<div dir="auto">In the context of elimination methods (like IRV, Coombs, Baldwin, rtc, as well as all of our "worst-elimination" methods) the temptation for a faction to bury (insincerely lower on their ballots relative to one or more other candidates) a candidate C in order to help some candidate A win instead of C ... this temptation arises when C defeats A pairwise, but the A supporters, by lowering C, get C eliminated at some earlier elimination step so A and C are not competing head to head.<div dir="auto"><br></div><div dir="auto">Note that this burial ploy will not work wih IRV elimination, because lowering C on a ballot where A is already preferred over C will not decrease C's first place support ... so it cannot get C eliminated earlier ... since IRV elimination prioritizes low first place support.</div><div dir="auto"><br></div><div dir="auto">Coombs elimination, on the other hand prioritizes high last place counts for early elimination, so the burial ploy has a good chance of succeeding under Coombs.</div><div dir="auto"><br></div><div dir="auto">Note that the feature that gives IRV immunity to burial is the same feature that makes it vulnerable to the Squeeze Effect.</div><div dir="auto"><br></div><div dir="auto">So is it possible to have immunity to burial and squeeze in the same method?</div><div dir="auto"><br></div><div dir="auto">Yes, our "worst-elimination" methods have immunity to both... immunity to squeeze because of Condorcet efficiency and immunity to burial because in the above ploy, to eliminate C earlier (whether by burial or some other means) must backfire as long as the ballot change preserves C's pairwise win over A.</div><div dir="auto"><br></div><div dir="auto">It does preserve C's pairwise win over A in the case of burial ... because A was already ranked ahead of C by the buriers before the burial.</div><div dir="auto"><br></div><div dir="auto">So how does this fact make C's elimination before A backfire?</div><div dir="auto"><br></div><div dir="auto">Because according to our method...when C reaches "worst" status .... it is eliminated only "after any and every candidate defeated by it [including A] is eliminated"</div><div dir="auto"><br></div><div dir="auto">In other words, if and when C reaches "worst" status (with or without the push downward from A supporters), it takes down A with it. So it doesn't matter if our nominal standard of worst is "fewest first" or "most last" or anything else ... if it speeds up C's demise, it also speeds up the demise of any candidate that C defeats pairwise.</div><div dir="auto"><br></div><div dir="auto">In the three candidate case ... C is the sincere CW, and wins if C is eliminated, sothe other candidate B is the sincere Condorcet Loser.</div><div dir="auto"><br></div><div dir="auto">The A faction buries C under B, which creates a beat cycle ABCA. </div><div dir="auto"><br></div><div dir="auto">A thinks this cycle gives it a chance at winning ... which it would under most elimination methods.</div><div dir="auto"><br></div><div dir="auto">But not under ours, because, on the one hand A cannot win unless B or C is "worst" ... and ...</div><div dir="auto"><br></div><div dir="auto">If B is worst it takes A down with it because B defeats A in the cycle ... then B defeats C.</div><div dir="auto"><br></div><div dir="auto">On the other hand, if C is worst, it takes A down with it, leaving B as winner.</div><div dir="auto"><br></div><div dir="auto">So burial of the A faction's second choice results in the election of their anti-favorite B ... a complete backfire of the burial gambit!</div><div dir="auto"><br></div><div dir="auto">I hope that.explanation clarifies the main reason for the clearing out of the candidates defeated by the pivot candidate, i.e. the nominally "worst" candidate, at each elimination stage ... see there really is a "method to our madness".</div><div dir="auto"><br></div><div dir="auto">You may remember I once proposed a Quick & Dirty method that simply said elect the "best" candidate that pairwise defeats the "worst" Smith candidate.</div><div dir="auto"><br></div><div dir="auto">That's a shortcut rule of thumb that will elect the same candidate as our "worst-elimination" methods do whenever there are no more than three Smith members ... but the short cut is not Landau efficient ... so I don't recommend it. </div><div dir="auto"><br></div><div dir="auto">The main defect of the shortcut is that it requires some knowledge of Smith ... which our "worst-elimination" methods do not require. </div><div dir="auto"><br></div><div dir="auto">So even though Q&D is shorter ... it is neither quite as good nor quite as simple.</div><div dir="auto"><br></div><div dir="auto">If you have any question about any other method that you would like to compare with its nearest "worst-elimination" method ... it could interest other readers of the EM list, too.</div><div dir="auto"><br></div><div dir="auto">Remember "worst" is a nominal, tentative judgment that can hardly go wrong ... since the direct pairwise comparisons trump the tentative judgments if there is any disagreement.</div><div dir="auto"><br></div><div dir="auto">Good sources for "worst" candidates are losers of other methods. </div><div dir="auto"><br></div><div dir="auto">Also losing candidates in strong pairwise defeats ... for any decent gauge of defeat strength.</div><div dir="auto"><br></div><div dir="auto">Enjoy!</div><div dir="auto"><br></div><div dir="auto">Forest</div><div dir="auto"><br></div><div dir="auto"><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Feb 26, 2023, 8:37 AM Forest Simmons <<a href="mailto:forest.simmons21@gmail.com">forest.simmons21@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="auto">The ElectoScope aka Yee Diagram makes clear both the problem with and the solution to the Center Squeeze phenomenon ... elimination methods that judge "worst" by size of the Voronoi regions ten to suffer from the defect.<div dir="auto"><br></div><div dir="auto">But the cure is easy and sure ... no eliminations of undefeated candidates.</div><div dir="auto"><br></div><div dir="auto">All Condorcet Efficient methods have the same Yee Diagram ... the win region for a candidate is its entire Voronoi polygon, no matter how small.</div><div dir="auto"><br></div><div dir="auto">Next ... burial ...</div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Feb 24, 2023, 3:49 PM Forest Simmons <<a href="mailto:forest.simmons21@gmail.com" target="_blank" rel="noreferrer">forest.simmons21@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="auto"><div dir="auto"><br></div><div dir="auto">Why can't we just have majority rule? Why all the fiss?</div><div dir="auto"><br></div><div dir="auto">Many a student of my "Math for Liberal Arts" class asked me that question during the decades I taught the Community College course by that name.<br></div><div dir="auto"><br></div><div dir="auto"> That's the reason Joe Malkovich's contribution to the textbook was so important ... his examples of ballot profiles for which no two of several different majority rule methods agreed on who should be elected.</div><div dir="auto"><br></div><div dir="auto">Most if not all of these methods start out with the phrase..."Elect the majority winner if there is one, otherwise cull out the weakest (meaning democratically weakest) candidates one by one until there is a majority winner among the remaining."</div><div dir="auto"><br></div><div dir="auto">But there is no agreement on what constitutes "democratically weak' ... and it makes a big difference!</div><div dir="auto"><br></div><div dir="auto">So what can we do?</div><div dir="auto"><br></div><div dir="auto">One thing we have tried without much success is to suggest that the next best thing, lacking a first preference majority winner ... is to elect the candidate unbeaten by any majority comparison with another candidate.</div><div dir="auto"><br></div><div dir="auto">But just as there is no guaranteed outright majority winner ... neither is there any guarantee of the existence of a pairwise unbeaten candidate.</div><div dir="auto"><br></div><div dir="auto">It turns out that the best we can guarantee along these lines is the existence of at least one candidate that can pairwise beat in two steps every candidate that he cannot defeat in one step (by a majority of the participating voters).</div><div dir="auto"><br></div><div dir="auto">Such a candidate is said to be "uncovered." We're going to need a better word than that if we want to get anybody on board with this minimum guaranteeable standard of "majority rule."</div><div dir="auto"><br></div><div dir="auto">Let's say a candidate is "democratically strong" if it has a beatpath to every other candidate ... and is "very strong majority pairwise" if it has a beatpath of one or two steps to each of the other candidates ... each step being a pairwise victory by a majority of the participating voters ... meaning voters expressing a preference.</div><div dir="auto"><br></div><div dir="auto">Then the "Strong Majority Pairwise Criterion" (SMPC) is satisfied only by methods that always elect uncovered candidates. </div><div dir="auto"><br></div><div dir="auto">Contrast that with the weaker, relatively impotent Condorcet Criterion which is satisfied by any method that elects an unbeaten candidate "when such a candidate exists" ... the copout escape clause in quotes letting the method off the hook whenever things start to get interesting.</div><div dir="auto"><br></div><div dir="auto">Another way to express compliance with this SMPC criterion is "Landau Efficient."</div><div dir="auto"><br></div><div dir="auto">Every method under the "Worst-Elimination" umbrella is seamlessly Landau Efficient ... it effortlessly (and without fanfare) satisfies the SMPC ... no matter what nominal standard of worst is instantiated into the umbrella template.</div><div dir="auto"><br></div><div dir="auto">Who can name even one commonly known election method that is Landau efficient?</div><div dir="auto"><br></div><div dir="auto">What's more ... no matter the nominal "worst" criterion, the method will be more or less burial resistant ... as I will explain presently.</div><div dir="auto"><br></div><div dir="auto">I suggest that proposals for any method under this umbrella, include verbiage to the effect ...</div><div dir="auto"><br></div><div dir="auto">"When there is no majority winner or any candidate that a majority of the participating voters rank ahead of each of the other candidates ... cull out one-by-one the nominally "worst" candidates as well as any democratically weaker candidates (as determined by majority ballot preferences) until there is a majority winner among the remaining candidates."</div><div dir="auto"><br></div><div dir="auto">This umbrella is so robust that the choice of nominal "worst" is not overly critical. The main thing is to keep it simple enough that (1) voters can easily understand and relate to it, and (2) it can be efficiently and transparently tallied by precinct without multiple passes through the ballots.</div><div dir="auto"><br></div><div dir="auto">Complicated "worst" criteria are the ones that tend to introduce crowding and teaming distortions ... smallest Borda score is a example of this kind of "worst" criterion ... pun intended.</div><div dir="auto"><br></div><div dir="auto">Anti-vote splitting can be easily ensured (in general) by allowing equal-top whole counting, and multiple truncations in large elections.</div><div dir="auto"><br></div><div dir="auto">In the continuation I will explain why this method tends to backfire on buriers.</div><div dir="auto"><br></div><div dir="auto">At some point those who have power to advocate for one method over another need to understand them beyond the surface heuristics that appeal to the impatient public.</div><div dir="auto"><br></div><div dir="auto">Among other things enlightened defenders of electoral democracy need to understand the "squeeze effect" and "burial ploys" ...</div><div dir="auto"><br></div><div dir="auto">To be continued ...</div><div dir="auto"><br></div><div dir="auto">-Forest</div><div dir="auto"><br></div></div>
</blockquote></div>
</blockquote></div>