<div dir="ltr"><div dir="ltr"><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Tue, Jan 16, 2024 at 7:27 AM C.Benham <<a href="mailto:cbenham@adam.com.au">cbenham@adam.com.au</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><u></u>
<div>
<p>Ted,<br>
<br>
</p><blockquote type="cite">
<pre style="white-space:pre-wrap;color:rgb(0,0,0);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial"> 3. Otherwise, drop ballots that don't contain ranks above last for
any member of the Smith Set.</pre>
</blockquote>
<p></p>
<p>Why not simply drop all ballots that make no distinction among
members of the Smith set?<br>
<br>
</p><blockquote type="cite">
<pre style="white-space:pre-wrap;color:rgb(0,0,0);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial">I believe it passes LNHelp.</pre>
</blockquote>
<br>
Douglas Woodall showed some time ago that Condorcet and LNHelp are
incompatible. I can't find<br>
his proof, but it says so here:<br>
<br>
<a href="https://en.wikipedia.org/wiki/Later-no-help_criterion" target="_blank">https://en.wikipedia.org/wiki/Later-no-help_criterion</a><br>
<br>
<blockquote type="cite"><span style="color:rgb(32,33,34);font-family:sans-serif;font-size:14px;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial;float:none;display:inline"><span> </span>The<span> </span></span><a href="https://en.wikipedia.org/wiki/Condorcet_criterion" title="Condorcet criterion" style="text-decoration:none;color:rgb(51,102,204);background:none rgb(255,255,255);font-family:sans-serif;font-size:14px;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal" target="_blank">Condorcet
criterion</a><span style="color:rgb(32,33,34);font-family:sans-serif;font-size:14px;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial;float:none;display:inline"><span> </span>is
incompatible with later-no-help.</span></blockquote>
<br>
From your post again:<br>
<blockquote type="cite">
<pre style="white-space:pre-wrap;color:rgb(0,0,0);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial">It probably fails Participation ..</pre>
</blockquote>
<br>
It has been known (for a longer time) that Condorcet and
Participation are incompatible.<br>
<br>
So since we know for sure that your method meets Condorcet, we
also know that it doesn't meet <br>
Later-no-Help or Participation.<br>
<br>
Using a multi-slot ratings ballot for a Condorcet method of
similar complexity I like:<br>
<br>
*Eliminate all candidates not in the Smith set.<br>
<br>
Interpret each ballot as giving approval to those remaining
candidates they rate above average (mean <br>
of the ratings given to Smith-set members).<br>
<br>
Now, using these approvals, elect the Margins-Sorted Approval
winner.*<br></div></blockquote><div><br></div><div>It seems to me that Smith//Implicit-Approval or Smith//implicit-approval-sorted-margins would be affected by a couple of factors:</div><div><ul><li>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?</li><li>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. </li></ul><div>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.</div></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div>
<br>
<blockquote type="cite">
<pre style="white-space:pre-wrap;color:rgb(0,0,0);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial">Has Smith//Median Rating been proposed before?</pre>
</blockquote>
Not that I know of.<br>
<br>
Chris Benham<br>
<br>
<br>
<br>
<blockquote type="cite">
<h1 style="color:rgb(0,0,0);font-family:"Times New Roman";font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial"><br>
</h1>
<b style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial">Ted
Stern</b><span style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial;float:none;display:inline"><span> </span></span><a href="mailto: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" title="[EM] Pairwise Median Rating" style="font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal" target="_blank">dodecatheon
at gmail.com</a><br style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial">
<i style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial">Tue
Jan 2 15:12:26 PST 2024</i><span style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial;float:none;display:inline"></span>
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<hr style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial">
<pre style="white-space:pre-wrap;color:rgb(0,0,0);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial">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 (
<a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html" target="_blank">http://lists.electorama.com/pipermail/election-methods-electorama.com/2016-October/000599.html</a>
):
><i>* 46 A>B
</i>*>* 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.
><i>* 25 A>B
</i>*>* 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.
><i>* 35 A
</i>*>* 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
(<a href="https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html" target="_blank">https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html</a>). 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.</pre>
</blockquote>
<br>
<p></p>
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</blockquote></div></div>