[EM] Pairwise Median Rating
Ted Stern
dodecatheon at gmail.com
Mon Jan 22 16:26:46 PST 2024
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
>>
>
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