<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.<br></p><p class="MsoNormal"><br></p><p class="MsoNormal">Forest<br></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
</div>
</div><br></div>