- 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.

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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,

On Mon, Jan 22, 2024, 18:57 C.Benham <cbenham@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. > Ah, you meant average *per ballot*. I thought you meant over all ballots. > 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@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@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* >>> >>> - 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> >>> - Next message (by thread): [EM] Pairwise Median Rating >>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/005218.html> >>> - *Messages sorted by:* [ date ] >>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/date.html#5216> >>> [ thread ] >>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/thread.html#5216> >>> [ subject ] >>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/subject.html#5216> >>> [ author ] >>> <http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-January/author.html#5216> >>> >>> ------------------------------ >>> >>> 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 >>> >>