Pairwise Median Rating
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:
TheCondorcet criterion
https://en.wikipedia.org/wiki/Condorcet_criterionis 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.*
Has Smith//Median Rating been proposed before?
Not that I know of.
Chris Benham
Ted Sterndodecatheon 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.
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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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
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.
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