<div dir="ltr"><div><div><div><div><div><div><div><div><div><div>Interesting. <br><br>Being rating-methods, both of these 2 methods meet FBC, don't they?<br><br></div>That's neat, four-slot XA, with ratings of 0, 1/3, 1/2, & 1. <br><br></div><div>That has a lot of appeal, and the ratings give it a kind of simplicity.<br></div><div><br></div>I've just now found the postings. I've only just read them, preliminarily, so this is just a quick preliminary reply.<br><br></div>Both methods, of course, would be a bit difficult to get most people to accept the definition of, to get someone to sit still for the entire definition.<br><br></div>XA is remarkable, in how it uses the rating information in a completely different way that, I guess, had never been considered until Jennings suggested it. But that great unfamiliarity won't help its acceptance.<br><br></div>That doesn't mean that offering it isn't possible, but, as you said, there will be the task of finding the easiest explanation and way of wording the definition.<br><br></div>I was wondering before, if XA's unique & unusual way of using the information could result in its having especially desirable properties that are otherwise difficult to attain, and evidently it does.<br><br></div>SARA, like 4-slot XA, has 4 ratings, but they're _qualitatively_ different ratings, each with a different kind of meaning. That's a complication for the public, knowing which of the four completely separate things to do. Without chicken dilemma, someone could just use the 4 XA ratings as sincere grades if they wanted to. I'd guess that, with any rating method, including those, the best strategy is to top-rate one's top-set.<br><br></div>The difference and the problem is what to do in a chicken dilemma situation. No doubt they both make that sitluation easier than Approval would.<br><br></div>Yes, continuing examination is needed, but it's promising. Maybe MMPO has a rival or two.<br><br></div>Michael Ossipoff<br><br><div><div><div><div><div><div><div><div><div><br><br><div><br><br><br><br></div></div></div></div></div></div></div></div></div></div></div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Oct 27, 2016 at 7:01 PM, Jameson Quinn <span dir="ltr"><<a href="mailto:jameson.quinn@gmail.com" target="_blank">jameson.quinn@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">I do see that this is better than Bucklin, as it is more robust to burial. However, all your scenario-building still relies on relatively high-information voters. Specifically, voters have to know which CD faction is the honest winner. Without that knowledge, the larger faction will (for safety) have to give enough cooperation to the larger to enable a betrayal.<div><br></div><div>Is there any way to fix this? There might be... I've had some thoughts, and none of them has worked yet, but I'm not convinced that none will.</div><div><br></div><div>Still, I think that the majority score (formerly known as SARA) solution to the CD is better. This is to eliminate candidate C (the minority threat) because it is rejected by a majority, and then compare candidates A and B (the subfactions) using some measure in which minimal cooperation cannot be distinguished from rejection.</div><div><br></div><div>(One could make majority score voting more robust to rejection by one CD subfaction, by doing eliminations in order from most- to least-rejected, and stopping before the final elimination. But I think this "robustness" is merely an illusion, because it would by making betrayal strategy safer, it could make it more common.) </div><div><br></div><div>Note that majority score has an additional safeguard against CD betrayal: a faction smaller than 25% cannot possibly win through such a betrayal. This means that some (exploitable) cooperation may needed in case of CD scenario where the majority is split more than two ways; but I think that's tolerable. </div><div><br></div><div>How often will there be a CD faction that is considering offensive strategy (that is, feels it might be smaller) yet is confident that it is more than 25%? I think that if the margin of error in polling is in the neighborhood of 4%, then such a faction would need to be at least 29%, and the opposing faction would need to be around 33%, leaving just 38% for the minority threat; and from 38% down to a non-threatening 33% is not that big a gap. So I think such situations would be rare and unstable; not worth the trouble of organizing strategy around.</div><div><br></div><div>I do see the allure of XA; it has the "no zero-information exaggeration incentive" property that Bucklin has, with this additional CD resistance that Forest points out. But I still think that majority score is better.</div><div><br></div><div>(Of course, I also still think that SODA has the best CD resistance of any well-defined single-winner method I know. But SODA requires participation by the candidates, which is not in all cases possible; and I think its strangeness makes it overall tougher to sell than something like majority score.)</div></div><div class="gmail_extra"><br><div class="gmail_quote"><div><div class="h5">2016-10-27 17:56 GMT-04:00 Forest Simmons <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span>:<br></div></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div><div class="h5"><div dir="ltr">It turns out that Chiastic Approval is a good method in the
context of the Chicken Dilemma, much better than ordinary Approval, Majority
Judgment, or plain Range.<div class="gmail_quote"><div dir="ltr"><p class="MsoNormal"><br></p>
<p class="MsoNormal">Ballots are score/range style ratings.<span> </span>Let x be the greatest number for which there
is some candidate that is given a rating of at least x percent on at least x
percent of the ballots.<span> </span>Elect the candidate
X that is given a rating of at least x percent on the greatest number of
ballots.</p><p class="MsoNormal"><br></p><p class="MsoNormal">The Greek letter Chi corresponds to the Roman letter X,,
hence the name Chiastic Approval or XA for short.<span> </span>Furthermore, when the method is described
graphically, the value of x is found by intersecting two graphs whose union
looks like the letter Chi.<br><span> </span></p><p class="MsoNormal"><br></p><p class="MsoNormal">Andy Jennings came up with XA while thinking about
how to improve Majority Judgement. Since we were both familiar with
ancient literary structures called Chiasms (identified in the Book of
Mormon about 15 decades after its first publication) the name came
naturally.<br></p><p class="MsoNormal"></p><p class="MsoNormal"><span><br> </span></p>
<p class="MsoNormal">Skip the following technical paragraph unless you are very curious
about the graphical description.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">[Let f be the function given by f(x) = the percentage of
ballots on which X is given a rating of at least x percent.<span> </span>Then f is a decreasing function whose graph
looks like the downward stroke of the letter Chi.<span> </span>The graph of y = x looks like the stroke with
positive slope.<span> </span>These two graphs cross
at the point (x, x) which yields the Chiastic Approval cutoff x.]</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Now consider the following ballot profile …</p>
<p class="MsoNormal">41 C</p>
<p class="MsoNormal">31 A>B(33%)</p>
<p class="MsoNormal">28 B>A(50%)</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Note that A is the only candidate with a rating of at least
50% on at least 50% of the ballots, so A is the XA winner.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">We could lower the 50% to 42%, and raise the 33% to 40%, and
A would still be the XA winner, as the only candidate with a rating of at least
42% on at least 42% of the ballots.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">In fact we could go further than that by splitting up the
the 28 B>A faction with some die hard defectors:</p>
<p class="MsoNormal">41 C</p>
<p class="MsoNormal">31 A>B(40%)</p>
<p class="MsoNormal">11 B>A(42%)</p>
<p class="MsoNormal">17 B</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Candidate A is still the only candidate given a rating of at
least 42% on at least 42 percent of the ballots.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">But if two more B faction voters defect, then C is elected
as the only candidate given a rating of at least 41 percent on at least 41
percent of the ballots:</p>
<p class="MsoNormal">41 C</p>
<p class="MsoNormal">31 A>B(40%)</p>
<p class="MsoNormal">8 B>A(42%)</p>
<p class="MsoNormal">20 B</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">In the general CD set up we have three factions with sincere
preference profiles</p>
<p class="MsoNormal">P: C</p>
<p class="MsoNormal">Q: A>B</p>
<p class="MsoNormal">R: B>A</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Where P > Q > R>0, and P+Q+R=100</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Under Chiastic Approval there is a Nash equilibrium that
protects the sincere CW candidate A :</p>
<p class="MsoNormal">P: C</p>
<p class="MsoNormal">Q: A>B(33%)</p>
<p class="MsoNormal">R: B>A(50%)</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Candidate A is the only candidate rated at a level of at
least 50% on at least 50% of the ballots.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">As in the first example, the equilibrium is preserved if the
33% is raised to any value less than P%, and/or the 50% is lowered to any value
greater than P percent.</p>
<p class="MsoNormal">P: C</p>
<p class="MsoNormal">Q: A>B(P%-epsilon)</p>
<p class="MsoNormal">R: B>A(P%+epsilon)</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Furthermore part of the B>A faction can defect without
destroying this equilibrium:</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">P: C</p>
<p class="MsoNormal">Q: A>B(P%-epsilon)</p>
<p class="MsoNormal">R1: B>A(P%+epsilon)</p>
<p class="MsoNormal">R2: B</p>
<p class="MsoNormal">For R=R1+R2 as long as R1 > P – Q .</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">So we see that XA has a rather robust Nash equilibrium that
protects the CWs in the context of a Chicken Dilemma threat.<span> </span>The threatened faction down-rates the
candidate of the potential defectors to any value less than P%.<span> </span>Since (in this context) P is always greater
than 33 (otherwise it could not be the largest of the three factions), the 33
percent rating can always be safely used to deter the defection.<span> </span>Mainly psychological reasons would make it
more satisfactory to raise that 33% closer to P%.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">So we see that high resolution ratings are not needed. Four levels will suffice nicely if they are 0, 33%, 50%, and 100%. Grade ballots like those used for Majority Judgement could be adapted to XA.</p><p class="MsoNormal"><br></p><p class="MsoNormal">As an approval variant like Bucklin, XA has no vulnerability to burial tactics.</p><p class="MsoNormal"><br></p><p class="MsoNormal">Unlike MMPO it also satisfies Plurality.</p><p class="MsoNormal"><br></p><p class="MsoNormal">It is monotone and clone independent (in the sense that Approval and Range are clone independent).</p><p class="MsoNormal"><br></p><p class="MsoNormal">It is efficiently summable, but is it precinct consistent? i.e. does a candidate that wins in every precinct win over-all?</p><p class="MsoNormal"><br></p><p class="MsoNormal">Does it satisfy Participation?</p><p class="MsoNormal"><br></p><p class="MsoNormal">We need to explore it, and learn how to explain it as simply as possible, so we can persuade people to use it.<span class="m_5889209740320896002HOEnZb"><font color="#888888"><br></font></span></p><span class="m_5889209740320896002HOEnZb"><font color="#888888"><p class="MsoNormal"><br></p><p class="MsoNormal">Forest<br></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
</font></span></div>
</div><br></div>
<br></div></div>----<br>
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