[EM] Pairwise Median Rating

Tue Jan 2 15:12:26 PST 2024

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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