<html><head></head><body><div style="color:#000; background-color:#fff; font-family:HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif;font-size:12px"><div id="yui_3_16_0_ym19_1_1487704621185_243219"><span id="yui_3_16_0_ym19_1_1487704621185_246604"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_246603">Hi Steve,</font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_243218"><span><font size="3"><br></font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945"><span id="yui_3_16_0_ym19_1_1487704621185_242952"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_242951">Well, it seems clear to me that page 15 and slides 158-163 are talking about the same concept of being "partially strategy-proof-in-ranking." In both places the explanation offered by the authors is purely talking about how many candidates (out of a pair of two) a single voter might be able to adjust (in terms of their final grade). I think Kristofer gave the same interpretation on Jan 4 (although at the time I didn't understand the point).</font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945"><span><font size="3"><br></font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span id="yui_3_16_0_ym19_1_1487704621185_243684"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_243683">That is to say, when you describe page 15 as suggesting that a voter is unlikely to be able to change a result, I feel this is not relevant to this page. It's mathematically not possible at all to contrive a scenario where a single voter who rated A>B can change his ballot in a way to adjust the final grades of both A and B when (in the original outcome) B had a higher final grade than A. They're saying it may be possible for a voter to affect the grade of one, but he won't ever be able to affect the grade of both.</font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945"><span><font size="3"><br></font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945"><span id="yui_3_16_0_ym19_1_1487704621185_243682"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_243681">On page 15 as you quote, they seem to conclude from this discussion that MJ "cuts in half the probability of manipulation." I find it confusing that they would call this "cut in half" and that they would call this "manipulation" without some additional explanation, but I don't see how page 15 offers any other interpretation of what they're referring to.</font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945"><span><font size="3"><br></font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945"><span id="yui_3_16_0_ym19_1_1487704621185_244127"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_244126">Regarding pages 197-198, I suspect these are related to slides 166-171. They both seem to discuss a naive estimate of the possibility that a voter could affect the outcome. That would seem to be a different issue from what's on page 15, yet both are called either "manipulability" or "probability of manipulation." I would note that they say the lowest possible probability is n + 1/2n, but they don't say what is the highest possible probability. We may guess 100%, but would any method actually correspond to that?</font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945"><span><font size="3"><br></font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span id="yui_3_16_0_ym19_1_1487704621185_244776"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_244775">I don't really grasp what they are showing with the math: If n is a count of voters, why would probability be expressed in terms of n? Is this really the percentage of voters that can change the result? And if so would that really be just over 50% in the <i id="yui_3_16_0_ym19_1_1487704621185_246601"><span id="yui_3_16_0_ym19_1_1487704621185_245132">best</span> case</i>? (I wonder if Kristofer has already explained all this and I just didn't read it...) </font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span><font size="3"><br></font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span id="yui_3_16_0_ym19_1_1487704621185_245171"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_245170">To clarify one item:</font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span style="font-size: 13.3333px;" id="yui_3_16_0_ym19_1_1487704621185_245192">>>K: This means that if the relative order of two candidates' final grades is not to the liking of some voter, that voter may be </span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span style="font-size: 13.3333px;" id="yui_3_16_0_ym19_1_1487704621185_245386">>>able to adjust the grade of one of the candidates, but not both. "One and not two" seems promising as something that has </span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span style="font-size: 13.3333px;" id="yui_3_16_0_ym19_1_1487704621185_245379">>>been cut by half.</span><span><font size="3"><br></font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span style="font-size: 13.3333px;" id="yui_3_16_0_ym19_1_1487704621185_245243">>S:</span><span style="font-size: 13.3333px;" id="yui_3_16_0_ym19_1_1487704621185_245244"> </span><span style="font-size: 13.3333px;" id="yui_3_16_0_ym19_1_1487704621185_245245">Does your last phrase indicate that you now accept the validity of B&L’s claim that MJ “cuts” manipulation in “half”?</span><span style="font-size: 13.3333px;"><br></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span style="font-size: 13.3333px;"><br></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span id="yui_3_16_0_ym19_1_1487704621185_245544"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_245543">It indicates that I think "one and not two" is where they are getting "half." If so, I accept that this has been "cut in half," but I'm skeptical that this concept is a good representation of the "manipulability" of an SGF.</font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><font size="3"><br></font></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span id="yui_3_16_0_ym19_1_1487704621185_245859"><font size="3" id="yui_3_16_0_ym19_1_1487704621185_245858">Kevin</font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><span style="font-family: Calibri;"><font size="3"><br></font></span></div><div id="yui_3_16_0_ym19_1_1487704621185_242945" dir="ltr"><br></div><div class="qtdSeparateBR"><br><br></div><div class="yahoo_quoted" id="yui_3_16_0_ym19_1_1487704621185_242723" style="display: block;"> <div style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 12px;" id="yui_3_16_0_ym19_1_1487704621185_242722"> <div style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 16px;" id="yui_3_16_0_ym19_1_1487704621185_242721"> <div dir="ltr" id="yui_3_16_0_ym19_1_1487704621185_242941"> <font size="2" face="Arial" id="yui_3_16_0_ym19_1_1487704621185_242943"> <hr size="1" id="yui_3_16_0_ym19_1_1487704621185_242942"> <b id="yui_3_16_0_ym19_1_1487704621185_246609"><span style="font-weight:bold;" id="yui_3_16_0_ym19_1_1487704621185_246608">De :</span></b> steve bosworth <stevebosworth@hotmail.com><br> <b><span style="font-weight: bold;">À :</span></b> "stepjak@yahoo.fr" <stepjak@yahoo.fr>; "election-methods@lists.electorama.com" <election-methods@lists.electorama.com> <br> <b><span style="font-weight: bold;">Envoyé le :</span></b> Jeudi 23 février 2017 8h53<br> <b><span style="font-weight: bold;">Objet :</span></b> Re: [EM] (3) ?goal of a better election method?<br> </font> </div> <div class="y_msg_container" id="yui_3_16_0_ym19_1_1487704621185_242720"><br><div id="yiv8039895279">
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<span style="color:black;font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_242728"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_242727">HI Kevin,<br>
Message: 2<br>
Date: Sun, 19 Feb 2017 19:20:34 +0000 (UTC)<br>
From: Kevin Venzke <stepjak@yahoo.fr><br>
To: steve bosworth <stevebosworth@hotmail.com>,<br>
"election-methods@lists.electorama.com"<br>
<election-methods@lists.electorama.com><br>
Subject: Re: [EM] goal of a better election method<br>
Message-ID: <1682602380.1138980.1487532034974@mail.yahoo.com><br>
Content-Type: text/plain; charset="utf-8"<br>
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K: Ok, I read the entire .pdf.<span style=""> </span>I don't think the "half the manipulability" claim is related to slide 150.</font></span></div>
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<span style="font-size:10pt;"> </span></div>
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<span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_242730">S:<span style="">
</span>Sorry, I did not mean that this claim is explicit on slide 150.<span style="">
</span>Slide 150 only</span><span style="color:black;font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_243010"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_243009"> again refers to the more
limited scope for “manipulation” offered by MJ‘s social grading function (SGF) in contrast to any method using “point-summing” which allows “all voters [to] manipulate:<span style="">
</span>“</font></span><span style="color:red;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_242732">If the mechanism is a point-summing method (the mean with respect to some parametrization), for almost all profiles, all voters can manipulate</span><span style="color:black;font-family:SFSS1095;">.”</span></div>
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<span style="color:black;font-size:10pt;"><font face="Calibri">Before slide 150, slide 142 promised that: “</font></span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_242733">We
are going to prove that majority judgement is </span><span style="color:red;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_243241">strategy-proof
</span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_243011">for a large class of utility functions. When it is not, it is shown that it
</span><span style="color:red;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_243149">combats manipulations
</span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_242735">in many well defined senses”.</span><span style="color:black;font-size:10pt;"><font face="Calibri">
</font></span></div>
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<span style="color:black;font-family:SFSS1095;">However, o</span><span style="color:black;font-size:12pt;" id="yui_3_16_0_ym19_1_1487704621185_242737"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_242736">n page 15 in Chapter 1 of
B&L’s book (<i style="">Majority Judgment</i>, 2011, MIT), they verbally illustrate why the above claims are true.<span style="">
</span>I will quote this passage shortly but please correct me if I am mistaken in seeing it as making the following simple point:<span style="">
</span>Using MJ for the French election, a voter honestly would give Royal a grade of at least “good” and Bayrou less than “good”.<span style="">
</span>If she tries to manipulate the result, she has practically no chance of increasing the probability of Royal being elected simply by giving Royal a “very good” or an ”excellent” instead, and giving Bayrou a “poor” or “reject”.<span style="">
</span>Whether she votes honestly or manipulatively, each candidate’s “median-grade” will almost certainly remain the same.<span style="" id="yui_3_16_0_ym19_1_1487704621185_243735">
</span>She would only have the best chance of changing the result if she instead were to give Royal a grade lower than “good” and Bayrou “higher” than “good”, i.e. voted in a way diametrically opposed to her own honest evaluations of the candidates, presumably
not her desire:</font></span></div>
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<span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245193">Page 15:<span style="">
</span>“How could a voter who graded Royal higher than Bayrou manipulate?<span style="">
</span>By changing</span></div>
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<span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245195">the grades assigned to try to lower Bayrou’s majority-gauge and to raise</span></div>
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<span style="font-size:10pt;">Royal’s majority-gauge. But the majority judgment is
</span><i id="yui_3_16_0_ym19_1_1487704621185_244830"><span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_244831">partially strategy-proof in-</span></i></div>
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<i><span style="font-size:10pt;">ranking</span></i><span style="font-size:10pt;">: those voters who can lower Bayrou’s
majority-gauge cannot raise</span></div>
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<span style="font-size:10pt;">Royal’s, and those who can raise Royal’s majority-gauge cannot lower Bayrou’s.</span></div>
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<span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245247">For suppose a voter can lower Bayrou’s. Then she must have given Bayrou</span></div>
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<span style="font-size:10pt;">a
</span><i><span style="font-size:10pt;">Good
</span></i><span style="font-size:10pt;">or better; but having preferred Royal to Bayrou, the voter gave a grade</span></div>
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<span style="font-size:10pt;">of better than
</span><i><span style="font-size:10pt;">Good
</span></i><span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245283">to Royal, so she cannot raise Royal’s majority-gauge (cannot</span></div>
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<span style="font-size:10pt;">raise her
</span><i><span style="font-family:MTMI;font-size:10pt;">p</span></i><span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245284">). Symmetrically, a voter who can raise Royal’s majority-gauge</span></div>
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<span style="font-size:10pt;">must have given her a
</span><i><span style="font-size:10pt;">Good
</span></i><span style="font-size:10pt;">or worse and thus to Bayrou a worse than
</span><i><span style="font-size:10pt;">Good</span></i><span style="font-size:10pt;">;</span></div>
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<span style="font-size:10pt;">so the voter cannot lower Bayrou’s majority-gauge (cannot increase his
</span><i><span style="font-family:MTMI;font-size:10pt;">q</span></i><span style="font-size:10pt;">).</span></div>
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<span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245218">Compared with other mechanisms, the majority judgment<span style="color:red;"> cuts in half the</span></span></div>
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<span style="color:red;font-size:10pt;">possibility of manipulation</span><span style="font-size:10pt;">, however bizarre
a voter’s motivations or whatever</span></div>
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<span style="font-size:10pt;">her utility function. The majority judgment resists manipulation in still other</span></div>
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<span style="font-size:10pt;">ways that other methods do not, but to see how requires information found</span></div>
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<div style="margin:0in 0in 8pt;" id="yui_3_16_0_ym19_1_1487704621185_243854"><span style="line-height:107%;font-size:10pt;">in voters’ individual ballots ….”</span></div>
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<div style="margin:0in 0in 8pt;line-height:12pt;" id="yui_3_16_0_ym19_1_1487704621185_243853"><span style="color:black;font-size:12pt;" id="yui_3_16_0_ym19_1_1487704621185_243852"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_243851">S:
With regard to B&L’s claim to offer an exact mathematic “proof” that MJ reduces by about “half” the scope for “manipulation” as that offered by the “point-summing” methods, can you please explain your own view of this claim.<span style="">
</span>More exactly, do the following quoted passages merely restate the above explanation by using only pure mathematic notations, or do they mathematically provide us will the exact relevant number close to one half?<span style="">
</span>Either way, I would very much appreciate it if you would help me to fully understand the case B&L make in their book in the pages leading up to pp.197-8 which contain the following formulaes:
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<div style="margin:0in 0in 8pt;line-height:12pt;" id="yui_3_16_0_ym19_1_1487704621185_243757"><span style="color:black;font-size:12pt;" id="yui_3_16_0_ym19_1_1487704621185_243858"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_243857">“For,
a priori, a judge who manipulates wishes to increase the final grade with probability ½ and wishes to decrease it with probability ½.<span style="">
</span>And, a priori, any one of the n judges may cheat.<span style="">
</span>Thus, when the input of grades is <b style="">
r</b>, the probability for a judge to manipulate successfully when using an SGF with aggregation function
<i style="">f</i> is</font></span></div>
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<div style="margin:0in 0in 8pt;" id="yui_3_16_0_ym19_1_1487704621185_243942"><font face="Calibri"><span style="color:black;font-size:20pt;">(</span><span style="color:black;font-size:12pt;">1/2</span><span style="color:black;font-size:20pt;">)</span><span style="color:black;font-size:10pt;">
</span><span style="color:black;font-size:24pt;">(</span></font><u><span style="color:black;font-family:CMMI10;">μ</span></u><u><sup><span style="color:black;font-family:CMSY8;font-size:10pt;">−</span></sup></u><font face="Calibri"><u><span style="color:black;font-size:12pt;">
(<i style="">f</i> (<b style="">r</b>)</span></u><span style="color:black;font-size:24pt;">)</span><span style="color:black;font-size:10pt;">
+ </span><span style="color:black;font-size:24pt;">(</span><span style="color:black;font-size:12pt;">1/2</span><span style="color:black;font-size:24pt;">)
(</span></font><u><span style="color:black;font-family:CMMI10;">μ</span></u><u id="yui_3_16_0_ym19_1_1487704621185_245310"><sup id="yui_3_16_0_ym19_1_1487704621185_245309"><span style="color:black;font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245308">+</span></sup></u><font face="Calibri"><u><span style="color:black;font-size:12pt;">
(<i style="">f</i> (<b style="">r</b>)</span></u><span style="color:black;font-size:24pt;">)</span><span style="color:black;font-size:10pt;">
=<span style=""> </span>(</span></font><u><span style="color:black;font-family:CMMI10;">μ</span></u><font face="Calibri"><u><span style="color:black;font-size:12pt;">
(<i style="">f</i> (<b style="">r</b>)</span></u><span style="color:black;font-size:10pt;">
)</span></font></div>
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<div style="margin:0in 0in 8pt;line-height:12pt;" id="yui_3_16_0_ym19_1_1487704621185_244099"><span style="color:black;font-size:10pt;"><font face="Calibri"><span style="">
</span>n<span style=""> </span><span style="">
</span>n<span style=""> </span>2n</font></span></div>
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<div style="margin:0in 0in 8pt;line-height:12pt;" id="yui_3_16_0_ym19_1_1487704621185_244098"><span style="color:black;font-size:12pt;" id="yui_3_16_0_ym19_1_1487704621185_244191"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_244190">p.198:<span style="">
</span>“One would like this probability to be as small as possible in the worst case; this amounts to finding an
<i style="">f</i> that solves</font></span></div>
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<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244770">
<span style="color:black;font-family:CMSS10;"><span style=""> </span></span><span style="color:black;font-family:SFSS1095;">min<span style="">
</span>max</span><span style="color:black;font-family:CMMI10;">_<span style="">
</span>μ</span><span style="color:black;font-size:12pt;" id="yui_3_16_0_ym19_1_1487704621185_245211"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_245210"> (<i style="" id="yui_3_16_0_ym19_1_1487704621185_245209">f</i>
(<b style="">r</b>))</font></span><span style="color:black;font-family:CMMI10;"> =<span style="">
</span></span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_245311">min
</span><span style="color:black;font-family:CMMI10;" id="yui_3_16_0_ym19_1_1487704621185_245312">μ</span><span style="color:black;font-size:12pt;"><font face="Calibri">
(<i style="">f</i> ).</font></span></div>
<font face="Times New Roman"></font>
<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244097">
<span style="color:black;font-family:CMMI10;"><span style="">
</span><i style="">f</i><span style="">
</span></span><span style="color:black;font-size:8pt;">r
</span><span style="color:black;font-size:8pt;">= (</span><span style="color:black;font-size:8pt;">r</span><span style="color:black;font-size:6pt;">1</span><span style="color:black;font-family:CMMI8;font-size:8pt;">,...,</span><span style="color:black;font-size:8pt;">r</span><span style="color:black;font-size:6pt;">n</span><span style="color:black;font-size:8pt;" id="yui_3_16_0_ym19_1_1487704621185_245213">)<span style="">
</span><span style="" id="yui_3_16_0_ym19_1_1487704621185_245212"> </span><span style=""> </span></span><i style=""><span style="color:black;font-family:CMMI10;">f</span></i></div>
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<div style="margin:0in 0in 8pt;line-height:12pt;" id="yui_3_16_0_ym19_1_1487704621185_244064"><span style="color:black;font-size:10pt;"><span style=""><font face="Calibri">
</font></span></span></div>
<font face="Times New Roman"></font>
<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244058">
<span style="color:black;font-size:12pt;" id="yui_3_16_0_ym19_1_1487704621185_244060"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_244059">that is, minimizes the manipulability.<span style="">
</span>Thus the minimum probability is slightly over one-half, n+1/2n, and is only achieved when
<i style="">f</i> is an order function.”<br>
</font></span><span style="font-size:12pt;"><span style="">
</span></span></div>
<font face="Times New Roman"></font>
<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_243939">
<span style="color:black;font-size:12pt;" id="yui_3_16_0_ym19_1_1487704621185_245215"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_245214">S:<span style="">
</span>Also, how exactly does the above relate to Laraki’s slide 170: </font></span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_244061"><span style=""> </span>"Given an aggregation function
</span><i style=""><span style="color:black;font-family:SFSI1095;">f</span></i><span style="color:black;font-family:SFSI1095;">
</span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_244062">and input
</span><span style="color:black;">r
</span><span style="color:black;font-family:CMSS10;" id="yui_3_16_0_ym19_1_1487704621185_244185">= (</span><span style="color:black;font-family:SFSI1095;" id="yui_3_16_0_ym19_1_1487704621185_244731">r</span><span style="color:black;font-size:8pt;" id="yui_3_16_0_ym19_1_1487704621185_244189">1</span><span style="color:black;font-family:CMMI10;" id="yui_3_16_0_ym19_1_1487704621185_244294">,
. . . , </span><span style="color:black;font-family:SFSI1095;">r</span><span style="color:black;font-size:8pt;">n</span><span style="color:black;font-family:CMSS10;">)</span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_244121">,
let</span></div>
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<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244186">
<span style="color:black;font-family:CMMI10;">μ</span><sup><span style="color:black;font-family:CMSY8;font-size:10pt;">−</span></sup><span style="color:black;font-family:CMSS10;">(</span><i style=""><span style="color:black;font-family:SFSI1095;">f</span></i><span style="color:black;font-family:SFSI1095;">
</span><span style="color:black;font-family:CMMI10;">,
</span><span style="color:black;">r</span><span style="color:black;font-family:CMSS10;">) =
</span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_245216">nbre of judges who can decrease the final grade,</span></div>
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<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244193">
<span style="color:black;font-family:CMMI10;">μ</span><sup><span style="color:black;font-size:10pt;">+</span></sup><span style="color:black;font-family:CMSS10;">(</span><i style=""><span style="color:black;font-family:SFSI1095;">f</span></i><span style="color:black;font-family:SFSI1095;">
</span><span style="color:black;font-family:CMMI10;">,
</span><span style="color:black;">r</span><span style="color:black;font-family:CMSS10;">) =
</span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_245316">nbre of judges who can increase the final grade,</span></div>
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<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244070">
<span style="color:black;font-family:SFSS1095;"> </span></div>
<font face="Times New Roman"></font>
<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244096">
<span style="color:black;font-family:SFSS1095;">Let
</span><span style="color:black;font-family:Symbol;">l</span><span style="color:black;font-family:CMMI10;">
</span><span style="color:black;font-family:CMSS10;">=
</span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_245313">probability a judge wishes to increase the final grade. The</span></div>
<font face="Times New Roman"></font>
<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244095">
<span style="color:red;font-family:SFSI1095;">probability of effective-manipulability
</span><span style="color:black;font-family:SFSS1095;">of
</span><i style=""><span style="color:black;font-family:SFSI1095;">f</span></i><span style="color:black;font-family:SFSI1095;">
</span><span style="color:black;font-family:SFSS1095;">is</span></div>
<font face="Times New Roman"></font>
<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244094">
<span style="color:black;font-family:SFSI1095;"> </span></div>
<font face="Times New Roman"></font>
<div style="margin:0in 0in 0pt;line-height:normal;" id="yui_3_16_0_ym19_1_1487704621185_244089">
<span style="color:black;font-family:SFSI1095;">EM</span><span style="color:black;font-family:CMSS10;">(</span><i style=""><span style="color:black;font-family:SFSI1095;">f</span></i><span style="color:black;font-family:SFSI1095;">
</span><span style="color:black;font-family:CMSS10;">) =
</span><span style="color:black;font-family:SFSS1095;">max<span style="">
</span></span><span style="color:black;font-size:8pt;"><span style=""> </span></span><span style="color:black;font-family:SFSS1095;" id="yui_3_16_0_ym19_1_1487704621185_245287">max</span><span style="color:black;font-family:CMMI10;">_<span style="">
</span></span><u><span style="color:black;font-family:Symbol;">l</span></u><u><span style="color:black;font-family:CMMI10;"> μ</span></u><u><sup><span style="color:black;font-size:10pt;">+</span></sup></u><u><span style="color:black;font-family:CMSS10;">(</span></u><i style=""><u><span style="color:black;font-family:SFSI1095;">f</span></u></i><u><span style="color:black;font-family:SFSI1095;">
</span></u><u><span style="color:black;font-family:CMMI10;">,
</span></u><u><span style="color:black;">r</span></u><u id="yui_3_16_0_ym19_1_1487704621185_245315"><span style="color:black;font-family:CMSS10;" id="yui_3_16_0_ym19_1_1487704621185_245314">) + (</span></u><u><span style="color:black;font-family:SFSS1095;">1
</span></u><u><span style="color:black;font-family:CMSY10;">−</span></u><u><span style="color:black;font-family:Symbol;">l</span></u><u><span style="color:black;font-family:CMSS10;">)
</span></u><u><span style="color:black;font-family:CMMI10;">μ</span></u><u><sup><span style="color:black;font-family:CMSY8;font-size:10pt;">−</span></sup></u><u><span style="color:black;font-family:CMSS10;">(</span></u><i style=""><u><span style="color:black;font-family:SFSI1095;">f</span></u></i><u><span style="color:black;font-family:SFSI1095;">
</span></u><u><span style="color:black;font-family:CMMI10;">,
</span></u><u><span style="color:black;">r</span></u><u><span style="color:black;font-family:CMSS10;">)</span></u><span style="color:black;font-family:CMSS10;">
</span></div>
<font face="Times New Roman"></font>
<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244312">
<span style="color:black;font-family:CMMI10;"><span style="">
</span><span style=""> </span></span><span style="color:black;font-size:8pt;">r
</span><span style="color:black;font-size:8pt;">= (</span><span style="color:black;font-size:8pt;">r</span><span style="color:black;font-size:6pt;">1</span><span style="color:black;font-family:CMMI8;font-size:8pt;">,...,</span><span style="color:black;font-size:8pt;">r</span><span style="color:black;font-size:6pt;">n</span><span style="color:black;font-size:8pt;">)<span style="">
</span></span><span style="color:black;font-size:8pt;">0</span><span style="color:black;font-family:CMSY8;font-size:8pt;">≤
</span><span style="color:black;font-family:Symbol;">l</span><span style="color:black;font-family:CMMI8;font-size:8pt;">
</span><span style="color:black;font-family:CMSY8;font-size:8pt;">≤</span><span style="color:black;font-size:8pt;">1<span style="">
</span></span><span style="color:black;font-family:SFSI1095;">n</span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244360">
<span style="color:black;font-size:10pt;"><font face="Calibri"> </font></span></div>
<font face="Times New Roman"></font>
<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244021">
<span style="color:black;font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_244842"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_244841">S:<span style="">
</span>I can send you the above formulaes as an attachment if they have been scrambled by your inbox.</font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244051">
<span style="color:black;font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_244068"><br>
<font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_244067">K: Leading up to slide 150, [Laraki] argues that *if* your goal is simply for candidate X to get a grade of 7, then under order functions like MJ, your best strategy is to rate that candidate a 7. Under Range this is not true, because if
that candidate is sitting at a 1, rating him a 10 would drag him towards a 7 rating faster than just rating him a 7 would. There is no "half" to be found here; it doesn't seem that any SGF could outperform MJ on this measur</font></span><span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_244050"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_244049">e.<br>
On slide 158 they describe the concept of being "partially strategy-proof-in-ranking." This means that if the relative order of two candidates' final grades is not to the liking of some voter, that voter may be able to adjust the grade of one of the candidates,
but not both. "One and not two" seems promising as something that has been cut by half.</font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244181">
<span style="font-size:10pt;"><font face="Calibri"> </font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_245205">
<span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245207"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_245206">S:<span style="">
</span>Does your last phrase indicate that you now accept the validity of B&L’s claim that MJ “cuts” manipulation in “half”?</font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244053">
<span style="font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_244363"><br>
<font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_245249">K: Then</font><span style="color:black;" id="yui_3_16_0_ym19_1_1487704621185_244362"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_244361"> on 163 they say that order functions are the only SGFs that have this property.<br>
What do you think, is there a better suggestion for something that has been halved? I would sort of hope so because if the answer is always either "one candidate" or "both candidates" that means there are only two outcomes for a test we're taking as a representation
of manipulability under SGFs.</font></span></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244836">
<span style="color:black;font-size:10pt;"><font face="Calibri"> </font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244834">
<span style="color:black;font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245204"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_245203">S:<span style="">
</span>I hope to understand the point you are making here after receiving your help and comments on all the above.<span style="">
</span>I look forward to your feedback.</font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244183">
<span style="color:black;font-size:10pt;" id="yui_3_16_0_ym19_1_1487704621185_245202"><font face="Calibri" id="yui_3_16_0_ym19_1_1487704621185_245201">Steve<br>
<br>
De?: steve bosworth <stevebosworth@hotmail.com><br>
??: "stepjak@yahoo.fr" <stepjak@yahoo.fr>; "election-methods@lists.electorama.com" <election-methods@lists.electorama.com>
<br>
Envoy? le : Dimanche 19 f?vrier 2017 3h03<br>
Objet?: [EM] goal of a better election method<br style="">
<br style="">
</font></span></div>
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<span style="color:black;font-size:10pt;"><font face="Calibri">Message: 2<br>
Date: Sun, 12 Feb 2017 17:31:37 +0000 (UTC)<br>
From: Kevin Venzke <stepjak@yahoo.fr><br>
To: steve bosworth <stevebosworth@hotmail.com>,? EM list<br>
??????? <election-methods@electorama.com><br>
Subject: Re: [EM] ?goal of a better election method?<br>
Message-ID: <1346455147.4984684.1486920697303@mail.yahoo.com><br>
Content-Type: text/plain; charset="utf-8"<br>
<br>
Hi Kevin:<br>
> S: Secondly, I believe that B&L ? prove? (pp. 15, 19, 186-198) that MJ provides only about ? half? the incentives? >or opportunities for anti-democratic ?strategic? voting to be successful. If you disagree, please explain the flaw in their argument.<br>
<br>
K: What I understood from Kristofer's Jan 4 explanation of page 15 is that?this halving of the manipulability is not meant to be a comparison to?any other methods. It's a comparison to a (rather strange) hypothetical situation. If so, this claim on its own
could be true but is of unclear value.<br>
Possibly this argument is used to build up to a larger argument. But when it gets stated on its own it feels misleading to me, because there's no way for the reader to understand what this "half" is half of.</font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_245198">
<span style="color:black;font-size:10pt;"><font face="Calibri"> </font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;">
<span style="color:black;font-size:10pt;"><font face="Calibri">S: No, B&L clearly explain why MJ provides only about
?half? of the incentives and opportunities to ?manipulate? the results as compared to all the methods that gain their results by ?summing? or ?averaging? all the votes.? In his reply to me, Kristofer also gave me a link to the following source in which B&L
concisely state the same claim on slide 150.?<span style=""> </span>
This is in the middle of their discussion of ?strategy?, i.e. between slides 143 & 185:</font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;">
<span style="color:black;font-size:10pt;"><font face="Calibri"> </font></span></div>
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<div style="background:white;margin:0in 0in 0pt;line-height:normal;vertical-align:bottom;" id="yui_3_16_0_ym19_1_1487704621185_244364">
<span style="color:black;font-size:10pt;"><font face="Calibri">Kristofer:<span style="">
</span>See this slide set by B&L for more on that:<br>
</font></span><a rel="nofollow" id="yiv8039895279LPlnk868326" target="_blank" href="http://igm.univ-mlv.fr/AlgoB/algoperm2012/01Laraki.pdf"><span style="color:blue;font-size:10pt;"><font face="Calibri">http://igm.univ-mlv.fr/AlgoB/algoperm2012/01Laraki.pdf</font></span></a><span style="color:black;font-size:10pt;"><font face="Calibri">
. <br>
| Majority Judgement - Measuring, Ranking and Electing igm.univ-mlv.frTraditional Methods and results Incompatibility Between Electing and Ranking Majority Judgement: Two Applications Majority Judgement Measuring, Ranking and Electing |<br>
<br>
<br>
I look forward to your feedback.</font></span></div>
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<span style="color:black;font-size:10pt;"><font face="Calibri">Steve</font></span></div>
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<span style="color:black;font-size:10pt;"><font face="Calibri">Kevin<br style="">
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<div style="margin:0in 0in 8pt;"><font face="Calibri">++++++++++++++++++++++</font><span style="color:black;font-size:10pt;"><br>
<font face="Calibri">2. Re: ?goal of a better election method? (Kevin Venzke)<br>
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Message: 2<br>
Date: Sun, 12 Feb 2017 17:31:37 +0000 (UTC)<br>
From: Kevin Venzke <stepjak@yahoo.fr><br>
To: steve bosworth <stevebosworth@hotmail.com>, EM list<br>
<election-methods@electorama.com><br>
Subject: Re: [EM] ?goal of a better election method?<br>
Message-ID: <1346455147.4984684.1486920697303@mail.yahoo.com><br>
Content-Type: text/plain; charset="utf-8"<br>
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Hi Kevin:<br>
> S: Secondly, I believe that B&L ? prove? (pp. 15, 19, 186-198) that MJ provides only about ?
<span style=""> </span>half? <span style=""> </span>the incentives? >or opportunities for anti-democratic ?strategic? voting to be successful. If you disagree, please explain the flaw in their argument.<br>
<br>
K: What I understood from Kristofer's Jan 4 explanation of page 15 is that? this halving of the manipulability is not meant to be a comparison to ?any other methods. It's a comparison to a (rather strange) hypothetical situation. If so, this claim on its own
could be true but is of unclear value.<br>
Possibly this argument is used to build up to a larger argument. But when it gets stated on its own it feels misleading to me, because there's no way for the reader to understand what this "half" is half of.<br>
<br>
S: No, B&L clearly explain why MJ provides only about ?half? of the incentives and opportunities to ?manipulate? the results as compared to all the methods that gain their results by ?summing? or ?averaging? all the votes. In his reply to me, Kristofer also
gave me a link to the following source in which B&L concisely state the same claim on slide 150. This is in the middle of their discussion of ?strategy?, i.e. between slides 143 & 185:<br>
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Kristofer: See this slide set by B&L for more on that:<br>
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Majority Judgement - Measuring, Ranking and Electing<</font></span><a rel="nofollow" target="_blank" href="http://igm.univ-mlv.fr/AlgoB/algoperm2012/01Laraki.pdf"><span style="color:blue;line-height:107%;font-size:10pt;"><font face="Calibri">http://igm.univ-mlv.fr/AlgoB/algoperm2012/01Laraki.pdf</font></span></a><span style="color:black;line-height:107%;font-size:10pt;"><font face="Calibri">><br>
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Traditional Methods and results Incompatibility Between Electing and Ranking Majority Judgement: Two Applications Majority Judgement Measuring, Ranking and Electing<br>
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I look forward to your feedback.<br>
<br>
Steve<br>
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Kevin<br style="">
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