<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_1474048662928_8721"><span>Hi Mike,</span></div><div id="yui_3_16_0_ym19_1_1474048662928_8642"><span><br></span></div><div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8641"><span id="yui_3_16_0_ym19_1_1474048662928_8640">If the cast ballots (in your first example) were A>B, B, and C, how could a possible equilibrium be A, B=A, and C? Only one faction showed any willingness to vote for A, correct?</span></div><div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8644"><span><br></span></div><div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8645"><span>Kevin</span></div><div class="qtdSeparateBR" id="yui_3_16_0_ym19_1_1474048662928_8471"><br><br></div><div class="yahoo_quoted" id="yui_3_16_0_ym19_1_1474048662928_8480" 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_1474048662928_8479"> <div style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 16px;" id="yui_3_16_0_ym19_1_1474048662928_8478"> <div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8477"> <font size="2" face="Arial" id="yui_3_16_0_ym19_1_1474048662928_8476"> <hr size="1" id="yui_3_16_0_ym19_1_1474048662928_8715"> <b><span style="font-weight:bold;">De :</span></b> Michael Ossipoff <email9648742@gmail.com><br> <b><span style="font-weight: bold;">À :</span></b> election-methods@electorama.com <br> <b><span style="font-weight: bold;">Envoyé le :</span></b> Vendredi 16 septembre 2016 10h47<br> <b><span style="font-weight: bold;">Objet :</span></b> [EM] EM equilibrium dfn. NEO properties-examples.<br> </font> </div> <div class="y_msg_container" id="yui_3_16_0_ym19_1_1474048662928_8512"><br><div id="yiv0186407285"><div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8649">EM has discussed Nash equilibrium a number of times, & this, if seems to me, is what EM agrees Nash equilibrium to mean:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8511">A "cohort" is a set of voters who prefer & vote the same as eachother. </div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8537">At EM, for voting-systems, a Nash equilibrium is an outcome that no cohort can improve for itself by changing its vote.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8559">(end of dfn)</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8514">NEO assumes that the voters' rankings are sincere, & indicate the voters' actual preferences & indifferences.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8513">Chicken dilemma:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8560">The usual example:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8724">3 candidates: A, B, & C.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8561">The A voters & B voters are a majority who greatly prefer A & B to C. (though NEO of course doesn't recognize unexpressed preferences)</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8562">Faction size relations:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8566">C > A > B</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8563">The C voters are indifferent between A & B, & dislike both</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8565">Rankings:<br></div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8564">A voters: A > B<br>
B voters: B<br>
C voters: C</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8594">Two Approval Nash equilibria:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8569">A,B<br>
B<br>
C</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8851">Electing B.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8702">and</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8852">A<br>
B,A<br>
C</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8576">Electing A.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8653">So, find the equilibria in an election with just A & B:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8652">A<br>
B</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8701">and</div>
<div dir="ltr">A<br>
B, A</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8656">Either way A wins.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8657">CD's requirement, that B not win, is met.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8854">Truncation against CWs:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8658">Instead of A, B, & C, I prefer:</div>
<div dir="ltr">Worst, Middle, & Favorite. </div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8659">W, M, & F.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8665">More expressive. Of course the W voters are the offensive strategizers.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8664">Rankings:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8662">W voters: W<br>
M voters: M>W<br>
F voters: F>M</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8661">Approval Nash Equilibrium:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8663">F voters: F, M<br>
M voters: M<br>
W voters: W</div>
<div dir="ltr">If the F voters don't approve M, that could only change the winner to W, worsening the outcome for them.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8681">If the M voters approve W, that could only change the winner to W, worsening the outcome for them.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8693">W voters gain nothing by approving M. That's another Nash equilibrium.</div>
<div dir="ltr">M wins in both Equilibria.</div>
<div dir="ltr">Burial & defensive truncation:</div>
<div dir="ltr">Rankings:</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8694">F voters: F>M<br>
M voters: M<br>
W voters: W>F</div>
<div dir="ltr">Approval Nash Equilibria;</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8698">F voters: F, M<br>
M voters: M<br>
W voters: W,F</div>
<div dir="ltr">That's a disequilibrium, because the F voters' approval of M could change the winner from F to M. They withdraw that Approval:</div>
<div dir="ltr">F voters : F<br>
M voters: M<br>
W voters: W, F</div>
<div dir="ltr">That's a Nash equilibrium.</div>
<div dir="ltr">(W voters are assumed to prefer F to M, due to their ranking)</div>
<div dir="ltr">F wins the NEO election. The burial is thwarted & penalized.</div>
<div dir="ltr" id="yui_3_16_0_ym19_1_1474048662928_8696">Michael Ossipoff<br><br><br><br><br><br><br><br><br><br><br><br><br><br></div></div><br>----<br>Election-Methods mailing list - see <a href="http://electorama.com/em" target="_blank">http://electorama.com/em </a>for list info<br><br><br></div> </div> </div> </div></div></body></html>