[EM] Pairwise Median Rating
C.Benham
cbenham at adam.com.au
Fri Jan 26 16:41:12 PST 2024
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 at 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 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/
>>>>
>>>> * 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>
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>>>> Rating
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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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