[EM] Pairwise Median Rating

Tue Jan 9 16:14:52 PST 2024

Thinking about this a bit more, I think Pairwise Median Rating needs a
slight revision after the Smith phase. See inline comments below.

On Tue, Jan 2, 2024 at 3:12 PM Ted Stern <dodecatheon at gmail.com> wrote:

> 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.
>
>
Step  Smith.4:

   1. For each ballot, eliminate non-Smith candidates, then normalize each
   ballot by collapsing ranks with only eliminated candidates. For example,
   with a Smith set of {A, B, C}, and a ballot with

D > E > F > B=C > G > [gap] > A

Normalization would take this to

B=C > [gap] > A

which can also be written as B=C >> A.

I don't think normalization is summable. It would require at least one more
ballot counting pass.

>    1. [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.
>    2. [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.
>    3. [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.
>

This example is already normalized.

> >* 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.
>
>
Also normalized

> >* 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.
>
>
Also normalized.

> 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)*
>
>
With {Abby, Brad, Dave, Erin} in the Smith set, ballot normalization means
that Cora is eliminated, and ballots change to

222:Abby>Erin>Dave>Brad      (98 Cora second place combines with 124
Cora first place)
 76:Brad>Abby>Erin>Dave      (64 + 12)
111:Brad>Erin>Abby>Dave      (98 + 13)
125:Brad>Erin>Dave>Abby
 76:Erin>Abby>Dave>Brad
160:Dave>Abby>Brad>Erin      (21 + 139)
 53:Dave>Brad>Abby>Erin      (30 + 23)
 98:Dave>Brad>Erin>Abby

 Abby:  222 First; 222 + 76 + 76 + 160 = 534 First + Second
 Brad:  412 First; 412 + 151 = 563 First + Second
  Erin:  76 First, 534 First + Second

{A, B, E} have highest Median Rating of Second Place. A>E, B>E, B > A:  B
wins.

Brad has a MR margin of 112 over 50% + 1 vote = 461 ballots. The 160
Dave>Abby voters could lower Brad by one level to eliminate Brad from
Second Place median rating. However, the 76 Brad>Abby voters could
similarly lower Abby's ranking by one to defend.

> 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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