[EM] Pairwise Median Rating
Chris Benham
cbenhamau at yahoo.com.au
Mon Jan 22 18:55:20 PST 2024
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 at 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 at 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.
>>
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>>
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