Pairwise Median Rating
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:
- 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). - 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. - [Smith]
- Compute the pairwise preference array
- The winner is the candidate who defeats each other candidate
pairwise. - Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.
- [Median Rating]
- Set the MR threshold to top rank.
- 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. - The winner is the single candidate that has a majority of
undropped ballots at or above the threshold.
- [Pairwise]
- 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. - [MR Score]
- 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.
Cloneproof in what sense? Perfect clones (voters rank them as tied)? Or effective clones (voters never rank another candidate between them)? For a system like this, those might not be equivalent.
Suppose two factions of the Purple Party each field a candidate -- one for New Purple, and one for traditional Purple. Voters might draw meaningful distinctions between them even if they never rank a non-Purple candidate between them
Step 4 in particular might be where this matters (which could affect subsequent steps)
On 1/2/24 18:12, Ted Stern wrote:
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).
...
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.
By cloneproof, I mean that turning a single candidate into either perfect
or effective clone groups will ever enable a candidate in that group to win
when they would lose as a single candidate. That is, resistant to cloning
strategy.
On Thu, Jan 4, 2024, 16:42 Joshua Boehme joshua.p.boehme@gmail.com wrote:
Cloneproof in what sense? Perfect clones (voters rank them as tied)? Or
effective clones (voters never rank another candidate between them)? For a
system like this, those might not be equivalent.
Suppose two factions of the Purple Party each field a candidate -- one for
New Purple, and one for traditional Purple. Voters might draw meaningful
distinctions between them even if they never rank a non-Purple candidate
between them
Step 4 in particular might be where this matters (which could affect
subsequent steps)
On 1/2/24 18:12, Ted Stern wrote:
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).
...
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.
info
Election-Methods mailing list - see https://electorama.com/em for list
info
I would think to be considered resistant to cloning strategy the opposite would need to be true as well; that is, cloning a winning candidate shouldn’t make them lose, either.
--pH
On Jan 4, 2024, at 10:47 PM, Ted Stern dodecatheon@gmail.com wrote:
By cloneproof, I mean that turning a single candidate into either perfect or effective clone groups will ever enable a candidate in that group to win when they would lose as a single candidate. That is, resistant to cloning strategy.
On Thu, Jan 4, 2024, 16:42 Joshua Boehme <joshua.p.boehme@gmail.commailto:joshua.p.boehme@gmail.com> wrote:
Cloneproof in what sense? Perfect clones (voters rank them as tied)? Or effective clones (voters never rank another candidate between them)? For a system like this, those might not be equivalent.
Suppose two factions of the Purple Party each field a candidate -- one for New Purple, and one for traditional Purple. Voters might draw meaningful distinctions between them even if they never rank a non-Purple candidate between them
Step 4 in particular might be where this matters (which could affect subsequent steps)
On 1/2/24 18:12, Ted Stern wrote:
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).
...
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.
Election-Methods mailing list - see https://electorama.com/em for list info
Election-Methods mailing list - see https://electorama.com/em for list info
The median rating step satisfies independence of clones, like Majority
Judgment. It just uses a different tiebreaker.
https://electowiki.org/wiki/Majority_Judgment
On Fri, Jan 5, 2024, 00:27 Hahn, Paul manynote@wustl.edu wrote:
I would think to be considered resistant to cloning strategy the opposite
would need to be true as well; that is, cloning a winning candidate
shouldn’t make them lose, either.
--pH
On Jan 4, 2024, at 10:47 PM, Ted Stern dodecatheon@gmail.com wrote:
By cloneproof, I mean that turning a single candidate into either perfect
or effective clone groups will ever enable a candidate in that group to win
when they would lose as a single candidate. That is, resistant to cloning
strategy.
On Thu, Jan 4, 2024, 16:42 Joshua Boehme joshua.p.boehme@gmail.com
wrote:
Cloneproof in what sense? Perfect clones (voters rank them as tied)? Or
effective clones (voters never rank another candidate between them)? For a
system like this, those might not be equivalent.
Suppose two factions of the Purple Party each field a candidate -- one
for New Purple, and one for traditional Purple. Voters might draw
meaningful distinctions between them even if they never rank a non-Purple
candidate between them
Step 4 in particular might be where this matters (which could affect
subsequent steps)
On 1/2/24 18:12, Ted Stern wrote:
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).
...
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.
info
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 where I went off-track was by reading this as applying to the median ranking instead of the median score.
In the second example, though...
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.
Wouldn't C>A>B be a natural outcome for many methods since B>C has the weakest margin of the three?
On 1/5/24 05:18, Ted Stern wrote:
The median rating step satisfies independence of clones, like Majority
Judgment. It just uses a different tiebreaker.
Thinking about this a bit more, I think Pairwise Median Rating needs a
slight revision after the Smith phase. See inline comments below.
On Tue, Jan 2, 2024 at 3:12 PM Ted Stern dodecatheon@gmail.com wrote:
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.
Step Smith.4:
- For each ballot, eliminate non-Smith candidates, then normalize each
ballot by collapsing ranks with only eliminated candidates. For example,
with a Smith set of {A, B, C}, and a ballot with
D > E > F > B=C > G > [gap] > A
Normalization would take this to
B=C > [gap] > A
which can also be written as B=C >> A.
I don't think normalization is summable. It would require at least one more
ballot counting pass.
1. [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.
2. [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.
3. [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.
This example is already normalized.
- 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.
Also normalized
- 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.
Also normalized.
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)*
With {Abby, Brad, Dave, Erin} in the Smith set, ballot normalization means
that Cora is eliminated, and ballots change to
222:Abby>Erin>Dave>Brad (98 Cora second place combines with 124
Cora first place)
76:Brad>Abby>Erin>Dave (64 + 12)
111:Brad>Erin>Abby>Dave (98 + 13)
125:Brad>Erin>Dave>Abby
76:Erin>Abby>Dave>Brad
160:Dave>Abby>Brad>Erin (21 + 139)
53:Dave>Brad>Abby>Erin (30 + 23)
98:Dave>Brad>Erin>Abby
Abby: 222 First; 222 + 76 + 76 + 160 = 534 First + Second
Brad: 412 First; 412 + 151 = 563 First + Second
Erin: 76 First, 534 First + Second
{A, B, E} have highest Median Rating of Second Place. A>E, B>E, B > A: B
wins.
Brad has a MR margin of 112 over 50% + 1 vote = 461 ballots. The 160
Dave>Abby voters could lower Brad by one level to eliminate Brad from
Second Place median rating. However, the 76 Brad>Abby voters could
similarly lower Abby's ranking by one to defend.
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.
Ted—
That’s too complicated for a first Condorcet proposal. When Condorcet in
some simpler form has been in use, so we already have it, & people are
familiar with it, then improvement becomes possible.
But imagine writing to a city council or county board of supervisors &
suggesting that they enact something that complicated, or that they order a
referendum on it.
…or describing it to a member of the general public, on a street-corner, to
get their signature for an initiative.
CW, Implicit-Approval (CW, IA) is simple, briefly-defined & intuitive.
…& I like that Vermont proposal that said to elect the highest topcount if
there isn’t a voted CW. It doesn’t get any simpler, more modest, or more
un-arbitrary than that.
It could be called CW, Fav (CW, F).
My elaboration of that loses some simplicity, & so maybe CW, Fav is the
better proposal.
It could be explained that electing someone who pairbeats the favorite
could deter offensive strategy by threatening to backfire by electing him
whom the strategists voted over the CW.
I’d call that Favorite Beating Favorite (FBF).
But needing explanation to get something better can often lose the audience
& doom the proposal.
What I would (…& hopefully will…) do is offer a list of Condorcet
proposals, starting with Vermont’s simplest…& then the next-simplest ones…
On Tue, Jan 2, 2024 at 15:13 Ted Stern dodecatheon@gmail.com wrote:
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
For a public proposal, Condorcet//Top-Favorite is great. I'd back that.
On Thu, Jan 11, 2024, 15:45 Michael Ossipoff email9648742@gmail.com wrote:
Ted—
That’s too complicated for a first Condorcet proposal. When Condorcet in
some simpler form has been in use, so we already have it, & people are
familiar with it, then improvement becomes possible.
But imagine writing to a city council or county board of supervisors &
suggesting that they enact something that complicated, or that they order a
referendum on it.
…or describing it to a member of the general public, on a street-corner,
to get their signature for an initiative.
CW, Implicit-Approval (CW, IA) is simple, briefly-defined & intuitive.
…& I like that Vermont proposal that said to elect the highest topcount if
there isn’t a voted CW. It doesn’t get any simpler, more modest, or more
un-arbitrary than that.
It could be called CW, Fav (CW, F).
My elaboration of that loses some simplicity, & so maybe CW, Fav is the
better proposal.
It could be explained that electing someone who pairbeats the favorite
could deter offensive strategy by threatening to backfire by electing him
whom the strategists voted over the CW.
I’d call that Favorite Beating Favorite (FBF).
But needing explanation to get something better can often lose the
audience & doom the proposal.
What I would (…& hopefully will…) do is offer a list of Condorcet
proposals, starting with Vermont’s simplest…& then the next-simplest ones…
On Tue, Jan 2, 2024 at 15:13 Ted Stern dodecatheon@gmail.com wrote:
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
On Fri, Jan 12, 2024 at 12:22 Ted Stern dodecatheon@gmail.com wrote:
For a public proposal, Condorcet//Top-Favorite is great. I'd back that.
👍
On Thu, Jan 11, 2024, 15:45 Michael Ossipoff email9648742@gmail.com
wrote:
Ted—
That’s too complicated for a first Condorcet proposal. When Condorcet in
some simpler form has been in use, so we already have it, & people are
familiar with it, then improvement becomes possible.
But imagine writing to a city council or county board of supervisors &
suggesting that they enact something that complicated, or that they order a
referendum on it.
…or describing it to a member of the general public, on a street-corner,
to get their signature for an initiative.
CW, Implicit-Approval (CW, IA) is simple, briefly-defined & intuitive.
…& I like that Vermont proposal that said to elect the highest topcount
if there isn’t a voted CW. It doesn’t get any simpler, more modest, or more
un-arbitrary than that.
It could be called CW, Fav (CW, F).
My elaboration of that loses some simplicity, & so maybe CW, Fav is the
better proposal.
It could be explained that electing someone who pairbeats the favorite
could deter offensive strategy by threatening to backfire by electing him
whom the strategists voted over the CW.
I’d call that Favorite Beating Favorite (FBF).
But needing explanation to get something better can often lose the
audience & doom the proposal.
What I would (…& hopefully will…) do is offer a list of Condorcet
proposals, starting with Vermont’s simplest…& then the next-simplest ones…
On Tue, Jan 2, 2024 at 15:13 Ted Stern dodecatheon@gmail.com wrote:
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