<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body dir="auto"><div><span></span></div><div><meta http-equiv="content-type" content="text/html; charset=utf-8"><div></div><div><br></div><div>Elimination methods may have a policy mandate but they do not have a scientific or knowledge mandate. They violate conservation of (preference) information. Elimination independent of scale of preference is precisely the fault, found over two centuries ago, by Pierre-Simon Laplace, according to JFS Ross, with Condorcet pairing elimination of candidates. It does not take into account the relative importance of higher to lower preferences. Why Laplace sided with Borda amounted to his opening the way to rational counts of preference ranges. Of which Gregory supplied the definitive statistical method to next preferences, of weighting in arithmetic proportion.</div><div><br></div><div>It is doubtful whether the Condorcet Winner (or conversely Loser) applies to more than single winner elections, the least democratic, lacking multi-member range of representation. This perhaps may be demonstrated by the consideration that individual candidates hold an over-lapping group or "party" of opinions. Voters support for parties themselves become less and less partisan with more particular choices of parties in proportionally elected multi-member constituencies. So, it may be the CW becomes less and less an absolute individual choice, within the wider context of personal proportional representation, in multi-member constituencies.</div><div><br></div><div>Regards,</div><div>Richard Lung.</div><div><br></div><div><br></div><div><br></div><div><br>On 1 Mar 2023, at 2:47 am, Forest Simmons <<a href="mailto:forest.simmons21@gmail.com">forest.simmons21@gmail.com</a>> wrote:<br><br></div><div><div dir="auto">After many thought experiments I'm leaning towards the following Worst Loser Elimination version:<div dir="auto"><br></div><div dir="auto">Elect the pairwise undefeated candidate if there is one ... otherwise ...</div><div dir="auto">until there is an undefeated candidate ... among the uneliminated candidates, eliminate the candidate whose maximum pairwise support is the smallest ... after first eliminating any candidates it defeats pairwise.</div><div dir="auto"><br></div><div dir="auto">One thing I like about this version is that it only needs as input the pairwise support information ... unlike elimination methods that depend on Top or Bottom ballot counts, it is applicable to tournaments.</div><div dir="auto"><br></div><div dir="auto">To me, that is the true spirit of Universal Domain methods ... binary choices based on ordinal information .. no inference of strength that is based on how high or low in the rankings ... a subtle influence that subverts the purity of binary comparisons as a basis for voting.</div><div dir="auto"><br></div><div dir="auto">I'm not against methods that do make use of expressions of intensity of preference ... only methods that pretend not to, but actuually do. </div><div dir="auto"><br></div><div dir="auto">The pairwise information is efficiently precinct summable with one pass through the ballots. The practical importance of this feature is hard to over estimate.</div><div dir="auto"><br></div><div dir="auto">Example1</div><div dir="auto"><br></div><div dir="auto">x: A>B>C (Sincere is A>C>B)</div><div dir="auto">y: B>C>A</div><div dir="auto">z: C>A>B</div><div dir="auto"><br></div><div dir="auto">Assume (y+z)>x>max(y,z), so the A faction is the largest, but not a majority of the electorate.</div><div dir="auto"><br></div><div dir="auto">The maximum pairwise support for C is its support against A by the B and C factions ... the winning votes of the defeat of A by C: y+z= n-x, where n=x+y+z.</div><div dir="auto"><br></div><div dir="auto">The max pairwise supports of the other two candidates A and B are n-y and n-z, respectively, which are both larger than n-x ... the smallest value of max pairwise support ... making C the candidate to eliminate (after eliminating the candidate defeated by it).</div><div dir="auto"><br></div><div dir="auto">So in the context of this kind of three faction cycle with complete ballots, the candidate that defeats the favorite of the largest faction is the first one eliminated ... after first eliminating the candidate it defeats.</div><div dir="auto"><br></div><div dir="auto">How does this discourage burial? </div><div dir="auto"><br></div><div dir="auto">The candidate on the losing end of the weakest wv defeat is the one with the greatest hope of successfully burying its victor... and when it is the favorite of the largest faction, it has the greatest hope of coming out on top of the cycle resulting from that burial.</div><div dir="auto"><br></div><div dir="auto">Example 2</div><div dir="auto"><br></div><div dir="auto">48 C</div><div dir="auto">28 A>B</div><div dir="auto">24 B (sincere B>A)</div><div dir="auto"><br></div><div dir="auto">Evidently a Chicken defection by the B faction ... putting the Sincere CW into a cycle in which its max pairwise support is 28 votes against C. The respective max supports of B and C in this cycle are 52 and 48, both larger than 28.</div><div dir="auto"><br></div><div dir="auto">So A is eliminated after the candidate it defeats ... the defector B.</div><div dir="auto"><br></div><div dir="auto">In example 1 the MinMaxPS candidate took down with it the candidate responsible for its burial.</div><div dir="auto"><br></div><div dir="auto">In example 2 the MinMaxPS candidate took down with it the defecting candidate that put it into a cycle.</div><div dir="auto"><br></div><div dir="auto">In both cases the MinMaxPS candidate was the Sincere CW.</div><div dir="auto"><br></div><div dir="auto">In both cases the manipulating candidate had high hopes of winning the cycle it created by its manipulation ... because almost all Condorcet efficient methods break cycles at weakest wv defeat in the beatcycle chain.</div><div dir="auto"><br></div><div dir="auto">Our method also eliminates that defeat ... but both winner and loser of that defeat ... making the manipulator's gambit fail, instead of rewarding the loser of the weakest wv defeat like naive Condorcet methods do ... which thereby reward the manipulator responsible for creating the cycle.</div><div dir="auto"><br></div><div dir="auto">So this method takes very precise aim at taking down Chicken Defectors and Buriers that hope to defeat the Sincere CW by creating a cycle in which it (the Sincere CW) has become the weakest link.</div><div dir="auto"><br></div><div dir="auto">Our method counters that strategy by taking out the entire weakest link ... both winner and loser, unlike RP, CSSD, River, MinMax(wv), etc.</div><div dir="auto"><br></div><div dir="auto">Can the general public appreciate this? </div><div dir="auto"><br></div><div dir="auto">How do we help them see?</div><div dir="auto"><br></div><div dir="auto">Through a wide variety of examples where ordinary Condorcet fails to elect the Sincere CW while rewarding the cycle creator?</div><div dir="auto"><br></div><div dir="auto">But also we need to make clear that burial of the Sincere CW is no problem for IRV .... only because IRV is not a Condorct method ... it's not even expected to elect the CW in the first place .... instead of burying the CW it eliminates it by the Squeeze effect ... a much more common problem than Burial ... one that is only overcome by serious insincere compromise ... which belies the IRV promise of sincere voting safety.</div><div dir="auto"><br></div><div dir="auto">The squeeze effect happens with sincere votes. Burial cannot happen with sincere ballots. The Squeeze distortion of the democratic will, will be more common than Burial ... as long as voters vote sincerely ... which ordinary voters tend to do when not urged to vote lesser evil above their favorite.</div><div dir="auto"><br></div><div dir="auto">So don't throw out the Condorcet baby with the burial bath water. Keep Condorcet ... but make sure it is Burial and Chicken resistant Condorcet ... not vulnerable to the most tempting insincere manipulations.</div><div dir="auto"><br></div><div dir="auto">If that required some elaborate multiple pass version of Condorcet ... like Smith//IRV, BTR-IRV, RCIPE, etc ... or even anything as elaborate as Ranked Pairs ... then it might not be worth it.</div><div dir="auto"><br></div><div dir="auto">But voters like (and practically demand) elimination methods ... like ours ... with the same simplicity as MinMax the simplest well known Condorcet method (but not Burial resistant or Landau efficient).</div><div dir="auto"><br></div><div dir="auto">Let's formulate MinMax(wv) as an elimination method to facilitate the comparison:</div><div dir="auto"><br></div><div dir="auto">Until there is a candidate that is undefeated among the uneliminated candidates, eliminate the candidate whose Max Pairwise Support is minimal.</div><div dir="auto"><br></div><div dir="auto">Here's our manipulation resistant version:</div><div dir="auto"><br></div><div dir="auto"><span style="font-family:sans-serif">Until there is an candidate that is undefeated among the uneliminated candidates, eliminate the candidate whose Max Pairwise Support is minimal, after first eliminating any candidates defeated by it.</span><br></div><div dir="auto"><span style="font-family:sans-serif"><br></span></div><div dir="auto">-Forest</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 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 Sun, Feb 26, 2023, 9:47 PM 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"><div>The Symmetric Gross Loser notion of nominal "worst" seems to work well for burial resistance ... but it is not very chicken resistant.</div><div dir="auto"><br></div><div dir="auto">Here's a better one ... one I mentioned before .... where the nominally worst candidate is the loser of the strongest defeat... where defeat strength is gauged by the number of ballots on which the victor outranks the loser plus the bottom count of the loser... we abbreviate this gauge as wv+lbc... winning votes plus losing bottom count.</div><div dir="auto"><br></div><div dir="auto">Remember that a candidate's bottom count is the number of ballots on which it outranks nobody.</div><div dir="auto"><br></div><div dir="auto">So the method is to elect the CW if there is one ... otherwise ...</div><div dir="auto">Until there is an unbeaten candidate among the remaining candidates, eliminate the nominally worst candidate after first eliminating the candidates it beats pairwise (if any).</div><div dir="auto"><br></div><div dir="auto">The nominally worst candidate is </div><div dir="auto">the loser of the single strongest defeat, gauged by av+lbc ... winning votes plus losing bottom count.</div><div dir="auto"><br></div><div dir="auto">This method seems very promising for both burial and chicken resistance according to initial hand counts of some standard test cases.</div><div dir="auto"><br></div><div dir="auto">Simulations will tell.</div><div dir="auto"><br></div><div dir="auto">Here's my question: do simulations carry any weight with the public? Or do they just care about choice of buzz words and phrases like democracy; majority rule, etc?</div><div dir="auto"><br></div><div dir="auto">-Forest</div><div dir="auto"><br><div class="gmail_quote" dir="auto"><div dir="ltr" class="gmail_attr">On Sun, Feb 26, 2023, 3:57 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">Here's the cleanest notion of "worst" in this contex:<div dir="auto"><br></div><div dir="auto">The "nominally worst" candidate is the Symmetric Gross Loser (SGL) defined as the pairwise loser between the candidate with the greatest pairwise opposition and the least pairwise support.</div><div dir="auto"><br></div><div dir="auto">I am suggesting specializing our "worst-loser" elimination method to the following:</div><div dir="auto"><br></div><div dir="auto">While there is more than one remaining candidate, from among them eliminate the current Symmetric Gross Loser SGL after first eliminating every candidate (if any) pairwise defeated by this SGL.</div><div dir="auto"><br></div><div dir="auto">Elect the last candidate to be left standing (or eliminated).</div><div dir="auto"><br></div><div dir="auto">This is the version I would like to see tested.</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, 11:45 AM Forest Simmons <<a href="mailto:forest.simmons21@gmail.com" rel="noreferrer noreferrer" target="_blank">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>Correction below ...<br><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Feb 26, 2023, 10:58 AM Forest Simmons <<a href="mailto:forest.simmons21@gmail.com" rel="noreferrer noreferrer noreferrer" target="_blank">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">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></blockquote></div></div><div dir="auto"><br></div><div dir="auto">Whoops... A defeats B.</div><div dir="auto"><br></div><div dir="auto">So the burial can succeed if it is enough to make B beat C, but not enough to make C nominally "worse" than B ... a delicate, hence risky balance.</div><div dir="auto"><br></div><div dir="auto">Which nominal standards of "worst" make this balance most precarious if not impossible?</div><div dir="auto"><br></div><div dir="auto"><div class="gmail_quote"><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">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" rel="noreferrer noreferrer noreferrer noreferrer" target="_blank">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" rel="noreferrer noreferrer noreferrer noreferrer noreferrer" target="_blank">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>
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