[EM] Pairwise Median Rating

Ted Stern dodecatheon at gmail.com
Wed Jan 17 15:43:21 PST 2024 

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.
>
>
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