[EM] Pairwise Median Rating
Ted Stern
dodecatheon at gmail.com
Mon Jan 29 10:31:57 PST 2024
On Fri, Jan 26, 2024 at 4:41 PM C.Benham <cbenham at adam.com.au> wrote:
> Of course! Thanks. Pardon the brain fade.
>
> Normally Smith sets don't have two members, but your example didn't
> mention a third member or say how many candidates there are or how many
> rating slots the ballots have.
>
> So maybe there is another candidate F (which is in the Smith set) and the
> ballots have 6 slots, so our Score ballot interpretation becomes
>
> D6, E5, A4, B2, F0.
>
> So then our calculation becomes 4+2+0/3 = 2.
>
> Only A is scored higher than 2, so the ballot only approves A.
>
I envisioned a couple of different scenarios:
D6, E5, A4, B2 Fa0, Fb0, Fc0
with A, B, and F* in Smith.
Then (4 + 2 + 0 + 0 + 0)/5 = 1.2, so both A and B are approved, with only
Fb and Fc clones crowded in.
Similarly, if A clones are added, one might have
D6, E5, Aa4, Ab4, Ac4, B2, F0
with A*, B, and F in Smith.
Then (4 + 4 + 4 + 2 + 0)/3 = 3.66..., so only A* clones are approved, not B.
>
> This is a nice technique, but it is not clone resistant. Adding Smith
> candidate clones can change the average and thus the approval cutoff would
> change. This seems unstable to me.
>
>
> With ratings ballots I think only candidates that get the same rating on
> every ballot are considered clones. And besides that I think that more
> than three candidates in the Smith set would be vanishingly rare.
>
Whether its rare or not, as soon as you lose clone independence, the method
is opened to criticism.
>
>
> Chris
>
>
>
> On 27/01/2024 5:40 am, Ted Stern wrote:
>
>
>
> On Mon, Jan 22, 2024 at 7:07 PM C.Benham <cbenham 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
>>>>
>>>> The Condorcet criterion
>>>> <https://en.wikipedia.org/wiki/Condorcet_criterion> is incompatible
>>>> with later-no-help.
>>>>
>>>>
>>>> From your post again:
>>>>
>>>> It probably fails Participation ..
>>>>
>>>>
>>>> It has been known (for a longer time) that Condorcet and Participation
>>>> are incompatible.
>>>>
>>>> So since we know for sure that your method meets Condorcet, we also
>>>> know that it doesn't meet
>>>> Later-no-Help or Participation.
>>>>
>>>> Using a multi-slot ratings ballot for a Condorcet method of similar
>>>> complexity I like:
>>>>
>>>> *Eliminate all candidates not in the Smith set.
>>>>
>>>> Interpret each ballot as giving approval to those remaining candidates
>>>> they rate above average (mean
>>>> of the ratings given to Smith-set members).
>>>>
>>>> Now, using these approvals, elect the Margins-Sorted Approval winner.*
>>>>
>>>
>>> It seems to me that Smith//Implicit-Approval or
>>> Smith//implicit-approval-sorted-margins would be affected by a couple of
>>> factors:
>>>
>>> - How is the average calculated? Do you normalize scores? In other
>>> words, if a ballot has non-Smith candidates in the first, say, three ranks,
>>> do you up-rank the Smith candidate scores on that ballot by three? Also, if
>>> there are ranks below the top that contain only non-Smith candidates, do
>>> you collapse those ranks or leave the relative rank spacing on the ballot
>>> between Smith candidates untouched?
>>> - Approving Smith Candidates with scores above the mean has
>>> similarities to Median Ratings. It would be more similar and probably more
>>> stable to use the trimmed mean -- drop the top and bottom 25% of scores.
>>> This would give you an average score closer to the median.
>>>
>>> What advantage does Approval Sorted Margins have over
>>> Smith//Implicit-Approval? I like ASM but fear it is probably too complex
>>> for any advantage it gives you.
>>>
>>>>
>>>> Has Smith//Median Rating been proposed before?
>>>>
>>>> Not that I know of.
>>>>
>>>> Chris Benham
>>>>
>>>>
>>>>
>>>>
>>>> *Ted Stern* dodecatheon at gmail.com
>>>> <election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Pairwise%20Median%20Rating&In-Reply-To=%3CCAHGFzOSm%3Deni2SuD5YRMrYBu4Gn9%2BYQ2NrqC_sXG8QFPrcVApQ%40mail.gmail.com%3E>
>>>> *Tue Jan 2 15:12:26 PST 2024*
>>>>
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>>>>
>>>> ------------------------------
>>>>
>>>> Continuing my search for a summable voting method that discourages burial
>>>> and defection, I've come across this hybrid of Condorcet and median ratings
>>>> that acts like Smith/Approval with an automatic approval cutoff. I'm
>>>> calling it Pairwise Median Rating (PMR), but it could also be described as
>>>> Smith//MR//Pairwise//MRScore:
>>>>
>>>> 1. Equal Ranking and ranking gap allowed (essentially a ratings method
>>>> with rank inferred). For purposes of this discussion, assume 6 slots (5
>>>> ranks above rejection).
>>>> 2. In rank notation for this method, '>>' refers to a gap. So 'A >> B'
>>>> means A gets top rank while B gets 3rd place. Similarly '>>>' means a gap
>>>> of two slots: 'A>>>B' means A is top ranked while B is in 4th place.
>>>> 3. [Smith]
>>>> 1. Compute the pairwise preference array
>>>> 2. The winner is the candidate who defeats each other candidate
>>>> pairwise.
>>>> 3. Otherwise, drop ballots that don't contain ranks above last for
>>>> any member of the Smith Set.
>>>> 4. [Median Rating]
>>>> 1. Set the MR threshold to top rank.
>>>> 2. While no Smith candidate has a majority of undropped ballots at or
>>>> above the threshold, set the threshold to the next lower rank,
>>>> until there
>>>> is no lower rank.
>>>> 3. The winner is the single candidate that has a majority of
>>>> undropped ballots at or above the threshold.
>>>> 5. [Pairwise]
>>>> 1. Otherwise, if more than one candidate passes the threshold, look for
>>>> a pairwise beats-all candidate among candidates meeting the MR threshold.
>>>> (i.e. Condorcet on just the MR threshold set).
>>>> 2. If there is one, you have a winner.
>>>> 6. [MR Score]
>>>> 1. Otherwise, the winner is the Smith set candidate with the largest
>>>> number of ballots at or above the Median Rating threshold (their MRscore).
>>>>
>>>> This method is essentially Smith//Approval(explicit) with the approval
>>>> cutoff automatically inferred via median ratings
>>>>
>>>> Step Smith.3, dropping non-Smith-candidate-voting ballots, could be
>>>> considered optional, but by doing that, you ensure Immunity from Irrelevant
>>>> Ballots (IIB), aka the zero ballot problem that affects other Median Rating
>>>> / Majority Judgment methods. In other words, the majority threshold is
>>>> unaffected by ballots that do not rank a viable candidate. It is possible
>>>> to do this summably if need be.
>>>>
>>>> PMR either passes the Chicken Dilemma criterion without adjustment, or
>>>> there is a downranking strategy for defending against defection.
>>>>
>>>> Consider the following examples from Chris Benham's post re MinLV(erw)
>>>> Sorted Margins (http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html
>>>> ):
>>>>
>>>> >** 46 A>B
>>>> **>* 44 B>C (sincere is B or B>A)
>>>> *>* 05 C>A
>>>> *>* 05 C>B
>>>> *>>* A>B 51-49, B>C 90-10, C>A 54-46.
>>>> *
>>>>
>>>> With sincere ballots, A is the Condorcet Winner (CW). With B's defection,
>>>> there is a cycle, and there is no CW. The Smith set is {A, B, C}. The MR
>>>> threshold is 2nd place, and A and B both pass the threshold. A defeats B,
>>>> so A is the winner and B's defection/burial fails.
>>>>
>>>> >** 25 A>B
>>>> **>* 26 B>C
>>>> *>* 23 C>A
>>>> *>* 26 C
>>>> *>>* C>A 75-25, A>B 48-26, B>C 51-49*
>>>>
>>>> C wins with PMR (MR threshold is first place). B would win with most
>>>> other Condorcet methods.
>>>>
>>>> >** 35 A
>>>> **>* 10 A=B
>>>> *>* 30 B>C (sincere B > A)
>>>> *>* 25 C
>>>> *>>* C>A 55-45, A>B 35-30 (10A=B not counted), B>C 40-25.
>>>>
>>>> *A wins with sincere voting. When B defects to try to win, which it
>>>> would do with most other Condorcet methods, B wins. With PMR, C wins,
>>>> an undesirable outcome for B.
>>>>
>>>> Here is another example from Rob LeGrand
>>>> (https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html). It's not a
>>>> good example for chicken dilemma resistance, but it does demonstrate
>>>> differences from Schulze, MMPO, RP and Bucklin:
>>>>
>>>> # example from method description page
>>>> 98:Abby>Cora>Erin>Dave>Brad
>>>> 64:Brad>Abby>Erin>Cora>Dave
>>>> 12:Brad>Abby>Erin>Dave>Cora
>>>> 98:Brad>Erin>Abby>Cora>Dave
>>>> 13:Brad>Erin>Abby>Dave>Cora
>>>> 125:Brad>Erin>Dave>Abby>Cora
>>>> 124:Cora>Abby>Erin>Dave>Brad
>>>> 76:Cora>Erin>Abby>Dave>Brad
>>>> 21:Dave>Abby>Brad>Erin>Cora
>>>> 30:Dave>Brad>Abby>Erin>Cora
>>>> 98:Dave>Brad>Erin>Cora>Abby
>>>> 139:Dave>Cora>Abby>Brad>Erin
>>>> 23:Dave>Cora>Brad>Abby>Erin
>>>>
>>>> The pairwise matrix:
>>>>
>>>> against
>>>> Abby Brad Cora Dave Erin
>>>> for Abby 458 461 485 511
>>>> Brad 463 461 312 623
>>>> Cora 460 460 460 460
>>>> Dave 436 609 461 311
>>>> Erin 410 298 461 610
>>>>
>>>> There is no Condorcet winner. The Smith set is {Abby, Brad, Dave, Erin}.
>>>>
>>>> Abby wins with Schulze, MMPO, while Brad wins with Ranked Pairs, Erin
>>>> wins with Bucklin.
>>>>
>>>> In PMR, with a threshold of 3rd place, Abby, Brad, and Erin all pass
>>>> the threshold. Brad defeats Abby and Erin to win. But Brad's threshold
>>>> score of 484 is only slightly over the 50% mark of 460.5, so the Dave
>>>> voters hold the balance of power. Dave defeats Brad pairwise, so Dave
>>>> voters might not be as happy with a Brad victory, and Abby might be
>>>> able to persuade Dave voters to downrank Brad but not Abby. If
>>>> successful, Brad drops 44 points in MRScore and is no longer in the MR
>>>> threshold set. Abby defeats Erin, so Abby wins.
>>>>
>>>> * 21:Dave>Abby>>Brad>Erin>Cora (Brad -> 4th place)*
>>>> 30:Dave>Brad>Abby>Erin>Cora
>>>> 98:Dave>Brad>Erin>Cora>Abby
>>>> 139:Dave>Cora>Abby>Brad>Erin
>>>> * 23:Dave>Cora>>Brad>Abby>Erin (Brad -> 4th place)*
>>>>
>>>> PMR passes Condorcet Winner, Condorcet Loser, IIB, and is cloneproof.
>>>> I believe it passes LNHelp. It probably fails Participation and IIA.
>>>> There are probably weird examples where changing one vote changes the
>>>> MR threshold. But overall, I think it has a good balance of incentive
>>>> to deter burial and deliberate cycles.
>>>>
>>>> Has Smith//Median Rating been proposed before? It seems like a simple
>>>> modification to MR on its own.
>>>>
>>>>
>>>> ----
>>>> Election-Methods mailing list - see https://electorama.com/em for list
>>>> info
>>>>
>>> ----
>> Election-Methods mailing list - see https://electorama.com/em for list
>> info
>>
>
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