Why can't we just have majority rule? Why all the fiss? 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. 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. 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." But there is no agreement on what constitutes "democratically weak' ... and it makes a big difference! So what can we do? 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. But just as there is no guaranteed outright majority winner ... neither is there any guarantee of the existence of a pairwise unbeaten candidate. 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). 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." 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. Then the "Strong Majority Pairwise Criterion" (SMPC) is satisfied only by methods that always elect uncovered candidates. 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. Another way to express compliance with this SMPC criterion is "Landau Efficient." 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. Who can name even one commonly known election method that is Landau efficient? What's more ... no matter the nominal "worst" criterion, the method will be more or less burial resistant ... as I will explain presently. I suggest that proposals for any method under this umbrella, include verbiage to the effect ... "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." 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. 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. Anti-vote splitting can be easily ensured (in general) by allowing equal-top whole counting, and multiple truncations in large elections. In the continuation I will explain why this method tends to backfire on buriers. 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. Among other things enlightened defenders of electoral democracy need to understand the "squeeze effect" and "burial ploys" ... To be continued ... -Forest

On 2/25/23 00:49, Forest Simmons wrote: > Who can name even one commonly known election method that is Landau > efficient? Copeland! And Copeland,X or Copeland//X. Aren't they? > What's more ... no matter the nominal "worst" criterion, the method will > be more or less burial resistant ... as I will explain presently. I hope someone can show that with simulation. My experiments seem to show that's a pretty rare condition :-) Unless it's like max A>B, where there's no burial but tons of compromising to make up for it. -km

Yes, that's the only one ... and it's the one I mentionedwhen Markus expressed doubt about the existence of a monotonic Landau efficient method. It took another decade to get a UD Landau method that was both monotone and clone independent. The first such method was Agenda Based Chain Climbing based on a truncation/abstention count agenda. I look at Friendly Voting methods as decloned versions of Copeland. They seem to have more class than ABCC in many ways, including good potential for Independence from Pareto Dominated Alternatives, but hard to make both Decisive and Clone Free in a satisfactory way without more than one pass through the ballots or explicit strong approval cutoffs (as opposed to implicit/ weak approval or strong equal-top rankings) as a good basis for Martin Harper Lottery Probabilities to replace the random ballot favorite probabilities (fpA, fpC, etc.) normally used in Friendly Voting to declone Copeland. [Without explicit strong approval cutoffs it takes one pass through the ballots to find out who the Smith candidates are ... and if the median ranked Smith candidate on a ballot should be an inclusive or exclusive approval cutoff candidate.] -Forest On Sat, Feb 25, 2023, 2:52 AM Kristofer Munsterhjelm <km_elmet@t-online.de> wrote: > On 2/25/23 00:49, Forest Simmons wrote: > > > Who can name even one commonly known election method that is Landau > > efficient? > > Copeland! And Copeland,X or Copeland//X. Aren't they? > > > What's more ... no matter the nominal "worst" criterion, the method will > > be more or less burial resistant ... as I will explain presently. > > I hope someone can show that with simulation. My experiments seem to > show that's a pretty rare condition :-) Unless it's like max A>B, where > there's no burial but tons of compromising to make up for it. > > -km >

On 2/26/23 05:25, Forest Simmons wrote: > Yes, that's the only one ... and it's the one I mentioned when Markus > expressed doubt about the existence of a monotonic Landau efficient method. Huh, isn't Landau,X always monotone for monotone X? Not cloneproof, of course. Hm, now I wonder if Landau,RP is monotone and cloneproof... and then there's that Short Ranked Pairs we never implemented. It would also be interesting to make an ISDA/IPDA non-summable monotone version of Friendly Voting. But that'll have to be another project; I wouldn't want to distract you from yours :-) -km

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. But the cure is easy and sure ... no eliminations of undefeated candidates. All Condorcet Efficient methods have the same Yee Diagram ... the win region for a candidate is its entire Voronoi polygon, no matter how small. Next ... burial ... On Fri, Feb 24, 2023, 3:49 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > > Why can't we just have majority rule? Why all the fiss? > > 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. > > 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. > > 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." > > But there is no agreement on what constitutes "democratically weak' ... > and it makes a big difference! > > So what can we do? > > 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. > > But just as there is no guaranteed outright majority winner ... neither is > there any guarantee of the existence of a pairwise unbeaten candidate. > > 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). > > 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." > > 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. > > Then the "Strong Majority Pairwise Criterion" (SMPC) is satisfied only by > methods that always elect uncovered candidates. > > 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. > > Another way to express compliance with this SMPC criterion is "Landau > Efficient." > > 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. > > Who can name even one commonly known election method that is Landau > efficient? > > What's more ... no matter the nominal "worst" criterion, the method will > be more or less burial resistant ... as I will explain presently. > > I suggest that proposals for any method under this umbrella, include > verbiage to the effect ... > > "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." > > 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. > > 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. > > Anti-vote splitting can be easily ensured (in general) by allowing > equal-top whole counting, and multiple truncations in large elections. > > In the continuation I will explain why this method tends to backfire on > buriers. > > 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. > > Among other things enlightened defenders of electoral democracy need to > understand the "squeeze effect" and "burial ploys" ... > > To be continued ... > > -Forest > >

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. 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. 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. Note that the feature that gives IRV immunity to burial is the same feature that makes it vulnerable to the Squeeze Effect. So is it possible to have immunity to burial and squeeze in the same method? 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. 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. So how does this fact make C's elimination before A backfire? 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" 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. 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. The A faction buries C under B, which creates a beat cycle ABCA. A thinks this cycle gives it a chance at winning ... which it would under most elimination methods. But not under ours, because, on the one hand A cannot win unless B or C is "worst" ... and ... If B is worst it takes A down with it because B defeats A in the cycle ... then B defeats C. On the other hand, if C is worst, it takes A down with it, leaving B as winner. 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! 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". You may remember I once proposed a Quick & Dirty method that simply said elect the "best" candidate that pairwise defeats the "worst" Smith candidate. 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. The main defect of the shortcut is that it requires some knowledge of Smith ... which our "worst-elimination" methods do not require. So even though Q&D is shorter ... it is neither quite as good nor quite as simple. 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. 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. Good sources for "worst" candidates are losers of other methods. Also losing candidates in strong pairwise defeats ... for any decent gauge of defeat strength. Enjoy! Forest On Sun, Feb 26, 2023, 8:37 AM Forest Simmons <forest.simmons21@gmail.com> wrote: > 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. > > But the cure is easy and sure ... no eliminations of undefeated candidates. > > All Condorcet Efficient methods have the same Yee Diagram ... the win > region for a candidate is its entire Voronoi polygon, no matter how small. > > Next ... burial ... > > > > > > > On Fri, Feb 24, 2023, 3:49 PM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> >> Why can't we just have majority rule? Why all the fiss? >> >> 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. >> >> 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. >> >> 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." >> >> But there is no agreement on what constitutes "democratically weak' ... >> and it makes a big difference! >> >> So what can we do? >> >> 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. >> >> But just as there is no guaranteed outright majority winner ... neither >> is there any guarantee of the existence of a pairwise unbeaten candidate. >> >> 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). >> >> 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." >> >> 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. >> >> Then the "Strong Majority Pairwise Criterion" (SMPC) is satisfied only by >> methods that always elect uncovered candidates. >> >> 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. >> >> Another way to express compliance with this SMPC criterion is "Landau >> Efficient." >> >> 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. >> >> Who can name even one commonly known election method that is Landau >> efficient? >> >> What's more ... no matter the nominal "worst" criterion, the method will >> be more or less burial resistant ... as I will explain presently. >> >> I suggest that proposals for any method under this umbrella, include >> verbiage to the effect ... >> >> "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." >> >> 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. >> >> 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. >> >> Anti-vote splitting can be easily ensured (in general) by allowing >> equal-top whole counting, and multiple truncations in large elections. >> >> In the continuation I will explain why this method tends to backfire on >> buriers. >> >> 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. >> >> Among other things enlightened defenders of electoral democracy need to >> understand the "squeeze effect" and "burial ploys" ... >> >> To be continued ... >> >> -Forest >> >>

Correction below ... On Sun, Feb 26, 2023, 10:58 AM Forest Simmons <forest.simmons21@gmail.com> wrote: > 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. > > 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. > > 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. > > Note that the feature that gives IRV immunity to burial is the same > feature that makes it vulnerable to the Squeeze Effect. > > So is it possible to have immunity to burial and squeeze in the same > method? > > 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. > > 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. > > So how does this fact make C's elimination before A backfire? > > 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" > > 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. > > 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. > > The A faction buries C under B, which creates a beat cycle ABCA. > > A thinks this cycle gives it a chance at winning ... which it would under > most elimination methods. > > But not under ours, because, on the one hand A cannot win unless B or C is > "worst" ... and ... > > If B is worst it takes A down with it because B defeats A in the cycle ... > then B defeats C. > Whoops... A defeats B. 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. Which nominal standards of "worst" make this balance most precarious if not impossible? > On the other hand, if C is worst, it takes A down with it, leaving B as > winner. > > 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! > > 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". > > You may remember I once proposed a Quick & Dirty method that simply said > elect the "best" candidate that pairwise defeats the "worst" Smith > candidate. > > 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. > > The main defect of the shortcut is that it requires some knowledge of > Smith ... which our "worst-elimination" methods do not require. > > So even though Q&D is shorter ... it is neither quite as good nor quite as > simple. > > 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. > > 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. > > Good sources for "worst" candidates are losers of other methods. > > Also losing candidates in strong pairwise defeats ... for any decent gauge > of defeat strength. > > Enjoy! > > Forest > > > > On Sun, Feb 26, 2023, 8:37 AM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> 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. >> >> But the cure is easy and sure ... no eliminations of undefeated >> candidates. >> >> All Condorcet Efficient methods have the same Yee Diagram ... the win >> region for a candidate is its entire Voronoi polygon, no matter how small. >> >> Next ... burial ... >> >> >> >> >> >> >> On Fri, Feb 24, 2023, 3:49 PM Forest Simmons <forest.simmons21@gmail.com> >> wrote: >> >>> >>> Why can't we just have majority rule? Why all the fiss? >>> >>> 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. >>> >>> 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. >>> >>> 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." >>> >>> But there is no agreement on what constitutes "democratically weak' ... >>> and it makes a big difference! >>> >>> So what can we do? >>> >>> 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. >>> >>> But just as there is no guaranteed outright majority winner ... neither >>> is there any guarantee of the existence of a pairwise unbeaten candidate. >>> >>> 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). >>> >>> 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." >>> >>> 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. >>> >>> Then the "Strong Majority Pairwise Criterion" (SMPC) is satisfied only >>> by methods that always elect uncovered candidates. >>> >>> 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. >>> >>> Another way to express compliance with this SMPC criterion is "Landau >>> Efficient." >>> >>> 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. >>> >>> Who can name even one commonly known election method that is Landau >>> efficient? >>> >>> What's more ... no matter the nominal "worst" criterion, the method will >>> be more or less burial resistant ... as I will explain presently. >>> >>> I suggest that proposals for any method under this umbrella, include >>> verbiage to the effect ... >>> >>> "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." >>> >>> 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. >>> >>> 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. >>> >>> Anti-vote splitting can be easily ensured (in general) by allowing >>> equal-top whole counting, and multiple truncations in large elections. >>> >>> In the continuation I will explain why this method tends to backfire on >>> buriers. >>> >>> 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. >>> >>> Among other things enlightened defenders of electoral democracy need to >>> understand the "squeeze effect" and "burial ploys" ... >>> >>> To be continued ... >>> >>> -Forest >>> >>>

Here's the cleanest notion of "worst" in this contex: 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. I am suggesting specializing our "worst-loser" elimination method to the following: 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. Elect the last candidate to be left standing (or eliminated). This is the version I would like to see tested. -Forest On Sun, Feb 26, 2023, 11:45 AM Forest Simmons <forest.simmons21@gmail.com> wrote: > Correction below ... > > On Sun, Feb 26, 2023, 10:58 AM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> 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. >> >> 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. >> >> 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. >> >> Note that the feature that gives IRV immunity to burial is the same >> feature that makes it vulnerable to the Squeeze Effect. >> >> So is it possible to have immunity to burial and squeeze in the same >> method? >> >> 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. >> >> 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. >> >> So how does this fact make C's elimination before A backfire? >> >> 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" >> >> 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. >> >> 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. >> >> The A faction buries C under B, which creates a beat cycle ABCA. >> >> A thinks this cycle gives it a chance at winning ... which it would under >> most elimination methods. >> >> But not under ours, because, on the one hand A cannot win unless B or C >> is "worst" ... and ... >> >> If B is worst it takes A down with it because B defeats A in the cycle >> ... then B defeats C. >> > > Whoops... A defeats B. > > 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. > > Which nominal standards of "worst" make this balance most precarious if > not impossible? > > >> On the other hand, if C is worst, it takes A down with it, leaving B as >> winner. >> >> 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! >> >> 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". >> >> You may remember I once proposed a Quick & Dirty method that simply said >> elect the "best" candidate that pairwise defeats the "worst" Smith >> candidate. >> >> 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. >> >> The main defect of the shortcut is that it requires some knowledge of >> Smith ... which our "worst-elimination" methods do not require. >> >> So even though Q&D is shorter ... it is neither quite as good nor quite >> as simple. >> >> 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. >> >> 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. >> >> Good sources for "worst" candidates are losers of other methods. >> >> Also losing candidates in strong pairwise defeats ... for any decent >> gauge of defeat strength. >> >> Enjoy! >> >> Forest >> >> >> >> On Sun, Feb 26, 2023, 8:37 AM Forest Simmons <forest.simmons21@gmail.com> >> wrote: >> >>> 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. >>> >>> But the cure is easy and sure ... no eliminations of undefeated >>> candidates. >>> >>> All Condorcet Efficient methods have the same Yee Diagram ... the win >>> region for a candidate is its entire Voronoi polygon, no matter how small. >>> >>> Next ... burial ... >>> >>> >>> >>> >>> >>> >>> On Fri, Feb 24, 2023, 3:49 PM Forest Simmons <forest.simmons21@gmail.com> >>> wrote: >>> >>>> >>>> Why can't we just have majority rule? Why all the fiss? >>>> >>>> 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. >>>> >>>> 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. >>>> >>>> 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." >>>> >>>> But there is no agreement on what constitutes "democratically weak' ... >>>> and it makes a big difference! >>>> >>>> So what can we do? >>>> >>>> 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. >>>> >>>> But just as there is no guaranteed outright majority winner ... neither >>>> is there any guarantee of the existence of a pairwise unbeaten candidate. >>>> >>>> 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). >>>> >>>> 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." >>>> >>>> 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. >>>> >>>> Then the "Strong Majority Pairwise Criterion" (SMPC) is satisfied only >>>> by methods that always elect uncovered candidates. >>>> >>>> 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. >>>> >>>> Another way to express compliance with this SMPC criterion is "Landau >>>> Efficient." >>>> >>>> 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. >>>> >>>> Who can name even one commonly known election method that is Landau >>>> efficient? >>>> >>>> What's more ... no matter the nominal "worst" criterion, the method >>>> will be more or less burial resistant ... as I will explain presently. >>>> >>>> I suggest that proposals for any method under this umbrella, include >>>> verbiage to the effect ... >>>> >>>> "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." >>>> >>>> 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. >>>> >>>> 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. >>>> >>>> Anti-vote splitting can be easily ensured (in general) by allowing >>>> equal-top whole counting, and multiple truncations in large elections. >>>> >>>> In the continuation I will explain why this method tends to backfire on >>>> buriers. >>>> >>>> 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. >>>> >>>> Among other things enlightened defenders of electoral democracy need to >>>> understand the "squeeze effect" and "burial ploys" ... >>>> >>>> To be continued ... >>>> >>>> -Forest >>>> >>>>

The Symmetric Gross Loser notion of nominal "worst" seems to work well for burial resistance ... but it is not very chicken resistant. 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. Remember that a candidate's bottom count is the number of ballots on which it outranks nobody. So the method is to elect the CW if there is one ... otherwise ... 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). The nominally worst candidate is the loser of the single strongest defeat, gauged by av+lbc ... winning votes plus losing bottom count. This method seems very promising for both burial and chicken resistance according to initial hand counts of some standard test cases. Simulations will tell. 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? -Forest On Sun, Feb 26, 2023, 3:57 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > Here's the cleanest notion of "worst" in this contex: > > 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. > > I am suggesting specializing our "worst-loser" elimination method to the > following: > > 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. > > Elect the last candidate to be left standing (or eliminated). > > This is the version I would like to see tested. > > -Forest > > > > On Sun, Feb 26, 2023, 11:45 AM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> Correction below ... >> >> On Sun, Feb 26, 2023, 10:58 AM Forest Simmons <forest.simmons21@gmail.com> >> wrote: >> >>> 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. >>> >>> 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. >>> >>> 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. >>> >>> Note that the feature that gives IRV immunity to burial is the same >>> feature that makes it vulnerable to the Squeeze Effect. >>> >>> So is it possible to have immunity to burial and squeeze in the same >>> method? >>> >>> 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. >>> >>> 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. >>> >>> So how does this fact make C's elimination before A backfire? >>> >>> 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" >>> >>> 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. >>> >>> 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. >>> >>> The A faction buries C under B, which creates a beat cycle ABCA. >>> >>> A thinks this cycle gives it a chance at winning ... which it would >>> under most elimination methods. >>> >>> But not under ours, because, on the one hand A cannot win unless B or C >>> is "worst" ... and ... >>> >>> If B is worst it takes A down with it because B defeats A in the cycle >>> ... then B defeats C. >>> >> >> Whoops... A defeats B. >> >> 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. >> >> Which nominal standards of "worst" make this balance most precarious if >> not impossible? >> >> >>> On the other hand, if C is worst, it takes A down with it, leaving B as >>> winner. >>> >>> 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! >>> >>> 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". >>> >>> You may remember I once proposed a Quick & Dirty method that simply said >>> elect the "best" candidate that pairwise defeats the "worst" Smith >>> candidate. >>> >>> 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. >>> >>> The main defect of the shortcut is that it requires some knowledge of >>> Smith ... which our "worst-elimination" methods do not require. >>> >>> So even though Q&D is shorter ... it is neither quite as good nor quite >>> as simple. >>> >>> 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. >>> >>> 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. >>> >>> Good sources for "worst" candidates are losers of other methods. >>> >>> Also losing candidates in strong pairwise defeats ... for any decent >>> gauge of defeat strength. >>> >>> Enjoy! >>> >>> Forest >>> >>> >>> >>> On Sun, Feb 26, 2023, 8:37 AM Forest Simmons <forest.simmons21@gmail.com> >>> wrote: >>> >>>> 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. >>>> >>>> But the cure is easy and sure ... no eliminations of undefeated >>>> candidates. >>>> >>>> All Condorcet Efficient methods have the same Yee Diagram ... the win >>>> region for a candidate is its entire Voronoi polygon, no matter how small. >>>> >>>> Next ... burial ... >>>> >>>> >>>> >>>> >>>> >>>> >>>> On Fri, Feb 24, 2023, 3:49 PM Forest Simmons < >>>> forest.simmons21@gmail.com> wrote: >>>> >>>>> >>>>> Why can't we just have majority rule? Why all the fiss? >>>>> >>>>> 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. >>>>> >>>>> 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. >>>>> >>>>> 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." >>>>> >>>>> But there is no agreement on what constitutes "democratically weak' >>>>> ... and it makes a big difference! >>>>> >>>>> So what can we do? >>>>> >>>>> 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. >>>>> >>>>> But just as there is no guaranteed outright majority winner ... >>>>> neither is there any guarantee of the existence of a pairwise unbeaten >>>>> candidate. >>>>> >>>>> 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). >>>>> >>>>> 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." >>>>> >>>>> 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. >>>>> >>>>> Then the "Strong Majority Pairwise Criterion" (SMPC) is satisfied only >>>>> by methods that always elect uncovered candidates. >>>>> >>>>> 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. >>>>> >>>>> Another way to express compliance with this SMPC criterion is "Landau >>>>> Efficient." >>>>> >>>>> 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. >>>>> >>>>> Who can name even one commonly known election method that is Landau >>>>> efficient? >>>>> >>>>> What's more ... no matter the nominal "worst" criterion, the method >>>>> will be more or less burial resistant ... as I will explain presently. >>>>> >>>>> I suggest that proposals for any method under this umbrella, include >>>>> verbiage to the effect ... >>>>> >>>>> "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." >>>>> >>>>> 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. >>>>> >>>>> 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. >>>>> >>>>> Anti-vote splitting can be easily ensured (in general) by allowing >>>>> equal-top whole counting, and multiple truncations in large elections. >>>>> >>>>> In the continuation I will explain why this method tends to backfire >>>>> on buriers. >>>>> >>>>> 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. >>>>> >>>>> Among other things enlightened defenders of electoral democracy need >>>>> to understand the "squeeze effect" and "burial ploys" ... >>>>> >>>>> To be continued ... >>>>> >>>>> -Forest >>>>> >>>>>

On 2/27/23 06:47, Forest Simmons wrote: > Simulations will tell. > > 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? I'm mostly thinking about simulations as a way to determine what criteria are actually (likely) to be passed -- it's a good way to make a shortcut through subtle arguments -- and to determine if the criteria generalize the way we think they do (e.g. JGA's nomination incentive and clone independence; sometimes they do, sometimes they don't). Proofs that a method may pass or fail a criterion may be very subtle and possibly contain similarly subtle errors or assumptions that we don't see. (I've had my share of disproven proofs, myself!) So, in particular given how many proposals there have been for different methods on EM since it started, simulations seem appealing because they'll give another perspective and let us check our blind spots. (Of course, it's not that easy: someone has to actually implement the method -- and the simulations!) I don't think that the public at large cares much about simulations. I think that a citizens' assembly, who would have more time to investigate the methods, would care more. I'm not sure what it takes to create public support in a method, but I have a hunch there's no substitute for plain old advocacy - doing the work to make the method known. Certainly FairVote has managed to accomplish quite a lot through their singleminded focus on advocacy and marketing... at the expense of the method's quality itself (which would get in the way of their story). -km