean
>> of the ratings given to Smith-set members).
>>
>> Now, using these approvals, elect the Margins-Sorted Approval
>> winner.*
>>
>>
>> It seems to me that Smith//Implicit-Approval or
>> Smith//implicit-approval-sorted-margins would be affected by a
>> couple of factors:
>>
>> * How is the average calculated? Do you normalize scores? In
>> other words, if a ballot has non-Smith candidates in the
>> first, say, three ranks, do you up-rank the Smith candidate
>> scores on that ballot by three? Also, if there are ranks
>> below the top that contain only non-Smith candidates, do you
>> collapse those ranks or leave the relative rank spacing on
>> the ballot between Smith candidates untouched?
>> * Approving Smith Candidates with scores above the mean has
>> similarities to Median Ratings. It would be more similar and
>> probably more stable to use the trimmed mean -- drop the top
>> and bottom 25% of scores. This would give you an average
>> score closer to the median.
>>
>> What advantage does Approval Sorted Margins have over
>> Smith//Implicit-Approval? I like ASM but fear it is probably too
>> complex for any advantage it gives you.
>>
>>
>>> Has Smith//Median Rating been proposed before?
>> Not that I know of.
>>
>> Chris Benham
>>
>>
>>
>>>
>>> *Ted Stern*dodecatheon at gmail.com
>>> <mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
>>> /Tue Jan 2 15:12:26 PST 2024/
>>>
>>> * Previous message (by thread):[EM] [Game Theory] Iterated
>>> Prisoners' Dilemma as a voting method metric
>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005215.html>
>>> * Next message (by thread):[EM] Pairwise Median Rating
>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005218.html>
>>> * *Messages sorted by:*[ date ]
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>>> thread ]
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>>> subject ]
>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/subject.html#5216>[
>>> author ]
>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/author.html#5216>
>>>
>>> ------------------------------------------------------------------------
>>> Continuing my search for a summable voting method that discourages burial
>>> and defection, I've come across this hybrid of Condorcet and median ratings
>>> that acts like Smith/Approval with an automatic approval cutoff. I'm
>>> calling it Pairwise Median Rating (PMR), but it could also be described as
>>> Smith//MR//Pairwise//MRScore:
>>>
>>> 1. Equal Ranking and ranking gap allowed (essentially a ratings method
>>> with rank inferred). For purposes of this discussion, assume 6 slots (5
>>> ranks above rejection).
>>> 2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
>>> means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
>>> of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
>>> 3. [Smith]
>>> 1. Compute the pairwise preference array
>>> 2. The winner is the candidate who defeats each other candidate
>>> pairwise.
>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>> any member of the Smith Set.
>>> 4. [Median Rating]
>>> 1. Set the MR threshold to top rank.
>>> 2. While no Smith candidate has a majority of undropped ballots at or
>>> above the threshold, set the threshold to the next lower rank,
>>> until there
>>> is no lower rank.
>>> 3. The winner is the single candidate that has a majority of
>>> undropped ballots at or above the threshold.
>>> 5. [Pairwise]
>>> 1. Otherwise, if more than one candidate passes the threshold, look for
>>> a pairwise beats-all candidate among candidates meeting the MR threshold.
>>> (i.e. Condorcet on just the MR threshold set).
>>> 2. If there is one, you have a winner.
>>> 6. [MR Score]
>>> 1. Otherwise, the winner is the Smith set candidate with the largest
>>> number of ballots at or above the Median Rating threshold (their MRscore).
>>>
>>> This method is essentially Smith//Approval(explicit) with the approval
>>> cutoff automatically inferred via median ratings
>>>
>>> Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
>>> considered optional, but by doing that, you ensure Immunity from Irrelevant
>>> Ballots (IIB), aka the zero ballot problem that affects other Median Rating
>>> / Majority Judgment methods. In other words, the majority threshold is
>>> unaffected by ballots that do not rank a viable candidate. It is possible
>>> to do this summably if need be.
>>>
>>> PMR either passes the Chicken Dilemma criterion without adjustment, or
>>> there is a downranking strategy for defending against defection.
>>>
>>> Consider the following examples from Chris Benham's post re MinLV(erw)
>>> Sorted Margins (
>>> http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
>>> ):
>>>
>>> >/* 46 A>B /*>* 44 B>C (sincere is B or B>A)
>>> *>* 05 C>A
>>> *>* 05 C>B
>>> *>>* A>B 51-49, B>C 90-10, C>A 54-46.
>>> *
>>>
>>> With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
>>> there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
>>> threshold is 2nd place, and A and B both pass the threshold. A defeats B,
>>> so A is the winner and B's defection/burial fails.
>>>
>>> >/* 25 A>B /*>* 26 B>C
>>> *>* 23 C>A
>>> *>* 26 C
>>> *>>* C>A 75-25, A>B 48-26, B>C 51-49*
>>>
>>> C wins with PMR (MR threshold is first place). B would win with most
>>> other Condorcet methods.
>>>
>>> >/* 35 A /*>* 10 A=B
>>> *>* 30 B>C (sincere B > A)
>>> *>* 25 C
>>> *>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
>>>
>>> *A wins with sincere voting. When B defects to try to win, which it
>>> would do with most other Condorcet methods, B wins. With PMR, C wins,
>>> an undesirable outcome for B.
>>>
>>> Here is another example from Rob LeGrand
>>> (https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
>>> good example for chicken dilemma resistance, but it does demonstrate
>>> differences from Schulze, MMPO, RP and Bucklin:
>>>
>>> # example from method description page
>>> 98:Abby>Cora>Erin>Dave>Brad
>>> 64:Brad>Abby>Erin>Cora>Dave
>>> 12:Brad>Abby>Erin>Dave>Cora
>>> 98:Brad>Erin>Abby>Cora>Dave
>>> 13:Brad>Erin>Abby>Dave>Cora
>>> 125:Brad>Erin>Dave>Abby>Cora
>>> 124:Cora>Abby>Erin>Dave>Brad
>>> 76:Cora>Erin>Abby>Dave>Brad
>>> 21:Dave>Abby>Brad>Erin>Cora
>>> 30:Dave>Brad>Abby>Erin>Cora
>>> 98:Dave>Brad>Erin>Cora>Abby
>>> 139:Dave>Cora>Abby>Brad>Erin
>>> 23:Dave>Cora>Brad>Abby>Erin
>>>
>>> The pairwise matrix:
>>>
>>> against
>>> Abby Brad Cora Dave Erin
>>> for Abby 458 461 485 511
>>> Brad 463 461 312 623
>>> Cora 460 460 460 460
>>> Dave 436 609 461 311
>>> Erin 410 298 461 610
>>>
>>> There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
>>>
>>> Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
>>> wins with Bucklin.
>>>
>>> In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
>>> the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
>>> score of 484 is only slightly over the 50% mark of 460.5, so the Dave
>>> voters hold the balance of power. Dave defeats Brad pairwise, so Dave
>>> voters might not be as happy with a Brad victory, and Abby might be
>>> able to persuade Dave voters to downrank Brad but not Abby. If
>>> successful, Brad drops 44 points in MRScore and is no longer in the MR
>>> threshold set. Abby defeats Erin, so Abby wins.
>>>
>>> * 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
>>> 30:Dave>Brad>Abby>Erin>Cora
>>> 98:Dave>Brad>Erin>Cora>Abby
>>> 139:Dave>Cora>Abby>Brad>Erin
>>> * 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
>>>
>>> PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
>>> I believe it passes LNHelp. It probably fails Participation and IIA.
>>> There are probably weird examples where changing one vote changes the
>>> MR threshold. But overall, I think it has a good balance of incentive
>>> to deter burial and deliberate cycles.
>>>
>>> Has Smith//Median Rating been proposed before? It seems like a simple
>>> modification to MR on its own.
>>
>> ----
>> Election-Methods mailing list - see https://electorama.com/em
>> for list info
>>
TS
Ted Stern
Fri, Jan 26, 2024 7:10 PM
On Mon, Jan 22, 2024 at 7:07 PM C.Benham cbenham@adam.com.au wrote:
Ted,
...you didn't comment on whether ballots with all Smith candidates below
top rating should have their ratings bumped up: i.e., D > E > A > blank >
B (A and B in Smith) would be recounted as A > blank > B.
I don't think I left anything ambiguous.
Assuming in your example we are using say 5-slot ratings ballots then we
interpret it as a score ballot thus: D5, E4, A3, B0.
If A and B are in Smith then the average score of candidates in the Smith
set is 3+0/2 = 1.5. Only A is scored above 1.5 so only A is approved.
This is a nice technique, but it is not clone resistant. Adding Smith
candidate clones can change the average and thus the approval cutoff would
change. This seems unstable to me.
I think a better technique is to either have an explicit cutoff which is
lowered per ballot so that max(Smith score) is approved, or to make the
cutoff such that candidates with max(Smith candidate score)/2 or greater
per ballot are approved. I prefer the former.
I also noticed that there were cases where Smith//ASM(implicit) would get
different results (better, IMO) than Smith//Implicit-approval.
What does "ASM" stand for?
Approval Sorted Margins
https://electowiki.org/wiki/Approval_Sorted_Margins
Chris
On 23/01/2024 10:56 am, Ted Stern wrote:
Chris:
Thanks for the clarifications, though you didn't comment on whether
ballots with all Smith candidates below top rating should have their
ratings bumped up: i.e., D > E > A > blank > B (A and B in Smith) would be
recounted as A > blank > B. I think this makes the most sense as a voter
whose favorites are eliminated would want to ensure that their highest
ranked Smith candidate is counted as approved.
In general I agree with your comments, though I think Condorcet//Approval
with all ranked ballots approved is probably not optimal, and Approval
Sorted Margins with explicit approval would be too complex for a public
proposal. I'd be happy with Condorcet//Top-ratings as a public proposal.
Smith//Implicit-approval seems to perform well in a number of situations,
but not appreciably better enough to make it worth the effort of trying to
get people to accept something more complicated than Condorcet/Top-ratings.
I also noticed that there were cases where Smith//ASM(implicit) would get
different results (better, IMO) than Smith//Implicit-approval.
On Fri, Jan 19, 2024 at 4:05 PM C.Benham cbenham@adam.com.au wrote:
How is the average calculated?
We interpret the ratings ballots as score ballots, giving zero points for
the bottom rating (which is default for unrated),
1 point for the next highest, 2 points for the next above that and so
on.
Then for any given ballot we add up the scores of the candidates in the
Smith set and divide that by the number of candidates
in the Smith set and interpret that ballot as approving those Smith set
candidates it scores higher than that average score.
That simulates the best approval strategy if the voters only know which
candidates are in the Smith set.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval?
Do you mean Approval Sorted Margins using ranking ballots with an
explicit approval cutoff?
Assuming yes, it uses a more expressive ballot, it is less vulnerable to
Defection strategy, and burial strategies are more
likely to have no effect rather than backfire.
In the method I proposed, omitting the ASM step and just electing the
candidate with the highest approval score (derived
as specified) would I concede make for a simpler method that is nearly as
good.
I worry a bit that with all methods that begin with eliminating or
disqualifying all candidates who aren't in the Smith set or
just "elect the CW if there is one", over time if there is never a top
cycle then the top-cycle resolution method could stop
being taken seriously.
An attractive feature of ASM is that it is a Condorcet method that a lot
of the time would work fine without anyone needing
to know if there is top cycle or not.
If the Approval order is A>B>C and A pairwise beats B and B pairwise
beats C no-one needs to enquire about the pairwise
result between A and C.
If we want something super simple to explain and sell, then
Condorcet//Top Ratings and Condorcet//Approval (voted above bottom)
are both not bad and much better than STAR.
Chris Benham
On 18/01/2024 10:13 am, Ted Stern wrote:
On Tue, Jan 16, 2024 at 7:27 AM C.Benham cbenham@adam.com.au wrote:
Ted,
- Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
Why not simply drop all ballots that make no distinction among members
of the Smith set?
I believe it passes LNHelp.
Douglas Woodall showed some time ago that Condorcet and LNHelp are
incompatible. I can't find
his proof, but it says so here:
https://en.wikipedia.org/wiki/Later-no-help_criterion
The Condorcet criterion
https://en.wikipedia.org/wiki/Condorcet_criterion is incompatible
with later-no-help.
From your post again:
It probably fails Participation ..
It has been known (for a longer time) that Condorcet and Participation
are incompatible.
So since we know for sure that your method meets Condorcet, we also know
that it doesn't meet
Later-no-Help or Participation.
Using a multi-slot ratings ballot for a Condorcet method of similar
complexity I like:
*Eliminate all candidates not in the Smith set.
Interpret each ballot as giving approval to those remaining candidates
they rate above average (mean
of the ratings given to Smith-set members).
Now, using these approvals, elect the Margins-Sorted Approval winner.*
It seems to me that Smith//Implicit-Approval or
Smith//implicit-approval-sorted-margins would be affected by a couple of
factors:
- How is the average calculated? Do you normalize scores? In other
words, if a ballot has non-Smith candidates in the first, say, three ranks,
do you up-rank the Smith candidate scores on that ballot by three? Also, if
there are ranks below the top that contain only non-Smith candidates, do
you collapse those ranks or leave the relative rank spacing on the ballot
between Smith candidates untouched?
- Approving Smith Candidates with scores above the mean has
similarities to Median Ratings. It would be more similar and probably more
stable to use the trimmed mean -- drop the top and bottom 25% of scores.
This would give you an average score closer to the median.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval? I like ASM but fear it is probably too complex
for any advantage it gives you.
Has Smith//Median Rating been proposed before?
Not that I know of.
Chris Benham
Ted Stern dodecatheon at gmail.com
<election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
Tue Jan 2 15:12:26 PST 2024
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Continuing my search for a summable voting method that discourages burial
and defection, I've come across this hybrid of Condorcet and median ratings
that acts like Smith/Approval with an automatic approval cutoff. I'm
calling it Pairwise Median Rating (PMR), but it could also be described as
Smith//MR//Pairwise//MRScore:
1. Equal Ranking and ranking gap allowed (essentially a ratings method
with rank inferred). For purposes of this discussion, assume 6 slots (5
ranks above rejection).
2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
3. [Smith]
1. Compute the pairwise preference array
2. The winner is the candidate who defeats each other candidate
pairwise.
3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
4. [Median Rating]
1. Set the MR threshold to top rank.
2. While no Smith candidate has a majority of undropped ballots at or
above the threshold, set the threshold to the next lower rank,
until there
is no lower rank.
3. The winner is the single candidate that has a majority of
undropped ballots at or above the threshold.
5. [Pairwise]
1. Otherwise, if more than one candidate passes the threshold, look for
a pairwise beats-all candidate among candidates meeting the MR threshold.
(i.e. Condorcet on just the MR threshold set).
2. If there is one, you have a winner.
6. [MR Score]
1. Otherwise, the winner is the Smith set candidate with the largest
number of ballots at or above the Median Rating threshold (their MRscore).
This method is essentially Smith//Approval(explicit) with the approval
cutoff automatically inferred via median ratings
Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
considered optional, but by doing that, you ensure Immunity from Irrelevant
Ballots (IIB), aka the zero ballot problem that affects other Median Rating
/ Majority Judgment methods. In other words, the majority threshold is
unaffected by ballots that do not rank a viable candidate. It is possible
to do this summably if need be.
PMR either passes the Chicken Dilemma criterion without adjustment, or
there is a downranking strategy for defending against defection.
Consider the following examples from Chris Benham's post re MinLV(erw)
Sorted Margins (http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
):
** 46 A>B
*> 44 B>C (sincere is B or B>A)
> 05 C>A
> 05 C>B
>> A>B 51-49, B>C 90-10, C>A 54-46.
*
With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
threshold is 2nd place, and A and B both pass the threshold. A defeats B,
so A is the winner and B's defection/burial fails.
** 25 A>B
> 26 B>C
> 23 C>A
> 26 C
>> C>A 75-25, A>B 48-26, B>C 51-49
C wins with PMR (MR threshold is first place). B would win with most
other Condorcet methods.
** 35 A
*> 10 A=B
> 30 B>C (sincere B > A)
> 25 C
>> C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
*A wins with sincere voting. When B defects to try to win, which it
would do with most other Condorcet methods, B wins. With PMR, C wins,
an undesirable outcome for B.
Here is another example from Rob LeGrand
(https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
good example for chicken dilemma resistance, but it does demonstrate
differences from Schulze, MMPO, RP and Bucklin:
example from method description page
98:Abby>Cora>Erin>Dave>Brad
64:Brad>Abby>Erin>Cora>Dave
12:Brad>Abby>Erin>Dave>Cora
98:Brad>Erin>Abby>Cora>Dave
13:Brad>Erin>Abby>Dave>Cora
125:Brad>Erin>Dave>Abby>Cora
124:Cora>Abby>Erin>Dave>Brad
76:Cora>Erin>Abby>Dave>Brad
21:Dave>Abby>Brad>Erin>Cora
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin
23:Dave>Cora>Brad>Abby>Erin
The pairwise matrix:
against
Abby Brad Cora Dave Erin
for Abby 458 461 485 511
Brad 463 461 312 623
Cora 460 460 460 460
Dave 436 609 461 311
Erin 410 298 461 610
There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
wins with Bucklin.
In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
score of 484 is only slightly over the 50% mark of 460.5, so the Dave
voters hold the balance of power. Dave defeats Brad pairwise, so Dave
voters might not be as happy with a Brad victory, and Abby might be
able to persuade Dave voters to downrank Brad but not Abby. If
successful, Brad drops 44 points in MRScore and is no longer in the MR
threshold set. Abby defeats Erin, so Abby wins.
- 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin - 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
I believe it passes LNHelp. It probably fails Participation and IIA.
There are probably weird examples where changing one vote changes the
MR threshold. But overall, I think it has a good balance of incentive
to deter burial and deliberate cycles.
Has Smith//Median Rating been proposed before? It seems like a simple
modification to MR on its own.
Election-Methods mailing list - see https://electorama.com/em for list
info
Election-Methods mailing list - see https://electorama.com/em for list
info
On Mon, Jan 22, 2024 at 7:07 PM C.Benham <cbenham@adam.com.au> wrote:
> Ted,
>
> ...you didn't comment on whether ballots with all Smith candidates below
> top rating should have their ratings bumped up: i.e., D > E > A > blank >
> B (A and B in Smith) would be recounted as A > blank > B.
>
>
> I don't think I left anything ambiguous.
>
> Assuming in your example we are using say 5-slot ratings ballots then we
> interpret it as a score ballot thus: D5, E4, A3, B0.
>
> If A and B are in Smith then the average score of candidates in the Smith
> set is 3+0/2 = 1.5. Only A is scored above 1.5 so only A is approved.
>
>
This is a nice technique, but it is not clone resistant. Adding Smith
candidate clones can change the average and thus the approval cutoff would
change. This seems unstable to me.
I think a better technique is to either have an explicit cutoff which is
lowered per ballot so that max(Smith score) is approved, or to make the
cutoff such that candidates with max(Smith candidate score)/2 or greater
per ballot are approved. I prefer the former.
> I also noticed that there were cases where Smith//ASM(implicit) would get
> different results (better, IMO) than Smith//Implicit-approval.
>
>
> What does "ASM" stand for?
>
Approval Sorted Margins
<https://electowiki.org/wiki/Approval_Sorted_Margins>
>
> Chris
>
>
> On 23/01/2024 10:56 am, Ted Stern wrote:
>
> Chris:
>
> Thanks for the clarifications, though you didn't comment on whether
> ballots with all Smith candidates below top rating should have their
> ratings bumped up: i.e., D > E > A > blank > B (A and B in Smith) would be
> recounted as A > blank > B. I think this makes the most sense as a voter
> whose favorites are eliminated would want to ensure that their highest
> ranked Smith candidate is counted as approved.
>
> In general I agree with your comments, though I think Condorcet//Approval
> with all ranked ballots approved is probably not optimal, and Approval
> Sorted Margins with explicit approval would be too complex for a public
> proposal. I'd be happy with Condorcet//Top-ratings as a public proposal.
>
> Smith//Implicit-approval seems to perform well in a number of situations,
> but not appreciably better enough to make it worth the effort of trying to
> get people to accept something more complicated than Condorcet/Top-ratings.
> I also noticed that there were cases where Smith//ASM(implicit) would get
> different results (better, IMO) than Smith//Implicit-approval.
>
>
> On Fri, Jan 19, 2024 at 4:05 PM C.Benham <cbenham@adam.com.au> wrote:
>
>> How is the average calculated?
>>
>>
>> We interpret the ratings ballots as score ballots, giving zero points for
>> the bottom rating (which is default for unrated),
>> 1 point for the next highest, 2 points for the next above that and so
>> on.
>>
>> Then for any given ballot we add up the scores of the candidates in the
>> Smith set and divide that by the number of candidates
>> in the Smith set and interpret that ballot as approving those Smith set
>> candidates it scores higher than that average score.
>>
>> That simulates the best approval strategy if the voters only know which
>> candidates are in the Smith set.
>>
>> What advantage does Approval Sorted Margins have over
>> Smith//Implicit-Approval?
>>
>>
>> Do you mean Approval Sorted Margins using ranking ballots with an
>> explicit approval cutoff?
>>
>> Assuming yes, it uses a more expressive ballot, it is less vulnerable to
>> Defection strategy, and burial strategies are more
>> likely to have no effect rather than backfire.
>>
>> In the method I proposed, omitting the ASM step and just electing the
>> candidate with the highest approval score (derived
>> as specified) would I concede make for a simpler method that is nearly as
>> good.
>>
>> I worry a bit that with all methods that begin with eliminating or
>> disqualifying all candidates who aren't in the Smith set or
>> just "elect the CW if there is one", over time if there is never a top
>> cycle then the top-cycle resolution method could stop
>> being taken seriously.
>>
>> An attractive feature of ASM is that it is a Condorcet method that a lot
>> of the time would work fine without anyone needing
>> to know if there is top cycle or not.
>>
>> If the Approval order is A>B>C and A pairwise beats B and B pairwise
>> beats C no-one needs to enquire about the pairwise
>> result between A and C.
>>
>> If we want something super simple to explain and sell, then
>> Condorcet//Top Ratings and Condorcet//Approval (voted above bottom)
>> are both not bad and much better than STAR.
>>
>> Chris Benham
>>
>>
>>
>> On 18/01/2024 10:13 am, Ted Stern wrote:
>>
>>
>>
>> On Tue, Jan 16, 2024 at 7:27 AM C.Benham <cbenham@adam.com.au> wrote:
>>
>>> Ted,
>>>
>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>> any member of the Smith Set.
>>>
>>> Why not simply drop all ballots that make no distinction among members
>>> of the Smith set?
>>>
>>> I believe it passes LNHelp.
>>>
>>>
>>> Douglas Woodall showed some time ago that Condorcet and LNHelp are
>>> incompatible. I can't find
>>> his proof, but it says so here:
>>>
>>> https://en.wikipedia.org/wiki/Later-no-help_criterion
>>>
>>> The Condorcet criterion
>>> <https://en.wikipedia.org/wiki/Condorcet_criterion> is incompatible
>>> with later-no-help.
>>>
>>>
>>> From your post again:
>>>
>>> It probably fails Participation ..
>>>
>>>
>>> It has been known (for a longer time) that Condorcet and Participation
>>> are incompatible.
>>>
>>> So since we know for sure that your method meets Condorcet, we also know
>>> that it doesn't meet
>>> Later-no-Help or Participation.
>>>
>>> Using a multi-slot ratings ballot for a Condorcet method of similar
>>> complexity I like:
>>>
>>> *Eliminate all candidates not in the Smith set.
>>>
>>> Interpret each ballot as giving approval to those remaining candidates
>>> they rate above average (mean
>>> of the ratings given to Smith-set members).
>>>
>>> Now, using these approvals, elect the Margins-Sorted Approval winner.*
>>>
>>
>> It seems to me that Smith//Implicit-Approval or
>> Smith//implicit-approval-sorted-margins would be affected by a couple of
>> factors:
>>
>> - How is the average calculated? Do you normalize scores? In other
>> words, if a ballot has non-Smith candidates in the first, say, three ranks,
>> do you up-rank the Smith candidate scores on that ballot by three? Also, if
>> there are ranks below the top that contain only non-Smith candidates, do
>> you collapse those ranks or leave the relative rank spacing on the ballot
>> between Smith candidates untouched?
>> - Approving Smith Candidates with scores above the mean has
>> similarities to Median Ratings. It would be more similar and probably more
>> stable to use the trimmed mean -- drop the top and bottom 25% of scores.
>> This would give you an average score closer to the median.
>>
>> What advantage does Approval Sorted Margins have over
>> Smith//Implicit-Approval? I like ASM but fear it is probably too complex
>> for any advantage it gives you.
>>
>>>
>>> Has Smith//Median Rating been proposed before?
>>>
>>> Not that I know of.
>>>
>>> Chris Benham
>>>
>>>
>>>
>>>
>>> *Ted Stern* dodecatheon at gmail.com
>>> <election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
>>> *Tue Jan 2 15:12:26 PST 2024*
>>>
>>> - Previous message (by thread): [EM] [Game Theory] Iterated
>>> Prisoners' Dilemma as a voting method metric
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>>> [ thread ]
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>>> [ subject ]
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>>> [ author ]
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>>>
>>> ------------------------------
>>>
>>> Continuing my search for a summable voting method that discourages burial
>>> and defection, I've come across this hybrid of Condorcet and median ratings
>>> that acts like Smith/Approval with an automatic approval cutoff. I'm
>>> calling it Pairwise Median Rating (PMR), but it could also be described as
>>> Smith//MR//Pairwise//MRScore:
>>>
>>> 1. Equal Ranking and ranking gap allowed (essentially a ratings method
>>> with rank inferred). For purposes of this discussion, assume 6 slots (5
>>> ranks above rejection).
>>> 2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
>>> means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
>>> of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
>>> 3. [Smith]
>>> 1. Compute the pairwise preference array
>>> 2. The winner is the candidate who defeats each other candidate
>>> pairwise.
>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>> any member of the Smith Set.
>>> 4. [Median Rating]
>>> 1. Set the MR threshold to top rank.
>>> 2. While no Smith candidate has a majority of undropped ballots at or
>>> above the threshold, set the threshold to the next lower rank,
>>> until there
>>> is no lower rank.
>>> 3. The winner is the single candidate that has a majority of
>>> undropped ballots at or above the threshold.
>>> 5. [Pairwise]
>>> 1. Otherwise, if more than one candidate passes the threshold, look for
>>> a pairwise beats-all candidate among candidates meeting the MR threshold.
>>> (i.e. Condorcet on just the MR threshold set).
>>> 2. If there is one, you have a winner.
>>> 6. [MR Score]
>>> 1. Otherwise, the winner is the Smith set candidate with the largest
>>> number of ballots at or above the Median Rating threshold (their MRscore).
>>>
>>> This method is essentially Smith//Approval(explicit) with the approval
>>> cutoff automatically inferred via median ratings
>>>
>>> Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
>>> considered optional, but by doing that, you ensure Immunity from Irrelevant
>>> Ballots (IIB), aka the zero ballot problem that affects other Median Rating
>>> / Majority Judgment methods. In other words, the majority threshold is
>>> unaffected by ballots that do not rank a viable candidate. It is possible
>>> to do this summably if need be.
>>>
>>> PMR either passes the Chicken Dilemma criterion without adjustment, or
>>> there is a downranking strategy for defending against defection.
>>>
>>> Consider the following examples from Chris Benham's post re MinLV(erw)
>>> Sorted Margins (http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
>>> ):
>>>
>>> >** 46 A>B
>>> **>* 44 B>C (sincere is B or B>A)
>>> *>* 05 C>A
>>> *>* 05 C>B
>>> *>>* A>B 51-49, B>C 90-10, C>A 54-46.
>>> *
>>>
>>> With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
>>> there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
>>> threshold is 2nd place, and A and B both pass the threshold. A defeats B,
>>> so A is the winner and B's defection/burial fails.
>>>
>>> >** 25 A>B
>>> **>* 26 B>C
>>> *>* 23 C>A
>>> *>* 26 C
>>> *>>* C>A 75-25, A>B 48-26, B>C 51-49*
>>>
>>> C wins with PMR (MR threshold is first place). B would win with most
>>> other Condorcet methods.
>>>
>>> >** 35 A
>>> **>* 10 A=B
>>> *>* 30 B>C (sincere B > A)
>>> *>* 25 C
>>> *>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
>>>
>>> *A wins with sincere voting. When B defects to try to win, which it
>>> would do with most other Condorcet methods, B wins. With PMR, C wins,
>>> an undesirable outcome for B.
>>>
>>> Here is another example from Rob LeGrand
>>> (https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
>>> good example for chicken dilemma resistance, but it does demonstrate
>>> differences from Schulze, MMPO, RP and Bucklin:
>>>
>>> # example from method description page
>>> 98:Abby>Cora>Erin>Dave>Brad
>>> 64:Brad>Abby>Erin>Cora>Dave
>>> 12:Brad>Abby>Erin>Dave>Cora
>>> 98:Brad>Erin>Abby>Cora>Dave
>>> 13:Brad>Erin>Abby>Dave>Cora
>>> 125:Brad>Erin>Dave>Abby>Cora
>>> 124:Cora>Abby>Erin>Dave>Brad
>>> 76:Cora>Erin>Abby>Dave>Brad
>>> 21:Dave>Abby>Brad>Erin>Cora
>>> 30:Dave>Brad>Abby>Erin>Cora
>>> 98:Dave>Brad>Erin>Cora>Abby
>>> 139:Dave>Cora>Abby>Brad>Erin
>>> 23:Dave>Cora>Brad>Abby>Erin
>>>
>>> The pairwise matrix:
>>>
>>> against
>>> Abby Brad Cora Dave Erin
>>> for Abby 458 461 485 511
>>> Brad 463 461 312 623
>>> Cora 460 460 460 460
>>> Dave 436 609 461 311
>>> Erin 410 298 461 610
>>>
>>> There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
>>>
>>> Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
>>> wins with Bucklin.
>>>
>>> In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
>>> the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
>>> score of 484 is only slightly over the 50% mark of 460.5, so the Dave
>>> voters hold the balance of power. Dave defeats Brad pairwise, so Dave
>>> voters might not be as happy with a Brad victory, and Abby might be
>>> able to persuade Dave voters to downrank Brad but not Abby. If
>>> successful, Brad drops 44 points in MRScore and is no longer in the MR
>>> threshold set. Abby defeats Erin, so Abby wins.
>>>
>>> * 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
>>> 30:Dave>Brad>Abby>Erin>Cora
>>> 98:Dave>Brad>Erin>Cora>Abby
>>> 139:Dave>Cora>Abby>Brad>Erin
>>> * 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
>>>
>>> PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
>>> I believe it passes LNHelp. It probably fails Participation and IIA.
>>> There are probably weird examples where changing one vote changes the
>>> MR threshold. But overall, I think it has a good balance of incentive
>>> to deter burial and deliberate cycles.
>>>
>>> Has Smith//Median Rating been proposed before? It seems like a simple
>>> modification to MR on its own.
>>>
>>>
>>> ----
>>> Election-Methods mailing list - see https://electorama.com/em for list
>>> info
>>>
>> ----
> Election-Methods mailing list - see https://electorama.com/em for list
> info
>
C
C.Benham
Sat, Jan 27, 2024 12:41 AM
Of course! Thanks. Pardon the brain fade.
Normally Smith sets don't have two members, but your example didn't
mention a third member or say how many candidates there are or how many
rating slots the ballots have.
So maybe there is another candidate F (which is in the Smith set) and
the ballots have 6 slots, so our Score ballot interpretation becomes
D6, E5, A4, B2, F0.
So then our calculation becomes 4+2+0/3 = 2.
Only A is scored higher than 2, so the ballot only approves A.
This is a nice technique, but it is not clone resistant. Adding Smith
candidate clones can change the average and thus the approval cutoff
would change. This seems unstable to me.
With ratings ballots I think only candidates that get the same rating on
every ballot are considered clones. And besides that I think that more
than three candidates in the Smith set would be vanishingly rare.
Chris
On 27/01/2024 5:40 am, Ted Stern wrote:
On Mon, Jan 22, 2024 at 7:07 PM C.Benham cbenham@adam.com.au wrote:
Ted,
...you didn't comment on whether ballots with all Smith
candidates below top rating should have their ratings bumped up:
i.e., D > E > A > blank > B (A and B in Smith) would be
recounted as A > blank > B.
I don't think I left anything ambiguous.
Assuming in your example we are using say 5-slot ratings ballots
then we interpret it as a score ballot thus: D5, E4, A3, B0.
If A and B are in Smith then the average score of candidates in
the Smith set is 3+0/2 = 1.5. Only A is scored above 1.5 so
only A is approved.
This is a nice technique, but it is not clone resistant. Adding Smith
candidate clones can change the average and thus the approval cutoff
would change. This seems unstable to me.
I think a better technique is to either have an explicit cutoff which
is lowered per ballot so that max(Smith score) is approved, or to make
the cutoff such that candidates with max(Smith candidate score)/2 or
greater per ballot are approved. I prefer the former.
I also noticed that there were cases where Smith//ASM(implicit)
would get different results (better, IMO) than
Smith//Implicit-approval.
What does "ASM" stand for?
Approval Sorted Margins
https://electowiki.org/wiki/Approval_Sorted_Margins
Chris
On 23/01/2024 10:56 am, Ted Stern wrote:
Chris:
Thanks for the clarifications, though you didn't comment on
whether ballots with all Smith candidates below top rating should
have their ratings bumped up: i.e., D > E > A > blank > B (A and
B in Smith) would be recounted as A > blank > B. I think this
makes the most sense as a voter whose favorites are eliminated
would want to ensure that their highest ranked Smith candidate is
counted as approved.
In general I agree with your comments, though I think
Condorcet//Approval with all ranked ballots approved is probably
not optimal, and Approval Sorted Margins with explicit approval
would be too complex for a public proposal. I'd be happy with
Condorcet//Top-ratings as a public proposal.
Smith//Implicit-approval seems to perform well in a number of
situations, but not appreciably better enough to make it worth
the effort of trying to get people to accept something more
complicated than Condorcet/Top-ratings. I also noticed that there
were cases where Smith//ASM(implicit) would get different results
(better, IMO) than Smith//Implicit-approval.
On Fri, Jan 19, 2024 at 4:05 PM C.Benham <cbenham@adam.com.au> wrote:
How is the average calculated?
We interpret the ratings ballots as score ballots, giving
zero points for the bottom rating (which is default for unrated),
1 point for the next highest, 2 points for the next above
that and so on.
Then for any given ballot we add up the scores of the
candidates in the Smith set and divide that by the number of
candidates
in the Smith set and interpret that ballot as approving those
Smith set candidates it scores higher than that average score.
That simulates the best approval strategy if the voters only
know which candidates are in the Smith set.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval?
Do you mean Approval Sorted Margins using ranking ballots
with an explicit approval cutoff?
Assuming yes, it uses a more expressive ballot, it is less
vulnerable to Defection strategy, and burial strategies are more
likely to have no effect rather than backfire.
In the method I proposed, omitting the ASM step and just
electing the candidate with the highest approval score (derived
as specified) would I concede make for a simpler method that
is nearly as good.
I worry a bit that with all methods that begin with
eliminating or disqualifying all candidates who aren't in the
Smith set or
just "elect the CW if there is one", over time if there is
never a top cycle then the top-cycle resolution method could stop
being taken seriously.
An attractive feature of ASM is that it is a Condorcet method
that a lot of the time would work fine without anyone needing
to know if there is top cycle or not.
If the Approval order is A>B>C and A pairwise beats B and B
pairwise beats C no-one needs to enquire about the pairwise
result between A and C.
If we want something super simple to explain and sell, then
Condorcet//Top Ratings and Condorcet//Approval (voted above
bottom)
are both not bad and much better than STAR.
Chris Benham
On 18/01/2024 10:13 am, Ted Stern wrote:
On Tue, Jan 16, 2024 at 7:27 AM C.Benham
<cbenham@adam.com.au> wrote:
Ted,
3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
Why not simply drop all ballots that make no distinction
among members of the Smith set?
I believe it passes LNHelp.
Douglas Woodall showed some time ago that Condorcet and
LNHelp are incompatible. I can't find
his proof, but it says so here:
https://en.wikipedia.org/wiki/Later-no-help_criterion
TheCondorcet criterion
<https://en.wikipedia.org/wiki/Condorcet_criterion>is
incompatible with later-no-help.
From your post again:
It probably fails Participation ..
It has been known (for a longer time) that Condorcet and
Participation are incompatible.
So since we know for sure that your method meets
Condorcet, we also know that it doesn't meet
Later-no-Help or Participation.
Using a multi-slot ratings ballot for a Condorcet method
of similar complexity I like:
*Eliminate all candidates not in the Smith set.
Interpret each ballot as giving approval to those
remaining candidates they rate above average (mean
of the ratings given to Smith-set members).
Now, using these approvals, elect the Margins-Sorted
Approval winner.*
It seems to me that Smith//Implicit-Approval or
Smith//implicit-approval-sorted-margins would be affected by
a couple of factors:
* How is the average calculated? Do you normalize scores?
In other words, if a ballot has non-Smith candidates in
the first, say, three ranks, do you up-rank the Smith
candidate scores on that ballot by three? Also, if there
are ranks below the top that contain only non-Smith
candidates, do you collapse those ranks or leave the
relative rank spacing on the ballot between Smith
candidates untouched?
* Approving Smith Candidates with scores above the mean
has similarities to Median Ratings. It would be more
similar and probably more stable to use the trimmed mean
-- drop the top and bottom 25% of scores. This would
give you an average score closer to the median.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval? I like ASM but fear it is probably
too complex for any advantage it gives you.
Has Smith//Median Rating been proposed before?
Not that I know of.
Chris Benham
*Ted Stern*dodecatheon at gmail.com
<mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
/Tue Jan 2 15:12:26 PST 2024/
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------------------------------------------------------------------------
Continuing my search for a summable voting method that discourages burial
and defection, I've come across this hybrid of Condorcet and median ratings
that acts like Smith/Approval with an automatic approval cutoff. I'm
calling it Pairwise Median Rating (PMR), but it could also be described as
Smith//MR//Pairwise//MRScore:
1. Equal Ranking and ranking gap allowed (essentially a ratings method
with rank inferred). For purposes of this discussion, assume 6 slots (5
ranks above rejection).
2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
3. [Smith]
1. Compute the pairwise preference array
2. The winner is the candidate who defeats each other candidate
pairwise.
3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
4. [Median Rating]
1. Set the MR threshold to top rank.
2. While no Smith candidate has a majority of undropped ballots at or
above the threshold, set the threshold to the next lower rank,
until there
is no lower rank.
3. The winner is the single candidate that has a majority of
undropped ballots at or above the threshold.
5. [Pairwise]
1. Otherwise, if more than one candidate passes the threshold, look for
a pairwise beats-all candidate among candidates meeting the MR threshold.
(i.e. Condorcet on just the MR threshold set).
2. If there is one, you have a winner.
6. [MR Score]
1. Otherwise, the winner is the Smith set candidate with the largest
number of ballots at or above the Median Rating threshold (their MRscore).
This method is essentially Smith//Approval(explicit) with the approval
cutoff automatically inferred via median ratings
Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
considered optional, but by doing that, you ensure Immunity from Irrelevant
Ballots (IIB), aka the zero ballot problem that affects other Median Rating
/ Majority Judgment methods. In other words, the majority threshold is
unaffected by ballots that do not rank a viable candidate. It is possible
to do this summably if need be.
PMR either passes the Chicken Dilemma criterion without adjustment, or
there is a downranking strategy for defending against defection.
Consider the following examples from Chris Benham's post re MinLV(erw)
Sorted Margins (
http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
):
/* 46 A>B /> 44 B>C (sincere is B or B>A)
*>* 05 C>A
*>* 05 C>B
*>>* A>B 51-49, B>C 90-10, C>A 54-46.
*
With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
threshold is 2nd place, and A and B both pass the threshold. A defeats B,
so A is the winner and B's defection/burial fails.
/* 25 A>B /> 26 B>C
*>* 23 C>A
*>* 26 C
*>>* C>A 75-25, A>B 48-26, B>C 51-49*
C wins with PMR (MR threshold is first place). B would win with most
other Condorcet methods.
/* 35 A /> 10 A=B
*>* 30 B>C (sincere B > A)
*>* 25 C
*>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
*A wins with sincere voting. When B defects to try to win, which it
would do with most other Condorcet methods, B wins. With PMR, C wins,
an undesirable outcome for B.
Here is another example from Rob LeGrand
(https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
good example for chicken dilemma resistance, but it does demonstrate
differences from Schulze, MMPO, RP and Bucklin:
# example from method description page
98:Abby>Cora>Erin>Dave>Brad
64:Brad>Abby>Erin>Cora>Dave
12:Brad>Abby>Erin>Dave>Cora
98:Brad>Erin>Abby>Cora>Dave
13:Brad>Erin>Abby>Dave>Cora
125:Brad>Erin>Dave>Abby>Cora
124:Cora>Abby>Erin>Dave>Brad
76:Cora>Erin>Abby>Dave>Brad
21:Dave>Abby>Brad>Erin>Cora
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin
23:Dave>Cora>Brad>Abby>Erin
The pairwise matrix:
against
Abby Brad Cora Dave Erin
for Abby 458 461 485 511
Brad 463 461 312 623
Cora 460 460 460 460
Dave 436 609 461 311
Erin 410 298 461 610
There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
wins with Bucklin.
In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
score of 484 is only slightly over the 50% mark of 460.5, so the Dave
voters hold the balance of power. Dave defeats Brad pairwise, so Dave
voters might not be as happy with a Brad victory, and Abby might be
able to persuade Dave voters to downrank Brad but not Abby. If
successful, Brad drops 44 points in MRScore and is no longer in the MR
threshold set. Abby defeats Erin, so Abby wins.
* 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin
* 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
I believe it passes LNHelp. It probably fails Participation and IIA.
There are probably weird examples where changing one vote changes the
MR threshold. But overall, I think it has a good balance of incentive
to deter burial and deliberate cycles.
Has Smith//Median Rating been proposed before? It seems like a simple
modification to MR on its own.
----
Election-Methods mailing list - see
https://electorama.com/em for list info
----
Election-Methods mailing list - see https://electorama.com/em for
list info
Of course! Thanks. Pardon the brain fade.
Normally Smith sets don't have two members, but your example didn't
mention a third member or say how many candidates there are or how many
rating slots the ballots have.
So maybe there is another candidate F (which is in the Smith set) and
the ballots have 6 slots, so our Score ballot interpretation becomes
D6, E5, A4, B2, F0.
So then our calculation becomes 4+2+0/3 = 2.
Only A is scored higher than 2, so the ballot only approves A.
> This is a nice technique, but it is not clone resistant. Adding Smith
> candidate clones can change the average and thus the approval cutoff
> would change. This seems unstable to me.
With ratings ballots I think only candidates that get the same rating on
every ballot are considered clones. And besides that I think that more
than three candidates in the Smith set would be vanishingly rare.
Chris
On 27/01/2024 5:40 am, Ted Stern wrote:
>
>
> On Mon, Jan 22, 2024 at 7:07 PM C.Benham <cbenham@adam.com.au> wrote:
>
> Ted,
>
>> ...you didn't comment on whether ballots with all Smith
>> candidates below top rating should have their ratings bumped up:
>> i.e., D > E > A > blank > B (A and B in Smith) would be
>> recounted as A > blank > B.
>
> I don't think I left anything ambiguous.
>
> Assuming in your example we are using say 5-slot ratings ballots
> then we interpret it as a score ballot thus: D5, E4, A3, B0.
>
> If A and B are in Smith then the average score of candidates in
> the Smith set is 3+0/2 = 1.5. Only A is scored above 1.5 so
> only A is approved.
>
>
> This is a nice technique, but it is not clone resistant. Adding Smith
> candidate clones can change the average and thus the approval cutoff
> would change. This seems unstable to me.
>
> I think a better technique is to either have an explicit cutoff which
> is lowered per ballot so that max(Smith score) is approved, or to make
> the cutoff such that candidates with max(Smith candidate score)/2 or
> greater per ballot are approved. I prefer the former.
>
>> I also noticed that there were cases where Smith//ASM(implicit)
>> would get different results (better, IMO) than
>> Smith//Implicit-approval.
>>
>
> What does "ASM" stand for?
>
>
> Approval Sorted Margins
> <https://electowiki.org/wiki/Approval_Sorted_Margins>
>
>
> Chris
>
>
> On 23/01/2024 10:56 am, Ted Stern wrote:
>> Chris:
>>
>> Thanks for the clarifications, though you didn't comment on
>> whether ballots with all Smith candidates below top rating should
>> have their ratings bumped up: i.e., D > E > A > blank > B (A and
>> B in Smith) would be recounted as A > blank > B. I think this
>> makes the most sense as a voter whose favorites are eliminated
>> would want to ensure that their highest ranked Smith candidate is
>> counted as approved.
>>
>> In general I agree with your comments, though I think
>> Condorcet//Approval with all ranked ballots approved is probably
>> not optimal, and Approval Sorted Margins with explicit approval
>> would be too complex for a public proposal. I'd be happy with
>> Condorcet//Top-ratings as a public proposal.
>>
>> Smith//Implicit-approval seems to perform well in a number of
>> situations, but not appreciably better enough to make it worth
>> the effort of trying to get people to accept something more
>> complicated than Condorcet/Top-ratings. I also noticed that there
>> were cases where Smith//ASM(implicit) would get different results
>> (better, IMO) than Smith//Implicit-approval.
>>
>>
>> On Fri, Jan 19, 2024 at 4:05 PM C.Benham <cbenham@adam.com.au> wrote:
>>
>>> How is the average calculated?
>>
>> We interpret the ratings ballots as score ballots, giving
>> zero points for the bottom rating (which is default for unrated),
>> 1 point for the next highest, 2 points for the next above
>> that and so on.
>>
>> Then for any given ballot we add up the scores of the
>> candidates in the Smith set and divide that by the number of
>> candidates
>> in the Smith set and interpret that ballot as approving those
>> Smith set candidates it scores higher than that average score.
>>
>> That simulates the best approval strategy if the voters only
>> know which candidates are in the Smith set.
>>
>>> What advantage does Approval Sorted Margins have over
>>> Smith//Implicit-Approval?
>>
>> Do you mean Approval Sorted Margins using ranking ballots
>> with an explicit approval cutoff?
>>
>> Assuming yes, it uses a more expressive ballot, it is less
>> vulnerable to Defection strategy, and burial strategies are more
>> likely to have no effect rather than backfire.
>>
>> In the method I proposed, omitting the ASM step and just
>> electing the candidate with the highest approval score (derived
>> as specified) would I concede make for a simpler method that
>> is nearly as good.
>>
>> I worry a bit that with all methods that begin with
>> eliminating or disqualifying all candidates who aren't in the
>> Smith set or
>> just "elect the CW if there is one", over time if there is
>> never a top cycle then the top-cycle resolution method could stop
>> being taken seriously.
>>
>> An attractive feature of ASM is that it is a Condorcet method
>> that a lot of the time would work fine without anyone needing
>> to know if there is top cycle or not.
>>
>> If the Approval order is A>B>C and A pairwise beats B and B
>> pairwise beats C no-one needs to enquire about the pairwise
>> result between A and C.
>>
>> If we want something super simple to explain and sell, then
>> Condorcet//Top Ratings and Condorcet//Approval (voted above
>> bottom)
>> are both not bad and much better than STAR.
>>
>> Chris Benham
>>
>>
>>
>> On 18/01/2024 10:13 am, Ted Stern wrote:
>>>
>>>
>>> On Tue, Jan 16, 2024 at 7:27 AM C.Benham
>>> <cbenham@adam.com.au> wrote:
>>>
>>> Ted,
>>>
>>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>>> any member of the Smith Set.
>>>
>>> Why not simply drop all ballots that make no distinction
>>> among members of the Smith set?
>>>
>>>> I believe it passes LNHelp.
>>>
>>> Douglas Woodall showed some time ago that Condorcet and
>>> LNHelp are incompatible. I can't find
>>> his proof, but it says so here:
>>>
>>> https://en.wikipedia.org/wiki/Later-no-help_criterion
>>>
>>>> TheCondorcet criterion
>>>> <https://en.wikipedia.org/wiki/Condorcet_criterion>is
>>>> incompatible with later-no-help.
>>>
>>> From your post again:
>>>> It probably fails Participation ..
>>>
>>> It has been known (for a longer time) that Condorcet and
>>> Participation are incompatible.
>>>
>>> So since we know for sure that your method meets
>>> Condorcet, we also know that it doesn't meet
>>> Later-no-Help or Participation.
>>>
>>> Using a multi-slot ratings ballot for a Condorcet method
>>> of similar complexity I like:
>>>
>>> *Eliminate all candidates not in the Smith set.
>>>
>>> Interpret each ballot as giving approval to those
>>> remaining candidates they rate above average (mean
>>> of the ratings given to Smith-set members).
>>>
>>> Now, using these approvals, elect the Margins-Sorted
>>> Approval winner.*
>>>
>>>
>>> It seems to me that Smith//Implicit-Approval or
>>> Smith//implicit-approval-sorted-margins would be affected by
>>> a couple of factors:
>>>
>>> * How is the average calculated? Do you normalize scores?
>>> In other words, if a ballot has non-Smith candidates in
>>> the first, say, three ranks, do you up-rank the Smith
>>> candidate scores on that ballot by three? Also, if there
>>> are ranks below the top that contain only non-Smith
>>> candidates, do you collapse those ranks or leave the
>>> relative rank spacing on the ballot between Smith
>>> candidates untouched?
>>> * Approving Smith Candidates with scores above the mean
>>> has similarities to Median Ratings. It would be more
>>> similar and probably more stable to use the trimmed mean
>>> -- drop the top and bottom 25% of scores. This would
>>> give you an average score closer to the median.
>>>
>>> What advantage does Approval Sorted Margins have over
>>> Smith//Implicit-Approval? I like ASM but fear it is probably
>>> too complex for any advantage it gives you.
>>>
>>>
>>>> Has Smith//Median Rating been proposed before?
>>> Not that I know of.
>>>
>>> Chris Benham
>>>
>>>
>>>
>>>>
>>>> *Ted Stern*dodecatheon at gmail.com
>>>> <mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
>>>> /Tue Jan 2 15:12:26 PST 2024/
>>>>
>>>> * Previous message (by thread):[EM] [Game Theory]
>>>> Iterated Prisoners' Dilemma as a voting method
>>>> metric
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005215.html>
>>>> * Next message (by thread):[EM] Pairwise Median
>>>> Rating
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005218.html>
>>>> * *Messages sorted by:*[ date ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/date.html#5216>[
>>>> thread ]
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>>>> subject ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/subject.html#5216>[
>>>> author ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/author.html#5216>
>>>>
>>>> ------------------------------------------------------------------------
>>>> Continuing my search for a summable voting method that discourages burial
>>>> and defection, I've come across this hybrid of Condorcet and median ratings
>>>> that acts like Smith/Approval with an automatic approval cutoff. I'm
>>>> calling it Pairwise Median Rating (PMR), but it could also be described as
>>>> Smith//MR//Pairwise//MRScore:
>>>>
>>>> 1. Equal Ranking and ranking gap allowed (essentially a ratings method
>>>> with rank inferred). For purposes of this discussion, assume 6 slots (5
>>>> ranks above rejection).
>>>> 2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
>>>> means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
>>>> of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
>>>> 3. [Smith]
>>>> 1. Compute the pairwise preference array
>>>> 2. The winner is the candidate who defeats each other candidate
>>>> pairwise.
>>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>>> any member of the Smith Set.
>>>> 4. [Median Rating]
>>>> 1. Set the MR threshold to top rank.
>>>> 2. While no Smith candidate has a majority of undropped ballots at or
>>>> above the threshold, set the threshold to the next lower rank,
>>>> until there
>>>> is no lower rank.
>>>> 3. The winner is the single candidate that has a majority of
>>>> undropped ballots at or above the threshold.
>>>> 5. [Pairwise]
>>>> 1. Otherwise, if more than one candidate passes the threshold, look for
>>>> a pairwise beats-all candidate among candidates meeting the MR threshold.
>>>> (i.e. Condorcet on just the MR threshold set).
>>>> 2. If there is one, you have a winner.
>>>> 6. [MR Score]
>>>> 1. Otherwise, the winner is the Smith set candidate with the largest
>>>> number of ballots at or above the Median Rating threshold (their MRscore).
>>>>
>>>> This method is essentially Smith//Approval(explicit) with the approval
>>>> cutoff automatically inferred via median ratings
>>>>
>>>> Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
>>>> considered optional, but by doing that, you ensure Immunity from Irrelevant
>>>> Ballots (IIB), aka the zero ballot problem that affects other Median Rating
>>>> / Majority Judgment methods. In other words, the majority threshold is
>>>> unaffected by ballots that do not rank a viable candidate. It is possible
>>>> to do this summably if need be.
>>>>
>>>> PMR either passes the Chicken Dilemma criterion without adjustment, or
>>>> there is a downranking strategy for defending against defection.
>>>>
>>>> Consider the following examples from Chris Benham's post re MinLV(erw)
>>>> Sorted Margins (
>>>> http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
>>>> ):
>>>>
>>>> >/* 46 A>B /*>* 44 B>C (sincere is B or B>A)
>>>> *>* 05 C>A
>>>> *>* 05 C>B
>>>> *>>* A>B 51-49, B>C 90-10, C>A 54-46.
>>>> *
>>>>
>>>> With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
>>>> there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
>>>> threshold is 2nd place, and A and B both pass the threshold. A defeats B,
>>>> so A is the winner and B's defection/burial fails.
>>>>
>>>> >/* 25 A>B /*>* 26 B>C
>>>> *>* 23 C>A
>>>> *>* 26 C
>>>> *>>* C>A 75-25, A>B 48-26, B>C 51-49*
>>>>
>>>> C wins with PMR (MR threshold is first place). B would win with most
>>>> other Condorcet methods.
>>>>
>>>> >/* 35 A /*>* 10 A=B
>>>> *>* 30 B>C (sincere B > A)
>>>> *>* 25 C
>>>> *>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
>>>>
>>>> *A wins with sincere voting. When B defects to try to win, which it
>>>> would do with most other Condorcet methods, B wins. With PMR, C wins,
>>>> an undesirable outcome for B.
>>>>
>>>> Here is another example from Rob LeGrand
>>>> (https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
>>>> good example for chicken dilemma resistance, but it does demonstrate
>>>> differences from Schulze, MMPO, RP and Bucklin:
>>>>
>>>> # example from method description page
>>>> 98:Abby>Cora>Erin>Dave>Brad
>>>> 64:Brad>Abby>Erin>Cora>Dave
>>>> 12:Brad>Abby>Erin>Dave>Cora
>>>> 98:Brad>Erin>Abby>Cora>Dave
>>>> 13:Brad>Erin>Abby>Dave>Cora
>>>> 125:Brad>Erin>Dave>Abby>Cora
>>>> 124:Cora>Abby>Erin>Dave>Brad
>>>> 76:Cora>Erin>Abby>Dave>Brad
>>>> 21:Dave>Abby>Brad>Erin>Cora
>>>> 30:Dave>Brad>Abby>Erin>Cora
>>>> 98:Dave>Brad>Erin>Cora>Abby
>>>> 139:Dave>Cora>Abby>Brad>Erin
>>>> 23:Dave>Cora>Brad>Abby>Erin
>>>>
>>>> The pairwise matrix:
>>>>
>>>> against
>>>> Abby Brad Cora Dave Erin
>>>> for Abby 458 461 485 511
>>>> Brad 463 461 312 623
>>>> Cora 460 460 460 460
>>>> Dave 436 609 461 311
>>>> Erin 410 298 461 610
>>>>
>>>> There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
>>>>
>>>> Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
>>>> wins with Bucklin.
>>>>
>>>> In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
>>>> the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
>>>> score of 484 is only slightly over the 50% mark of 460.5, so the Dave
>>>> voters hold the balance of power. Dave defeats Brad pairwise, so Dave
>>>> voters might not be as happy with a Brad victory, and Abby might be
>>>> able to persuade Dave voters to downrank Brad but not Abby. If
>>>> successful, Brad drops 44 points in MRScore and is no longer in the MR
>>>> threshold set. Abby defeats Erin, so Abby wins.
>>>>
>>>> * 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
>>>> 30:Dave>Brad>Abby>Erin>Cora
>>>> 98:Dave>Brad>Erin>Cora>Abby
>>>> 139:Dave>Cora>Abby>Brad>Erin
>>>> * 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
>>>>
>>>> PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
>>>> I believe it passes LNHelp. It probably fails Participation and IIA.
>>>> There are probably weird examples where changing one vote changes the
>>>> MR threshold. But overall, I think it has a good balance of incentive
>>>> to deter burial and deliberate cycles.
>>>>
>>>> Has Smith//Median Rating been proposed before? It seems like a simple
>>>> modification to MR on its own.
>>>
>>> ----
>>> Election-Methods mailing list - see
>>> https://electorama.com/em for list info
>>>
> ----
> Election-Methods mailing list - see https://electorama.com/em for
> list info
>
C
C.Benham
Sat, Jan 27, 2024 8:07 PM
I think a better technique is to either have an explicit cutoff which
is lowered per ballot so that max(Smith score) is approved, ..
If we have an explicit cutoff, I would think that for ballots that make
some ranking distinction among Smith-set members but no approval
distinction,
we should move the approval cutoff the minimum distance needed to make
some approval distinction.
But an explicit approval cutoff is an extra complication that isn't
shared by any of the current electoral reform proposals with any
traction or profile.
IRV/RCV uses plain strict ranking (from the top) ballots, while STAR and
"Majority Judgement" (I think both) use 6-slot ratings ballots. And
"Score"
advocates (at least used to) propose 0-99 score ballots (in effect
100-slot ratings ballots).
...or to make the cutoff such that candidates with max(Smith candidate
score)/2 or greater per ballot are approved.
I don't see that as any improvement on my suggestion. It looks a bit
crude and arbitrary.
Adding Smith candidate clones can change the average and thus the
approval cutoff would change. This seems unstable to me.
With enough ratings slots I can't see how it could be "unstable".
I also noticed that there were cases where Smith//ASM(implicit) would
get different results (better, IMO) than Smith//Implicit-approval.
That doesn't surprise me.
Chris
On 27/01/2024 5:40 am, Ted Stern wrote:
On Mon, Jan 22, 2024 at 7:07 PM C.Benham cbenham@adam.com.au wrote:
Ted,
...you didn't comment on whether ballots with all Smith
candidates below top rating should have their ratings bumped up:
i.e., D > E > A > blank > B (A and B in Smith) would be
recounted as A > blank > B.
I don't think I left anything ambiguous.
Assuming in your example we are using say 5-slot ratings ballots
then we interpret it as a score ballot thus: D5, E4, A3, B0.
If A and B are in Smith then the average score of candidates in
the Smith set is 3+0/2 = 1.5. Only A is scored above 1.5 so
only A is approved.
This is a nice technique, but it is not clone resistant. Adding Smith
candidate clones can change the average and thus the approval cutoff
would change. This seems unstable to me.
I think a better technique is to either have an explicit cutoff which
is lowered per ballot so that max(Smith score) is approved, or to make
the cutoff such that candidates with max(Smith candidate score)/2 or
greater per ballot are approved. I prefer the former.
I also noticed that there were cases where Smith//ASM(implicit)
would get different results (better, IMO) than
Smith//Implicit-approval.
What does "ASM" stand for?
Approval Sorted Margins
https://electowiki.org/wiki/Approval_Sorted_Margins
Chris
On 23/01/2024 10:56 am, Ted Stern wrote:
Chris:
Thanks for the clarifications, though you didn't comment on
whether ballots with all Smith candidates below top rating should
have their ratings bumped up: i.e., D > E > A > blank > B (A and
B in Smith) would be recounted as A > blank > B. I think this
makes the most sense as a voter whose favorites are eliminated
would want to ensure that their highest ranked Smith candidate is
counted as approved.
In general I agree with your comments, though I think
Condorcet//Approval with all ranked ballots approved is probably
not optimal, and Approval Sorted Margins with explicit approval
would be too complex for a public proposal. I'd be happy with
Condorcet//Top-ratings as a public proposal.
Smith//Implicit-approval seems to perform well in a number of
situations, but not appreciably better enough to make it worth
the effort of trying to get people to accept something more
complicated than Condorcet/Top-ratings. I also noticed that there
were cases where Smith//ASM(implicit) would get different results
(better, IMO) than Smith//Implicit-approval.
On Fri, Jan 19, 2024 at 4:05 PM C.Benham <cbenham@adam.com.au> wrote:
How is the average calculated?
We interpret the ratings ballots as score ballots, giving
zero points for the bottom rating (which is default for unrated),
1 point for the next highest, 2 points for the next above
that and so on.
Then for any given ballot we add up the scores of the
candidates in the Smith set and divide that by the number of
candidates
in the Smith set and interpret that ballot as approving those
Smith set candidates it scores higher than that average score.
That simulates the best approval strategy if the voters only
know which candidates are in the Smith set.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval?
Do you mean Approval Sorted Margins using ranking ballots
with an explicit approval cutoff?
Assuming yes, it uses a more expressive ballot, it is less
vulnerable to Defection strategy, and burial strategies are more
likely to have no effect rather than backfire.
In the method I proposed, omitting the ASM step and just
electing the candidate with the highest approval score (derived
as specified) would I concede make for a simpler method that
is nearly as good.
I worry a bit that with all methods that begin with
eliminating or disqualifying all candidates who aren't in the
Smith set or
just "elect the CW if there is one", over time if there is
never a top cycle then the top-cycle resolution method could stop
being taken seriously.
An attractive feature of ASM is that it is a Condorcet method
that a lot of the time would work fine without anyone needing
to know if there is top cycle or not.
If the Approval order is A>B>C and A pairwise beats B and B
pairwise beats C no-one needs to enquire about the pairwise
result between A and C.
If we want something super simple to explain and sell, then
Condorcet//Top Ratings and Condorcet//Approval (voted above
bottom)
are both not bad and much better than STAR.
Chris Benham
On 18/01/2024 10:13 am, Ted Stern wrote:
On Tue, Jan 16, 2024 at 7:27 AM C.Benham
<cbenham@adam.com.au> wrote:
Ted,
3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
Why not simply drop all ballots that make no distinction
among members of the Smith set?
I believe it passes LNHelp.
Douglas Woodall showed some time ago that Condorcet and
LNHelp are incompatible. I can't find
his proof, but it says so here:
https://en.wikipedia.org/wiki/Later-no-help_criterion
TheCondorcet criterion
<https://en.wikipedia.org/wiki/Condorcet_criterion>is
incompatible with later-no-help.
From your post again:
It probably fails Participation ..
It has been known (for a longer time) that Condorcet and
Participation are incompatible.
So since we know for sure that your method meets
Condorcet, we also know that it doesn't meet
Later-no-Help or Participation.
Using a multi-slot ratings ballot for a Condorcet method
of similar complexity I like:
*Eliminate all candidates not in the Smith set.
Interpret each ballot as giving approval to those
remaining candidates they rate above average (mean
of the ratings given to Smith-set members).
Now, using these approvals, elect the Margins-Sorted
Approval winner.*
It seems to me that Smith//Implicit-Approval or
Smith//implicit-approval-sorted-margins would be affected by
a couple of factors:
* How is the average calculated? Do you normalize scores?
In other words, if a ballot has non-Smith candidates in
the first, say, three ranks, do you up-rank the Smith
candidate scores on that ballot by three? Also, if there
are ranks below the top that contain only non-Smith
candidates, do you collapse those ranks or leave the
relative rank spacing on the ballot between Smith
candidates untouched?
* Approving Smith Candidates with scores above the mean
has similarities to Median Ratings. It would be more
similar and probably more stable to use the trimmed mean
-- drop the top and bottom 25% of scores. This would
give you an average score closer to the median.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval? I like ASM but fear it is probably
too complex for any advantage it gives you.
Has Smith//Median Rating been proposed before?
Not that I know of.
Chris Benham
*Ted Stern*dodecatheon at gmail.com
<mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
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------------------------------------------------------------------------
Continuing my search for a summable voting method that discourages burial
and defection, I've come across this hybrid of Condorcet and median ratings
that acts like Smith/Approval with an automatic approval cutoff. I'm
calling it Pairwise Median Rating (PMR), but it could also be described as
Smith//MR//Pairwise//MRScore:
1. Equal Ranking and ranking gap allowed (essentially a ratings method
with rank inferred). For purposes of this discussion, assume 6 slots (5
ranks above rejection).
2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
3. [Smith]
1. Compute the pairwise preference array
2. The winner is the candidate who defeats each other candidate
pairwise.
3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
4. [Median Rating]
1. Set the MR threshold to top rank.
2. While no Smith candidate has a majority of undropped ballots at or
above the threshold, set the threshold to the next lower rank,
until there
is no lower rank.
3. The winner is the single candidate that has a majority of
undropped ballots at or above the threshold.
5. [Pairwise]
1. Otherwise, if more than one candidate passes the threshold, look for
a pairwise beats-all candidate among candidates meeting the MR threshold.
(i.e. Condorcet on just the MR threshold set).
2. If there is one, you have a winner.
6. [MR Score]
1. Otherwise, the winner is the Smith set candidate with the largest
number of ballots at or above the Median Rating threshold (their MRscore).
This method is essentially Smith//Approval(explicit) with the approval
cutoff automatically inferred via median ratings
Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
considered optional, but by doing that, you ensure Immunity from Irrelevant
Ballots (IIB), aka the zero ballot problem that affects other Median Rating
/ Majority Judgment methods. In other words, the majority threshold is
unaffected by ballots that do not rank a viable candidate. It is possible
to do this summably if need be.
PMR either passes the Chicken Dilemma criterion without adjustment, or
there is a downranking strategy for defending against defection.
Consider the following examples from Chris Benham's post re MinLV(erw)
Sorted Margins (
http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
):
/* 46 A>B /> 44 B>C (sincere is B or B>A)
*>* 05 C>A
*>* 05 C>B
*>>* A>B 51-49, B>C 90-10, C>A 54-46.
*
With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
threshold is 2nd place, and A and B both pass the threshold. A defeats B,
so A is the winner and B's defection/burial fails.
/* 25 A>B /> 26 B>C
*>* 23 C>A
*>* 26 C
*>>* C>A 75-25, A>B 48-26, B>C 51-49*
C wins with PMR (MR threshold is first place). B would win with most
other Condorcet methods.
/* 35 A /> 10 A=B
*>* 30 B>C (sincere B > A)
*>* 25 C
*>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
*A wins with sincere voting. When B defects to try to win, which it
would do with most other Condorcet methods, B wins. With PMR, C wins,
an undesirable outcome for B.
Here is another example from Rob LeGrand
(https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
good example for chicken dilemma resistance, but it does demonstrate
differences from Schulze, MMPO, RP and Bucklin:
# example from method description page
98:Abby>Cora>Erin>Dave>Brad
64:Brad>Abby>Erin>Cora>Dave
12:Brad>Abby>Erin>Dave>Cora
98:Brad>Erin>Abby>Cora>Dave
13:Brad>Erin>Abby>Dave>Cora
125:Brad>Erin>Dave>Abby>Cora
124:Cora>Abby>Erin>Dave>Brad
76:Cora>Erin>Abby>Dave>Brad
21:Dave>Abby>Brad>Erin>Cora
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin
23:Dave>Cora>Brad>Abby>Erin
The pairwise matrix:
against
Abby Brad Cora Dave Erin
for Abby 458 461 485 511
Brad 463 461 312 623
Cora 460 460 460 460
Dave 436 609 461 311
Erin 410 298 461 610
There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
wins with Bucklin.
In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
score of 484 is only slightly over the 50% mark of 460.5, so the Dave
voters hold the balance of power. Dave defeats Brad pairwise, so Dave
voters might not be as happy with a Brad victory, and Abby might be
able to persuade Dave voters to downrank Brad but not Abby. If
successful, Brad drops 44 points in MRScore and is no longer in the MR
threshold set. Abby defeats Erin, so Abby wins.
* 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin
* 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
I believe it passes LNHelp. It probably fails Participation and IIA.
There are probably weird examples where changing one vote changes the
MR threshold. But overall, I think it has a good balance of incentive
to deter burial and deliberate cycles.
Has Smith//Median Rating been proposed before? It seems like a simple
modification to MR on its own.
----
Election-Methods mailing list - see
https://electorama.com/em for list info
----
Election-Methods mailing list - see https://electorama.com/em for
list info
> I think a better technique is to either have an explicit cutoff which
> is lowered per ballot so that max(Smith score) is approved, ..
If we have an explicit cutoff, I would think that for ballots that make
some ranking distinction among Smith-set members but no approval
distinction,
we should move the approval cutoff the minimum distance needed to make
some approval distinction.
But an explicit approval cutoff is an extra complication that isn't
shared by any of the current electoral reform proposals with any
traction or profile.
IRV/RCV uses plain strict ranking (from the top) ballots, while STAR and
"Majority Judgement" (I think both) use 6-slot ratings ballots. And
"Score"
advocates (at least used to) propose 0-99 score ballots (in effect
100-slot ratings ballots).
> ...or to make the cutoff such that candidates with max(Smith candidate
> score)/2 or greater per ballot are approved.
I don't see that as any improvement on my suggestion. It looks a bit
crude and arbitrary.
> Adding Smith candidate clones can change the average and thus the
> approval cutoff would change. This seems unstable to me.
With enough ratings slots I can't see how it could be "unstable".
> I also noticed that there were cases where Smith//ASM(implicit) would
> get different results (better, IMO) than Smith//Implicit-approval.
That doesn't surprise me.
Chris
On 27/01/2024 5:40 am, Ted Stern wrote:
>
>
> On Mon, Jan 22, 2024 at 7:07 PM C.Benham <cbenham@adam.com.au> wrote:
>
> Ted,
>
>> ...you didn't comment on whether ballots with all Smith
>> candidates below top rating should have their ratings bumped up:
>> i.e., D > E > A > blank > B (A and B in Smith) would be
>> recounted as A > blank > B.
>
> I don't think I left anything ambiguous.
>
> Assuming in your example we are using say 5-slot ratings ballots
> then we interpret it as a score ballot thus: D5, E4, A3, B0.
>
> If A and B are in Smith then the average score of candidates in
> the Smith set is 3+0/2 = 1.5. Only A is scored above 1.5 so
> only A is approved.
>
>
> This is a nice technique, but it is not clone resistant. Adding Smith
> candidate clones can change the average and thus the approval cutoff
> would change. This seems unstable to me.
>
> I think a better technique is to either have an explicit cutoff which
> is lowered per ballot so that max(Smith score) is approved, or to make
> the cutoff such that candidates with max(Smith candidate score)/2 or
> greater per ballot are approved. I prefer the former.
>
>> I also noticed that there were cases where Smith//ASM(implicit)
>> would get different results (better, IMO) than
>> Smith//Implicit-approval.
>>
>
> What does "ASM" stand for?
>
>
> Approval Sorted Margins
> <https://electowiki.org/wiki/Approval_Sorted_Margins>
>
>
> Chris
>
>
> On 23/01/2024 10:56 am, Ted Stern wrote:
>> Chris:
>>
>> Thanks for the clarifications, though you didn't comment on
>> whether ballots with all Smith candidates below top rating should
>> have their ratings bumped up: i.e., D > E > A > blank > B (A and
>> B in Smith) would be recounted as A > blank > B. I think this
>> makes the most sense as a voter whose favorites are eliminated
>> would want to ensure that their highest ranked Smith candidate is
>> counted as approved.
>>
>> In general I agree with your comments, though I think
>> Condorcet//Approval with all ranked ballots approved is probably
>> not optimal, and Approval Sorted Margins with explicit approval
>> would be too complex for a public proposal. I'd be happy with
>> Condorcet//Top-ratings as a public proposal.
>>
>> Smith//Implicit-approval seems to perform well in a number of
>> situations, but not appreciably better enough to make it worth
>> the effort of trying to get people to accept something more
>> complicated than Condorcet/Top-ratings. I also noticed that there
>> were cases where Smith//ASM(implicit) would get different results
>> (better, IMO) than Smith//Implicit-approval.
>>
>>
>> On Fri, Jan 19, 2024 at 4:05 PM C.Benham <cbenham@adam.com.au> wrote:
>>
>>> How is the average calculated?
>>
>> We interpret the ratings ballots as score ballots, giving
>> zero points for the bottom rating (which is default for unrated),
>> 1 point for the next highest, 2 points for the next above
>> that and so on.
>>
>> Then for any given ballot we add up the scores of the
>> candidates in the Smith set and divide that by the number of
>> candidates
>> in the Smith set and interpret that ballot as approving those
>> Smith set candidates it scores higher than that average score.
>>
>> That simulates the best approval strategy if the voters only
>> know which candidates are in the Smith set.
>>
>>> What advantage does Approval Sorted Margins have over
>>> Smith//Implicit-Approval?
>>
>> Do you mean Approval Sorted Margins using ranking ballots
>> with an explicit approval cutoff?
>>
>> Assuming yes, it uses a more expressive ballot, it is less
>> vulnerable to Defection strategy, and burial strategies are more
>> likely to have no effect rather than backfire.
>>
>> In the method I proposed, omitting the ASM step and just
>> electing the candidate with the highest approval score (derived
>> as specified) would I concede make for a simpler method that
>> is nearly as good.
>>
>> I worry a bit that with all methods that begin with
>> eliminating or disqualifying all candidates who aren't in the
>> Smith set or
>> just "elect the CW if there is one", over time if there is
>> never a top cycle then the top-cycle resolution method could stop
>> being taken seriously.
>>
>> An attractive feature of ASM is that it is a Condorcet method
>> that a lot of the time would work fine without anyone needing
>> to know if there is top cycle or not.
>>
>> If the Approval order is A>B>C and A pairwise beats B and B
>> pairwise beats C no-one needs to enquire about the pairwise
>> result between A and C.
>>
>> If we want something super simple to explain and sell, then
>> Condorcet//Top Ratings and Condorcet//Approval (voted above
>> bottom)
>> are both not bad and much better than STAR.
>>
>> Chris Benham
>>
>>
>>
>> On 18/01/2024 10:13 am, Ted Stern wrote:
>>>
>>>
>>> On Tue, Jan 16, 2024 at 7:27 AM C.Benham
>>> <cbenham@adam.com.au> wrote:
>>>
>>> Ted,
>>>
>>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>>> any member of the Smith Set.
>>>
>>> Why not simply drop all ballots that make no distinction
>>> among members of the Smith set?
>>>
>>>> I believe it passes LNHelp.
>>>
>>> Douglas Woodall showed some time ago that Condorcet and
>>> LNHelp are incompatible. I can't find
>>> his proof, but it says so here:
>>>
>>> https://en.wikipedia.org/wiki/Later-no-help_criterion
>>>
>>>> TheCondorcet criterion
>>>> <https://en.wikipedia.org/wiki/Condorcet_criterion>is
>>>> incompatible with later-no-help.
>>>
>>> From your post again:
>>>> It probably fails Participation ..
>>>
>>> It has been known (for a longer time) that Condorcet and
>>> Participation are incompatible.
>>>
>>> So since we know for sure that your method meets
>>> Condorcet, we also know that it doesn't meet
>>> Later-no-Help or Participation.
>>>
>>> Using a multi-slot ratings ballot for a Condorcet method
>>> of similar complexity I like:
>>>
>>> *Eliminate all candidates not in the Smith set.
>>>
>>> Interpret each ballot as giving approval to those
>>> remaining candidates they rate above average (mean
>>> of the ratings given to Smith-set members).
>>>
>>> Now, using these approvals, elect the Margins-Sorted
>>> Approval winner.*
>>>
>>>
>>> It seems to me that Smith//Implicit-Approval or
>>> Smith//implicit-approval-sorted-margins would be affected by
>>> a couple of factors:
>>>
>>> * How is the average calculated? Do you normalize scores?
>>> In other words, if a ballot has non-Smith candidates in
>>> the first, say, three ranks, do you up-rank the Smith
>>> candidate scores on that ballot by three? Also, if there
>>> are ranks below the top that contain only non-Smith
>>> candidates, do you collapse those ranks or leave the
>>> relative rank spacing on the ballot between Smith
>>> candidates untouched?
>>> * Approving Smith Candidates with scores above the mean
>>> has similarities to Median Ratings. It would be more
>>> similar and probably more stable to use the trimmed mean
>>> -- drop the top and bottom 25% of scores. This would
>>> give you an average score closer to the median.
>>>
>>> What advantage does Approval Sorted Margins have over
>>> Smith//Implicit-Approval? I like ASM but fear it is probably
>>> too complex for any advantage it gives you.
>>>
>>>
>>>> Has Smith//Median Rating been proposed before?
>>> Not that I know of.
>>>
>>> Chris Benham
>>>
>>>
>>>
>>>>
>>>> *Ted Stern*dodecatheon at gmail.com
>>>> <mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
>>>> /Tue Jan 2 15:12:26 PST 2024/
>>>>
>>>> * Previous message (by thread):[EM] [Game Theory]
>>>> Iterated Prisoners' Dilemma as a voting method
>>>> metric
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005215.html>
>>>> * Next message (by thread):[EM] Pairwise Median
>>>> Rating
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005218.html>
>>>> * *Messages sorted by:*[ date ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/date.html#5216>[
>>>> thread ]
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>>>> subject ]
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>>>> author ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/author.html#5216>
>>>>
>>>> ------------------------------------------------------------------------
>>>> Continuing my search for a summable voting method that discourages burial
>>>> and defection, I've come across this hybrid of Condorcet and median ratings
>>>> that acts like Smith/Approval with an automatic approval cutoff. I'm
>>>> calling it Pairwise Median Rating (PMR), but it could also be described as
>>>> Smith//MR//Pairwise//MRScore:
>>>>
>>>> 1. Equal Ranking and ranking gap allowed (essentially a ratings method
>>>> with rank inferred). For purposes of this discussion, assume 6 slots (5
>>>> ranks above rejection).
>>>> 2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
>>>> means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
>>>> of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
>>>> 3. [Smith]
>>>> 1. Compute the pairwise preference array
>>>> 2. The winner is the candidate who defeats each other candidate
>>>> pairwise.
>>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>>> any member of the Smith Set.
>>>> 4. [Median Rating]
>>>> 1. Set the MR threshold to top rank.
>>>> 2. While no Smith candidate has a majority of undropped ballots at or
>>>> above the threshold, set the threshold to the next lower rank,
>>>> until there
>>>> is no lower rank.
>>>> 3. The winner is the single candidate that has a majority of
>>>> undropped ballots at or above the threshold.
>>>> 5. [Pairwise]
>>>> 1. Otherwise, if more than one candidate passes the threshold, look for
>>>> a pairwise beats-all candidate among candidates meeting the MR threshold.
>>>> (i.e. Condorcet on just the MR threshold set).
>>>> 2. If there is one, you have a winner.
>>>> 6. [MR Score]
>>>> 1. Otherwise, the winner is the Smith set candidate with the largest
>>>> number of ballots at or above the Median Rating threshold (their MRscore).
>>>>
>>>> This method is essentially Smith//Approval(explicit) with the approval
>>>> cutoff automatically inferred via median ratings
>>>>
>>>> Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
>>>> considered optional, but by doing that, you ensure Immunity from Irrelevant
>>>> Ballots (IIB), aka the zero ballot problem that affects other Median Rating
>>>> / Majority Judgment methods. In other words, the majority threshold is
>>>> unaffected by ballots that do not rank a viable candidate. It is possible
>>>> to do this summably if need be.
>>>>
>>>> PMR either passes the Chicken Dilemma criterion without adjustment, or
>>>> there is a downranking strategy for defending against defection.
>>>>
>>>> Consider the following examples from Chris Benham's post re MinLV(erw)
>>>> Sorted Margins (
>>>> http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
>>>> ):
>>>>
>>>> >/* 46 A>B /*>* 44 B>C (sincere is B or B>A)
>>>> *>* 05 C>A
>>>> *>* 05 C>B
>>>> *>>* A>B 51-49, B>C 90-10, C>A 54-46.
>>>> *
>>>>
>>>> With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
>>>> there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
>>>> threshold is 2nd place, and A and B both pass the threshold. A defeats B,
>>>> so A is the winner and B's defection/burial fails.
>>>>
>>>> >/* 25 A>B /*>* 26 B>C
>>>> *>* 23 C>A
>>>> *>* 26 C
>>>> *>>* C>A 75-25, A>B 48-26, B>C 51-49*
>>>>
>>>> C wins with PMR (MR threshold is first place). B would win with most
>>>> other Condorcet methods.
>>>>
>>>> >/* 35 A /*>* 10 A=B
>>>> *>* 30 B>C (sincere B > A)
>>>> *>* 25 C
>>>> *>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
>>>>
>>>> *A wins with sincere voting. When B defects to try to win, which it
>>>> would do with most other Condorcet methods, B wins. With PMR, C wins,
>>>> an undesirable outcome for B.
>>>>
>>>> Here is another example from Rob LeGrand
>>>> (https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
>>>> good example for chicken dilemma resistance, but it does demonstrate
>>>> differences from Schulze, MMPO, RP and Bucklin:
>>>>
>>>> # example from method description page
>>>> 98:Abby>Cora>Erin>Dave>Brad
>>>> 64:Brad>Abby>Erin>Cora>Dave
>>>> 12:Brad>Abby>Erin>Dave>Cora
>>>> 98:Brad>Erin>Abby>Cora>Dave
>>>> 13:Brad>Erin>Abby>Dave>Cora
>>>> 125:Brad>Erin>Dave>Abby>Cora
>>>> 124:Cora>Abby>Erin>Dave>Brad
>>>> 76:Cora>Erin>Abby>Dave>Brad
>>>> 21:Dave>Abby>Brad>Erin>Cora
>>>> 30:Dave>Brad>Abby>Erin>Cora
>>>> 98:Dave>Brad>Erin>Cora>Abby
>>>> 139:Dave>Cora>Abby>Brad>Erin
>>>> 23:Dave>Cora>Brad>Abby>Erin
>>>>
>>>> The pairwise matrix:
>>>>
>>>> against
>>>> Abby Brad Cora Dave Erin
>>>> for Abby 458 461 485 511
>>>> Brad 463 461 312 623
>>>> Cora 460 460 460 460
>>>> Dave 436 609 461 311
>>>> Erin 410 298 461 610
>>>>
>>>> There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
>>>>
>>>> Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
>>>> wins with Bucklin.
>>>>
>>>> In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
>>>> the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
>>>> score of 484 is only slightly over the 50% mark of 460.5, so the Dave
>>>> voters hold the balance of power. Dave defeats Brad pairwise, so Dave
>>>> voters might not be as happy with a Brad victory, and Abby might be
>>>> able to persuade Dave voters to downrank Brad but not Abby. If
>>>> successful, Brad drops 44 points in MRScore and is no longer in the MR
>>>> threshold set. Abby defeats Erin, so Abby wins.
>>>>
>>>> * 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
>>>> 30:Dave>Brad>Abby>Erin>Cora
>>>> 98:Dave>Brad>Erin>Cora>Abby
>>>> 139:Dave>Cora>Abby>Brad>Erin
>>>> * 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
>>>>
>>>> PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
>>>> I believe it passes LNHelp. It probably fails Participation and IIA.
>>>> There are probably weird examples where changing one vote changes the
>>>> MR threshold. But overall, I think it has a good balance of incentive
>>>> to deter burial and deliberate cycles.
>>>>
>>>> Has Smith//Median Rating been proposed before? It seems like a simple
>>>> modification to MR on its own.
>>>
>>> ----
>>> Election-Methods mailing list - see
>>> https://electorama.com/em for list info
>>>
> ----
> Election-Methods mailing list - see https://electorama.com/em for
> list info
>
TS
Ted Stern
Mon, Jan 29, 2024 6:31 PM
On Fri, Jan 26, 2024 at 4:41 PM C.Benham cbenham@adam.com.au wrote:
Of course! Thanks. Pardon the brain fade.
Normally Smith sets don't have two members, but your example didn't
mention a third member or say how many candidates there are or how many
rating slots the ballots have.
So maybe there is another candidate F (which is in the Smith set) and the
ballots have 6 slots, so our Score ballot interpretation becomes
D6, E5, A4, B2, F0.
So then our calculation becomes 4+2+0/3 = 2.
Only A is scored higher than 2, so the ballot only approves A.
I envisioned a couple of different scenarios:
D6, E5, A4, B2 Fa0, Fb0, Fc0
with A, B, and F* in Smith.
Then (4 + 2 + 0 + 0 + 0)/5 = 1.2, so both A and B are approved, with only
Fb and Fc clones crowded in.
Similarly, if A clones are added, one might have
D6, E5, Aa4, Ab4, Ac4, B2, F0
with A*, B, and F in Smith.
Then (4 + 4 + 4 + 2 + 0)/3 = 3.66..., so only A* clones are approved, not B.
This is a nice technique, but it is not clone resistant. Adding Smith
candidate clones can change the average and thus the approval cutoff would
change. This seems unstable to me.
With ratings ballots I think only candidates that get the same rating on
every ballot are considered clones. And besides that I think that more
than three candidates in the Smith set would be vanishingly rare.
Whether its rare or not, as soon as you lose clone independence, the method
is opened to criticism.
Chris
On 27/01/2024 5:40 am, Ted Stern wrote:
On Mon, Jan 22, 2024 at 7:07 PM C.Benham cbenham@adam.com.au wrote:
Ted,
...you didn't comment on whether ballots with all Smith candidates below
top rating should have their ratings bumped up: i.e., D > E > A > blank >
B (A and B in Smith) would be recounted as A > blank > B.
I don't think I left anything ambiguous.
Assuming in your example we are using say 5-slot ratings ballots then we
interpret it as a score ballot thus: D5, E4, A3, B0.
If A and B are in Smith then the average score of candidates in the Smith
set is 3+0/2 = 1.5. Only A is scored above 1.5 so only A is approved.
This is a nice technique, but it is not clone resistant. Adding Smith
candidate clones can change the average and thus the approval cutoff would
change. This seems unstable to me.
I think a better technique is to either have an explicit cutoff which is
lowered per ballot so that max(Smith score) is approved, or to make the
cutoff such that candidates with max(Smith candidate score)/2 or greater
per ballot are approved. I prefer the former.
I also noticed that there were cases where Smith//ASM(implicit) would get
different results (better, IMO) than Smith//Implicit-approval.
What does "ASM" stand for?
Approval Sorted Margins
https://electowiki.org/wiki/Approval_Sorted_Margins
Chris
On 23/01/2024 10:56 am, Ted Stern wrote:
Chris:
Thanks for the clarifications, though you didn't comment on whether
ballots with all Smith candidates below top rating should have their
ratings bumped up: i.e., D > E > A > blank > B (A and B in Smith) would be
recounted as A > blank > B. I think this makes the most sense as a voter
whose favorites are eliminated would want to ensure that their highest
ranked Smith candidate is counted as approved.
In general I agree with your comments, though I think Condorcet//Approval
with all ranked ballots approved is probably not optimal, and Approval
Sorted Margins with explicit approval would be too complex for a public
proposal. I'd be happy with Condorcet//Top-ratings as a public proposal.
Smith//Implicit-approval seems to perform well in a number of situations,
but not appreciably better enough to make it worth the effort of trying to
get people to accept something more complicated than Condorcet/Top-ratings.
I also noticed that there were cases where Smith//ASM(implicit) would get
different results (better, IMO) than Smith//Implicit-approval.
On Fri, Jan 19, 2024 at 4:05 PM C.Benham cbenham@adam.com.au wrote:
How is the average calculated?
We interpret the ratings ballots as score ballots, giving zero points
for the bottom rating (which is default for unrated),
1 point for the next highest, 2 points for the next above that and so
on.
Then for any given ballot we add up the scores of the candidates in the
Smith set and divide that by the number of candidates
in the Smith set and interpret that ballot as approving those Smith set
candidates it scores higher than that average score.
That simulates the best approval strategy if the voters only know which
candidates are in the Smith set.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval?
Do you mean Approval Sorted Margins using ranking ballots with an
explicit approval cutoff?
Assuming yes, it uses a more expressive ballot, it is less vulnerable to
Defection strategy, and burial strategies are more
likely to have no effect rather than backfire.
In the method I proposed, omitting the ASM step and just electing the
candidate with the highest approval score (derived
as specified) would I concede make for a simpler method that is nearly
as good.
I worry a bit that with all methods that begin with eliminating or
disqualifying all candidates who aren't in the Smith set or
just "elect the CW if there is one", over time if there is never a top
cycle then the top-cycle resolution method could stop
being taken seriously.
An attractive feature of ASM is that it is a Condorcet method that a lot
of the time would work fine without anyone needing
to know if there is top cycle or not.
If the Approval order is A>B>C and A pairwise beats B and B pairwise
beats C no-one needs to enquire about the pairwise
result between A and C.
If we want something super simple to explain and sell, then
Condorcet//Top Ratings and Condorcet//Approval (voted above bottom)
are both not bad and much better than STAR.
Chris Benham
On 18/01/2024 10:13 am, Ted Stern wrote:
On Tue, Jan 16, 2024 at 7:27 AM C.Benham cbenham@adam.com.au wrote:
Ted,
- Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
Why not simply drop all ballots that make no distinction among members
of the Smith set?
I believe it passes LNHelp.
Douglas Woodall showed some time ago that Condorcet and LNHelp are
incompatible. I can't find
his proof, but it says so here:
https://en.wikipedia.org/wiki/Later-no-help_criterion
The Condorcet criterion
https://en.wikipedia.org/wiki/Condorcet_criterion is incompatible
with later-no-help.
From your post again:
It probably fails Participation ..
It has been known (for a longer time) that Condorcet and Participation
are incompatible.
So since we know for sure that your method meets Condorcet, we also
know that it doesn't meet
Later-no-Help or Participation.
Using a multi-slot ratings ballot for a Condorcet method of similar
complexity I like:
*Eliminate all candidates not in the Smith set.
Interpret each ballot as giving approval to those remaining candidates
they rate above average (mean
of the ratings given to Smith-set members).
Now, using these approvals, elect the Margins-Sorted Approval winner.*
It seems to me that Smith//Implicit-Approval or
Smith//implicit-approval-sorted-margins would be affected by a couple of
factors:
- How is the average calculated? Do you normalize scores? In other
words, if a ballot has non-Smith candidates in the first, say, three ranks,
do you up-rank the Smith candidate scores on that ballot by three? Also, if
there are ranks below the top that contain only non-Smith candidates, do
you collapse those ranks or leave the relative rank spacing on the ballot
between Smith candidates untouched?
- Approving Smith Candidates with scores above the mean has
similarities to Median Ratings. It would be more similar and probably more
stable to use the trimmed mean -- drop the top and bottom 25% of scores.
This would give you an average score closer to the median.
What advantage does Approval Sorted Margins have over
Smith//Implicit-Approval? I like ASM but fear it is probably too complex
for any advantage it gives you.
Has Smith//Median Rating been proposed before?
Not that I know of.
Chris Benham
Ted Stern dodecatheon at gmail.com
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Continuing my search for a summable voting method that discourages burial
and defection, I've come across this hybrid of Condorcet and median ratings
that acts like Smith/Approval with an automatic approval cutoff. I'm
calling it Pairwise Median Rating (PMR), but it could also be described as
Smith//MR//Pairwise//MRScore:
1. Equal Ranking and ranking gap allowed (essentially a ratings method
with rank inferred). For purposes of this discussion, assume 6 slots (5
ranks above rejection).
2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
3. [Smith]
1. Compute the pairwise preference array
2. The winner is the candidate who defeats each other candidate
pairwise.
3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
4. [Median Rating]
1. Set the MR threshold to top rank.
2. While no Smith candidate has a majority of undropped ballots at or
above the threshold, set the threshold to the next lower rank,
until there
is no lower rank.
3. The winner is the single candidate that has a majority of
undropped ballots at or above the threshold.
5. [Pairwise]
1. Otherwise, if more than one candidate passes the threshold, look for
a pairwise beats-all candidate among candidates meeting the MR threshold.
(i.e. Condorcet on just the MR threshold set).
2. If there is one, you have a winner.
6. [MR Score]
1. Otherwise, the winner is the Smith set candidate with the largest
number of ballots at or above the Median Rating threshold (their MRscore).
This method is essentially Smith//Approval(explicit) with the approval
cutoff automatically inferred via median ratings
Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
considered optional, but by doing that, you ensure Immunity from Irrelevant
Ballots (IIB), aka the zero ballot problem that affects other Median Rating
/ Majority Judgment methods. In other words, the majority threshold is
unaffected by ballots that do not rank a viable candidate. It is possible
to do this summably if need be.
PMR either passes the Chicken Dilemma criterion without adjustment, or
there is a downranking strategy for defending against defection.
Consider the following examples from Chris Benham's post re MinLV(erw)
Sorted Margins (http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
):
** 46 A>B
*> 44 B>C (sincere is B or B>A)
> 05 C>A
> 05 C>B
>> A>B 51-49, B>C 90-10, C>A 54-46.
*
With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
threshold is 2nd place, and A and B both pass the threshold. A defeats B,
so A is the winner and B's defection/burial fails.
** 25 A>B
> 26 B>C
> 23 C>A
> 26 C
>> C>A 75-25, A>B 48-26, B>C 51-49
C wins with PMR (MR threshold is first place). B would win with most
other Condorcet methods.
** 35 A
*> 10 A=B
> 30 B>C (sincere B > A)
> 25 C
>> C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
*A wins with sincere voting. When B defects to try to win, which it
would do with most other Condorcet methods, B wins. With PMR, C wins,
an undesirable outcome for B.
Here is another example from Rob LeGrand
(https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
good example for chicken dilemma resistance, but it does demonstrate
differences from Schulze, MMPO, RP and Bucklin:
example from method description page
98:Abby>Cora>Erin>Dave>Brad
64:Brad>Abby>Erin>Cora>Dave
12:Brad>Abby>Erin>Dave>Cora
98:Brad>Erin>Abby>Cora>Dave
13:Brad>Erin>Abby>Dave>Cora
125:Brad>Erin>Dave>Abby>Cora
124:Cora>Abby>Erin>Dave>Brad
76:Cora>Erin>Abby>Dave>Brad
21:Dave>Abby>Brad>Erin>Cora
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin
23:Dave>Cora>Brad>Abby>Erin
The pairwise matrix:
against
Abby Brad Cora Dave Erin
for Abby 458 461 485 511
Brad 463 461 312 623
Cora 460 460 460 460
Dave 436 609 461 311
Erin 410 298 461 610
There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
wins with Bucklin.
In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
score of 484 is only slightly over the 50% mark of 460.5, so the Dave
voters hold the balance of power. Dave defeats Brad pairwise, so Dave
voters might not be as happy with a Brad victory, and Abby might be
able to persuade Dave voters to downrank Brad but not Abby. If
successful, Brad drops 44 points in MRScore and is no longer in the MR
threshold set. Abby defeats Erin, so Abby wins.
- 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
30:Dave>Brad>Abby>Erin>Cora
98:Dave>Brad>Erin>Cora>Abby
139:Dave>Cora>Abby>Brad>Erin - 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
I believe it passes LNHelp. It probably fails Participation and IIA.
There are probably weird examples where changing one vote changes the
MR threshold. But overall, I think it has a good balance of incentive
to deter burial and deliberate cycles.
Has Smith//Median Rating been proposed before? It seems like a simple
modification to MR on its own.
Election-Methods mailing list - see https://electorama.com/em for list
info
Election-Methods mailing list - see https://electorama.com/em for list
info
On Fri, Jan 26, 2024 at 4:41 PM C.Benham <cbenham@adam.com.au> wrote:
> Of course! Thanks. Pardon the brain fade.
>
> Normally Smith sets don't have two members, but your example didn't
> mention a third member or say how many candidates there are or how many
> rating slots the ballots have.
>
> So maybe there is another candidate F (which is in the Smith set) and the
> ballots have 6 slots, so our Score ballot interpretation becomes
>
> D6, E5, A4, B2, F0.
>
> So then our calculation becomes 4+2+0/3 = 2.
>
> Only A is scored higher than 2, so the ballot only approves A.
>
I envisioned a couple of different scenarios:
D6, E5, A4, B2 Fa0, Fb0, Fc0
with A, B, and F* in Smith.
Then (4 + 2 + 0 + 0 + 0)/5 = 1.2, so both A and B are approved, with only
Fb and Fc clones crowded in.
Similarly, if A clones are added, one might have
D6, E5, Aa4, Ab4, Ac4, B2, F0
with A*, B, and F in Smith.
Then (4 + 4 + 4 + 2 + 0)/3 = 3.66..., so only A* clones are approved, not B.
>
> This is a nice technique, but it is not clone resistant. Adding Smith
> candidate clones can change the average and thus the approval cutoff would
> change. This seems unstable to me.
>
>
> With ratings ballots I think only candidates that get the same rating on
> every ballot are considered clones. And besides that I think that more
> than three candidates in the Smith set would be vanishingly rare.
>
Whether its rare or not, as soon as you lose clone independence, the method
is opened to criticism.
>
>
> Chris
>
>
>
> On 27/01/2024 5:40 am, Ted Stern wrote:
>
>
>
> On Mon, Jan 22, 2024 at 7:07 PM C.Benham <cbenham@adam.com.au> wrote:
>
>> Ted,
>>
>> ...you didn't comment on whether ballots with all Smith candidates below
>> top rating should have their ratings bumped up: i.e., D > E > A > blank >
>> B (A and B in Smith) would be recounted as A > blank > B.
>>
>>
>> I don't think I left anything ambiguous.
>>
>> Assuming in your example we are using say 5-slot ratings ballots then we
>> interpret it as a score ballot thus: D5, E4, A3, B0.
>>
>> If A and B are in Smith then the average score of candidates in the Smith
>> set is 3+0/2 = 1.5. Only A is scored above 1.5 so only A is approved.
>>
>>
> This is a nice technique, but it is not clone resistant. Adding Smith
> candidate clones can change the average and thus the approval cutoff would
> change. This seems unstable to me.
>
> I think a better technique is to either have an explicit cutoff which is
> lowered per ballot so that max(Smith score) is approved, or to make the
> cutoff such that candidates with max(Smith candidate score)/2 or greater
> per ballot are approved. I prefer the former.
>
>> I also noticed that there were cases where Smith//ASM(implicit) would get
>> different results (better, IMO) than Smith//Implicit-approval.
>>
>>
>> What does "ASM" stand for?
>>
>
> Approval Sorted Margins
> <https://electowiki.org/wiki/Approval_Sorted_Margins>
>
>
>>
>> Chris
>>
>>
>> On 23/01/2024 10:56 am, Ted Stern wrote:
>>
>> Chris:
>>
>> Thanks for the clarifications, though you didn't comment on whether
>> ballots with all Smith candidates below top rating should have their
>> ratings bumped up: i.e., D > E > A > blank > B (A and B in Smith) would be
>> recounted as A > blank > B. I think this makes the most sense as a voter
>> whose favorites are eliminated would want to ensure that their highest
>> ranked Smith candidate is counted as approved.
>>
>> In general I agree with your comments, though I think Condorcet//Approval
>> with all ranked ballots approved is probably not optimal, and Approval
>> Sorted Margins with explicit approval would be too complex for a public
>> proposal. I'd be happy with Condorcet//Top-ratings as a public proposal.
>>
>> Smith//Implicit-approval seems to perform well in a number of situations,
>> but not appreciably better enough to make it worth the effort of trying to
>> get people to accept something more complicated than Condorcet/Top-ratings.
>> I also noticed that there were cases where Smith//ASM(implicit) would get
>> different results (better, IMO) than Smith//Implicit-approval.
>>
>>
>> On Fri, Jan 19, 2024 at 4:05 PM C.Benham <cbenham@adam.com.au> wrote:
>>
>>> How is the average calculated?
>>>
>>>
>>> We interpret the ratings ballots as score ballots, giving zero points
>>> for the bottom rating (which is default for unrated),
>>> 1 point for the next highest, 2 points for the next above that and so
>>> on.
>>>
>>> Then for any given ballot we add up the scores of the candidates in the
>>> Smith set and divide that by the number of candidates
>>> in the Smith set and interpret that ballot as approving those Smith set
>>> candidates it scores higher than that average score.
>>>
>>> That simulates the best approval strategy if the voters only know which
>>> candidates are in the Smith set.
>>>
>>> What advantage does Approval Sorted Margins have over
>>> Smith//Implicit-Approval?
>>>
>>>
>>> Do you mean Approval Sorted Margins using ranking ballots with an
>>> explicit approval cutoff?
>>>
>>> Assuming yes, it uses a more expressive ballot, it is less vulnerable to
>>> Defection strategy, and burial strategies are more
>>> likely to have no effect rather than backfire.
>>>
>>> In the method I proposed, omitting the ASM step and just electing the
>>> candidate with the highest approval score (derived
>>> as specified) would I concede make for a simpler method that is nearly
>>> as good.
>>>
>>> I worry a bit that with all methods that begin with eliminating or
>>> disqualifying all candidates who aren't in the Smith set or
>>> just "elect the CW if there is one", over time if there is never a top
>>> cycle then the top-cycle resolution method could stop
>>> being taken seriously.
>>>
>>> An attractive feature of ASM is that it is a Condorcet method that a lot
>>> of the time would work fine without anyone needing
>>> to know if there is top cycle or not.
>>>
>>> If the Approval order is A>B>C and A pairwise beats B and B pairwise
>>> beats C no-one needs to enquire about the pairwise
>>> result between A and C.
>>>
>>> If we want something super simple to explain and sell, then
>>> Condorcet//Top Ratings and Condorcet//Approval (voted above bottom)
>>> are both not bad and much better than STAR.
>>>
>>> Chris Benham
>>>
>>>
>>>
>>> On 18/01/2024 10:13 am, Ted Stern wrote:
>>>
>>>
>>>
>>> On Tue, Jan 16, 2024 at 7:27 AM C.Benham <cbenham@adam.com.au> wrote:
>>>
>>>> Ted,
>>>>
>>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>>> any member of the Smith Set.
>>>>
>>>> Why not simply drop all ballots that make no distinction among members
>>>> of the Smith set?
>>>>
>>>> I believe it passes LNHelp.
>>>>
>>>>
>>>> Douglas Woodall showed some time ago that Condorcet and LNHelp are
>>>> incompatible. I can't find
>>>> his proof, but it says so here:
>>>>
>>>> https://en.wikipedia.org/wiki/Later-no-help_criterion
>>>>
>>>> The Condorcet criterion
>>>> <https://en.wikipedia.org/wiki/Condorcet_criterion> is incompatible
>>>> with later-no-help.
>>>>
>>>>
>>>> From your post again:
>>>>
>>>> It probably fails Participation ..
>>>>
>>>>
>>>> It has been known (for a longer time) that Condorcet and Participation
>>>> are incompatible.
>>>>
>>>> So since we know for sure that your method meets Condorcet, we also
>>>> know that it doesn't meet
>>>> Later-no-Help or Participation.
>>>>
>>>> Using a multi-slot ratings ballot for a Condorcet method of similar
>>>> complexity I like:
>>>>
>>>> *Eliminate all candidates not in the Smith set.
>>>>
>>>> Interpret each ballot as giving approval to those remaining candidates
>>>> they rate above average (mean
>>>> of the ratings given to Smith-set members).
>>>>
>>>> Now, using these approvals, elect the Margins-Sorted Approval winner.*
>>>>
>>>
>>> It seems to me that Smith//Implicit-Approval or
>>> Smith//implicit-approval-sorted-margins would be affected by a couple of
>>> factors:
>>>
>>> - How is the average calculated? Do you normalize scores? In other
>>> words, if a ballot has non-Smith candidates in the first, say, three ranks,
>>> do you up-rank the Smith candidate scores on that ballot by three? Also, if
>>> there are ranks below the top that contain only non-Smith candidates, do
>>> you collapse those ranks or leave the relative rank spacing on the ballot
>>> between Smith candidates untouched?
>>> - Approving Smith Candidates with scores above the mean has
>>> similarities to Median Ratings. It would be more similar and probably more
>>> stable to use the trimmed mean -- drop the top and bottom 25% of scores.
>>> This would give you an average score closer to the median.
>>>
>>> What advantage does Approval Sorted Margins have over
>>> Smith//Implicit-Approval? I like ASM but fear it is probably too complex
>>> for any advantage it gives you.
>>>
>>>>
>>>> Has Smith//Median Rating been proposed before?
>>>>
>>>> Not that I know of.
>>>>
>>>> Chris Benham
>>>>
>>>>
>>>>
>>>>
>>>> *Ted Stern* dodecatheon at gmail.com
>>>> <election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
>>>> *Tue Jan 2 15:12:26 PST 2024*
>>>>
>>>> - Previous message (by thread): [EM] [Game Theory] Iterated
>>>> Prisoners' Dilemma as a voting method metric
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005215.html>
>>>> - Next message (by thread): [EM] Pairwise Median Rating
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005218.html>
>>>> - *Messages sorted by:* [ date ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/date.html#5216>
>>>> [ thread ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/thread.html#5216>
>>>> [ subject ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/subject.html#5216>
>>>> [ author ]
>>>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/author.html#5216>
>>>>
>>>> ------------------------------
>>>>
>>>> Continuing my search for a summable voting method that discourages burial
>>>> and defection, I've come across this hybrid of Condorcet and median ratings
>>>> that acts like Smith/Approval with an automatic approval cutoff. I'm
>>>> calling it Pairwise Median Rating (PMR), but it could also be described as
>>>> Smith//MR//Pairwise//MRScore:
>>>>
>>>> 1. Equal Ranking and ranking gap allowed (essentially a ratings method
>>>> with rank inferred). For purposes of this discussion, assume 6 slots (5
>>>> ranks above rejection).
>>>> 2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
>>>> means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
>>>> of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
>>>> 3. [Smith]
>>>> 1. Compute the pairwise preference array
>>>> 2. The winner is the candidate who defeats each other candidate
>>>> pairwise.
>>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>>> any member of the Smith Set.
>>>> 4. [Median Rating]
>>>> 1. Set the MR threshold to top rank.
>>>> 2. While no Smith candidate has a majority of undropped ballots at or
>>>> above the threshold, set the threshold to the next lower rank,
>>>> until there
>>>> is no lower rank.
>>>> 3. The winner is the single candidate that has a majority of
>>>> undropped ballots at or above the threshold.
>>>> 5. [Pairwise]
>>>> 1. Otherwise, if more than one candidate passes the threshold, look for
>>>> a pairwise beats-all candidate among candidates meeting the MR threshold.
>>>> (i.e. Condorcet on just the MR threshold set).
>>>> 2. If there is one, you have a winner.
>>>> 6. [MR Score]
>>>> 1. Otherwise, the winner is the Smith set candidate with the largest
>>>> number of ballots at or above the Median Rating threshold (their MRscore).
>>>>
>>>> This method is essentially Smith//Approval(explicit) with the approval
>>>> cutoff automatically inferred via median ratings
>>>>
>>>> Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
>>>> considered optional, but by doing that, you ensure Immunity from Irrelevant
>>>> Ballots (IIB), aka the zero ballot problem that affects other Median Rating
>>>> / Majority Judgment methods. In other words, the majority threshold is
>>>> unaffected by ballots that do not rank a viable candidate. It is possible
>>>> to do this summably if need be.
>>>>
>>>> PMR either passes the Chicken Dilemma criterion without adjustment, or
>>>> there is a downranking strategy for defending against defection.
>>>>
>>>> Consider the following examples from Chris Benham's post re MinLV(erw)
>>>> Sorted Margins (http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
>>>> ):
>>>>
>>>> >** 46 A>B
>>>> **>* 44 B>C (sincere is B or B>A)
>>>> *>* 05 C>A
>>>> *>* 05 C>B
>>>> *>>* A>B 51-49, B>C 90-10, C>A 54-46.
>>>> *
>>>>
>>>> With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
>>>> there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
>>>> threshold is 2nd place, and A and B both pass the threshold. A defeats B,
>>>> so A is the winner and B's defection/burial fails.
>>>>
>>>> >** 25 A>B
>>>> **>* 26 B>C
>>>> *>* 23 C>A
>>>> *>* 26 C
>>>> *>>* C>A 75-25, A>B 48-26, B>C 51-49*
>>>>
>>>> C wins with PMR (MR threshold is first place). B would win with most
>>>> other Condorcet methods.
>>>>
>>>> >** 35 A
>>>> **>* 10 A=B
>>>> *>* 30 B>C (sincere B > A)
>>>> *>* 25 C
>>>> *>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
>>>>
>>>> *A wins with sincere voting. When B defects to try to win, which it
>>>> would do with most other Condorcet methods, B wins. With PMR, C wins,
>>>> an undesirable outcome for B.
>>>>
>>>> Here is another example from Rob LeGrand
>>>> (https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
>>>> good example for chicken dilemma resistance, but it does demonstrate
>>>> differences from Schulze, MMPO, RP and Bucklin:
>>>>
>>>> # example from method description page
>>>> 98:Abby>Cora>Erin>Dave>Brad
>>>> 64:Brad>Abby>Erin>Cora>Dave
>>>> 12:Brad>Abby>Erin>Dave>Cora
>>>> 98:Brad>Erin>Abby>Cora>Dave
>>>> 13:Brad>Erin>Abby>Dave>Cora
>>>> 125:Brad>Erin>Dave>Abby>Cora
>>>> 124:Cora>Abby>Erin>Dave>Brad
>>>> 76:Cora>Erin>Abby>Dave>Brad
>>>> 21:Dave>Abby>Brad>Erin>Cora
>>>> 30:Dave>Brad>Abby>Erin>Cora
>>>> 98:Dave>Brad>Erin>Cora>Abby
>>>> 139:Dave>Cora>Abby>Brad>Erin
>>>> 23:Dave>Cora>Brad>Abby>Erin
>>>>
>>>> The pairwise matrix:
>>>>
>>>> against
>>>> Abby Brad Cora Dave Erin
>>>> for Abby 458 461 485 511
>>>> Brad 463 461 312 623
>>>> Cora 460 460 460 460
>>>> Dave 436 609 461 311
>>>> Erin 410 298 461 610
>>>>
>>>> There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
>>>>
>>>> Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
>>>> wins with Bucklin.
>>>>
>>>> In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
>>>> the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
>>>> score of 484 is only slightly over the 50% mark of 460.5, so the Dave
>>>> voters hold the balance of power. Dave defeats Brad pairwise, so Dave
>>>> voters might not be as happy with a Brad victory, and Abby might be
>>>> able to persuade Dave voters to downrank Brad but not Abby. If
>>>> successful, Brad drops 44 points in MRScore and is no longer in the MR
>>>> threshold set. Abby defeats Erin, so Abby wins.
>>>>
>>>> * 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
>>>> 30:Dave>Brad>Abby>Erin>Cora
>>>> 98:Dave>Brad>Erin>Cora>Abby
>>>> 139:Dave>Cora>Abby>Brad>Erin
>>>> * 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
>>>>
>>>> PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
>>>> I believe it passes LNHelp. It probably fails Participation and IIA.
>>>> There are probably weird examples where changing one vote changes the
>>>> MR threshold. But overall, I think it has a good balance of incentive
>>>> to deter burial and deliberate cycles.
>>>>
>>>> Has Smith//Median Rating been proposed before? It seems like a simple
>>>> modification to MR on its own.
>>>>
>>>>
>>>> ----
>>>> Election-Methods mailing list - see https://electorama.com/em for list
>>>> info
>>>>
>>> ----
>> Election-Methods mailing list - see https://electorama.com/em for list
>> info
>>
>
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