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<p>Of course! Thanks. Pardon the brain fade.<br>
<br>
Normally Smith sets don't have two members, but your example
didn't mention a third member or say how many candidates there are
or how many<br>
rating slots the ballots have.<br>
<br>
So maybe there is another candidate F (which is in the Smith set)
and the ballots have 6 slots, so our Score ballot interpretation
becomes<br>
<br>
D6, E5, A4, B2, F0.<br>
<br>
So then our calculation becomes 4+2+0/3 = 2.<br>
<br>
Only A is scored higher than 2, so the ballot only approves A.<br>
<br>
<blockquote type="cite">This is a nice technique, but it is not
clone resistant. Adding Smith candidate clones can change the
average and thus the approval cutoff would change. This seems
unstable to me.</blockquote>
<br>
With ratings ballots I think only candidates that get the same
rating on every ballot are considered clones. And besides that I
think that more than three candidates in the Smith set would be
vanishingly rare.<br>
<br>
Chris<br>
<br>
<br>
<br>
</p>
<div class="moz-cite-prefix">On 27/01/2024 5:40 am, Ted Stern wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CAHGFzORefGEv8wXQ0035vCboT4Wi=f6-TEYaDOzicYQzf+pqcw@mail.gmail.com">
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
<div dir="ltr">
<div dir="ltr"><br>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Mon, Jan 22, 2024 at
7:07 PM C.Benham <<a href="mailto:cbenham@adam.com.au"
moz-do-not-send="true" class="moz-txt-link-freetext">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">
<div>
<div lang="x-unicode">
<div lang="x-unicode">
<p>Ted,<br>
<br>
</p>
<blockquote type="cite">...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. </blockquote>
<br>
I don't think I left anything ambiguous.<br>
<br>
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. <br>
<br>
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.<br>
<br>
</div>
</div>
</div>
</blockquote>
<div><br>
</div>
<div>This is a nice technique, but it is not clone resistant.
Adding Smith candidate clones can change the average and
thus the approval cutoff would change. This seems unstable
to me.</div>
<div><br>
</div>
<div>I think a better technique is to either have an explicit
cutoff which is lowered per ballot so that max(Smith score)
is approved, or to make the cutoff such that candidates with
max(Smith candidate score)/2 or greater per ballot are
approved. I prefer the former.</div>
<blockquote class="gmail_quote"
style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<div>
<div lang="x-unicode">
<div lang="x-unicode">
<blockquote type="cite">I also noticed that there were
cases where Smith//ASM(implicit) would get different
results (better, IMO) than Smith//Implicit-approval.
<div><br>
</div>
</blockquote>
<br>
What does "ASM" stand for?<br>
</div>
</div>
</div>
</blockquote>
<div><br>
</div>
<div><a
href="https://electowiki.org/wiki/Approval_Sorted_Margins"
moz-do-not-send="true">Approval Sorted Margins</a></div>
<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>
<div lang="x-unicode">
<div lang="x-unicode"> <br>
Chris<br>
<br>
<br>
<div>On 23/01/2024 10:56 am, Ted Stern wrote:<br>
</div>
<blockquote type="cite">
<div dir="ltr">Chris:
<div><br>
</div>
<div>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.</div>
<div><br>
</div>
<div>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.<br>
</div>
<div><br>
</div>
<div>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.</div>
<div><br>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Fri, Jan 19,
2024 at 4:05 PM C.Benham <<a
href="mailto:cbenham@adam.com.au"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">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">
<div>
<p> </p>
<blockquote type="cite">How is the average
calculated?</blockquote>
<br>
We interpret the ratings ballots as score
ballots, giving zero points for the bottom
rating (which is default for unrated),<br>
1 point for the next highest, 2 points for the
next above that and so on. <br>
<br>
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<br>
in the Smith set and interpret that ballot as
approving those Smith set candidates it scores
higher than that average score.<br>
<p>That simulates the best approval strategy
if the voters only know which candidates are
in the Smith set.<br>
<br>
</p>
<blockquote type="cite">What advantage does
Approval Sorted Margins have over
Smith//Implicit-Approval?</blockquote>
<br>
Do you mean Approval Sorted Margins using
ranking ballots with an explicit approval
cutoff? <br>
<br>
Assuming yes, it uses a more expressive
ballot, it is less vulnerable to Defection
strategy, and burial strategies are more<br>
likely to have no effect rather than backfire.<br>
<br>
In the method I proposed, omitting the ASM
step and just electing the candidate with the
highest approval score (derived<br>
as specified) would I concede make for a
simpler method that is nearly as good.<br>
<br>
I worry a bit that with all methods that begin
with eliminating or disqualifying all
candidates who aren't in the Smith set or <br>
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<br>
being taken seriously.<br>
<br>
An attractive feature of ASM is that it is a
Condorcet method that a lot of the time would
work fine without anyone needing<br>
to know if there is top cycle or not.<br>
<br>
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<br>
result between A and C.<br>
<br>
If we want something super simple to explain
and sell, then Condorcet//Top Ratings and
Condorcet//Approval (voted above bottom)<br>
are both not bad and much better than STAR.<br>
<br>
Chris Benham<br>
<br>
<p><br>
<br>
</p>
<div>On 18/01/2024 10:13 am, Ted Stern wrote:<br>
</div>
<blockquote type="cite">
<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"
target="_blank"
moz-do-not-send="true"
class="moz-txt-link-freetext">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">
<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>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"
moz-do-not-send="true"
class="moz-txt-link-freetext">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"
moz-do-not-send="true">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"
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target="_blank"
moz-do-not-send="true">dodecatheon
at gmail.com</a><br
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Jan 2 15:12:26 PST 2024</i><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"
moz-do-not-send="true"
class="moz-txt-link-freetext">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"
moz-do-not-send="true"
class="moz-txt-link-freetext">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>
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