At each stage, among the remaining candidates let MaxTop and MaxBot, respectively, be the candidates with the largest Top and Bottom counts ***. Let COACC be the candidate with the greater of these two max counts. Eliminate every candidate X that is outranked by the COACC candidate on a majority of those ballots that show any preference between them .... between X and the COACC, that is. Also eliminate COACC itself if its Bottom Count is greater than its Top Count ... otherwise keep it. ***[At any stage a candidate's Top count is the number of ballots on which it is not outranked by any remaining candidate. Similarly, its Bottom count is the number of ballots on which it outranks no remaining candidate. These Top and Bottom counts are supposed to be updated (by vote transfers from eliminated candidates to those remaining) between all elimination stages.] Example 48 C 28 A>B 24 B MaxTop and MaxBot are C and A, respectively, with counts of 48 and 72. Since 72 is larger than 48, the obvious approval cutoff COACC is A, and because A's bottom count is greater than its top count, A itself will be eliminated along with the candidate B, that is disapproved because it is outranked by A=SOACC on a majority (28 to 24) of the ballots that show any preference ... leaving C as the winner. Note that the cutoff candidate SOACC will always be someone that should obviously be approved (most Top) or else someone obviously disapproved (most bottom). Of these two obvious choices, the more obvious of the two, the one with the greater max count, is the SOACC cutoff. This completely resolves the two most difficult approval questions ... where to place the cutoff between approved and disapproved, and whether or not to approve candidates ranked exactly on the cutoff boundary. The voters don't have to worry about it at all unless they want to over-ride the SOACC cutoff for some personal reason. Anybody find any significant fault with this method compared to their previous favorite? Easy or hard to understand? Likely or unlikely to elect the "best" candidate? Easy to sell or hard to sell? Best selling point? Biggest drawback? Thanks! -Forest

Here's an even simpler method in the same spirit that accomplishes the sam thing. First some notation: Friends(X) is the set of candidates that do not beat X . the complement of Enemies(X). Note that Friends(X) includes X itself ... "Love thy neighbor as thyself." PW(X,Y) is the Pairwise Winner in the contest between X and Y. Similarly, PL(X,Y) denotes the Pairwise Loser of the same contest. Here's the method: While more than one candidate remains ... eliminate Friends(PL(MaxTop,MaxBot)) EndWhile Then elect the remaining candidate. If a burial has taken place, at some point the respective burier and buried are likely to take on the roles of MaxToo and MaxBot, respectively. If so the burier will lose the pairwise contest, and be eliminated along with the rest of the burier's friends, including the one under whom the buried candidate was buried. Remember MaxTop and MaxBot are the respective candidates with the greatest Top and Bottom counts. MaxTop=Argmax TopCount(X) MaxBot=Argmax BotCount(X) TopCount(X) is the number of ballots on which no other candidate outranks X. BotCount(X) is the number of ballots on which X outranks no candidate. It is easy to see that this method is Landau efficient because every pivot candidate is among its friends, and all of the eliminated candidates are friends of the pivots of the stages that eliminated them. In other words, each candidate is a friend of a friend of the winner when not a direct friend of the winner itself ... the very definition of Landau efficiency in the language of friends! In fact, the set of pivot candidates forms a totally ordered chain that covers every non pivot candidate ... it cannot be extended further above the winner ... which means the method is Banks efficient. This message is not intended for the lay person. Somebody with extra time on there hands can write it up for general audiences. ... leaving out the technical information that means nothing to voters ... and putting in things helpful to voters that technical readers take for granted. That's plenty for now. Next time a tournament friendly version that is efficiently precinct summable ... or why not now? While more than one candidate remains ... eliminate Friends(PL(MinMaxPS,MaxMinPS)) EndWhile Elect the remaining candidate. MinMaxPS is the candidate whose Maximal Pairwise Support is minimal. MaxMinPS is the candidate whose Minimal Pairwise support is maximal. -Forest On Sat, Mar 18, 2023, 7:02 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > > > At each stage, among the remaining candidates let MaxTop and MaxBot, > respectively, be the candidates with the largest Top and Bottom counts ***. > > Let COACC be the candidate with the greater of these two max counts. > > Eliminate every candidate X that is outranked by the COACC candidate on a > majority of those ballots that show any preference between them .... > between X and the COACC, that is. > > Also eliminate COACC itself if its Bottom Count is greater than its Top > Count ... otherwise keep it. > > ***[At any stage a candidate's Top count is the number of ballots on which > it is not outranked by any remaining candidate. Similarly, its Bottom count > is the number of ballots on which it outranks no remaining candidate. These > Top and Bottom counts are supposed to be updated (by vote transfers from > eliminated candidates to those remaining) between all elimination stages.] > > Example > > 48 C > 28 A>B > 24 B > > MaxTop and MaxBot are C and A, respectively, with counts of 48 and 72. > > Since 72 is larger than 48, the obvious approval cutoff COACC is A, and > because A's bottom count is greater than its top count, A itself will be > eliminated along with the candidate B, that is disapproved because it is > outranked by A=SOACC on a majority (28 to 24) of the ballots that show any > preference ... leaving C as the winner. > > Note that the cutoff candidate SOACC will always be someone that should > obviously be approved (most Top) or else someone obviously disapproved > (most bottom). Of these two obvious choices, the more obvious of the two, > the one with the greater max count, is the SOACC cutoff. > > This completely resolves the two most difficult approval questions ... > where to place the cutoff between approved and disapproved, and whether or > not to approve candidates ranked exactly on the cutoff boundary. The voters > don't have to worry about it at all unless they want to over-ride the SOACC > cutoff for some personal reason. > > Anybody find any significant fault with this method compared to their > previous favorite? > > Easy or hard to understand? > > Likely or unlikely to elect the "best" candidate? > > Easy to sell or hard to sell? > > Best selling point? > > Biggest drawback? > > Thanks! > > -Forest > > > > > > > >

Another possibility: Until only one candidate remains ... Eliminate from among the remaining ... Friends(PL(MaxMaxPO,minminPO)) EndUntil Elect the only remaining candidate Example1 48C 28 AB 24 B MaxMaxPO=MaxTop=MaxMinPS=C minminPO=MaxBot=MinMaxPS=A PL(C,A)=A Friends(A)=A,B Eliminate A,B --> Elect C Example 2 a ABC b BCA c CAB Suppose a>b>c. C=MaxBot=mMPS=MMPO A=MaxTop=MmPS=mmPO PL(A,C)=A, Friends(A)=B,A --> C wins Check me ... this is tricky! -Forest On Sun, Mar 19, 2023, 9:31 AM Forest Simmons <forest.simmons21@gmail.com> wrote: > Here's an even simpler method in the same spirit that accomplishes the sam > thing. > > First some notation: > > Friends(X) is the set of candidates that do not beat X . the complement of > Enemies(X). > > Note that Friends(X) includes X itself ... "Love thy neighbor as thyself." > > PW(X,Y) is the Pairwise Winner in the contest between X and Y. > > Similarly, PL(X,Y) denotes the Pairwise Loser of the same contest. > > Here's the method: > > While more than one candidate remains ... eliminate > Friends(PL(MaxTop,MaxBot)) > EndWhile > Then elect the remaining candidate. > > If a burial has taken place, at some point the respective burier and > buried are likely to take on the roles of MaxToo and MaxBot, respectively. > > If so the burier will lose the pairwise contest, and be eliminated along > with the rest of the burier's friends, including the one under whom the > buried candidate was buried. > > Remember MaxTop and MaxBot are the respective candidates with the greatest > Top and Bottom counts. > > MaxTop=Argmax TopCount(X) > MaxBot=Argmax BotCount(X) > > TopCount(X) is the number of ballots on which no other candidate outranks > X. > > BotCount(X) is the number of ballots on which X outranks no candidate. > > It is easy to see that this method is Landau efficient because every pivot > candidate is among its friends, and all of the eliminated candidates are > friends of the pivots of the stages that eliminated them. In other words, > each candidate is a friend of a friend of the winner when not a direct > friend of the winner itself ... the very definition of Landau efficiency in > the language of friends! > > In fact, the set of pivot candidates forms a totally ordered chain that > covers every non pivot candidate ... it cannot be extended further above > the winner ... which means the method is Banks efficient. > > This message is not intended for the lay person. Somebody with extra time > on there hands can write it up for general audiences. ... leaving out the > technical information that means nothing to voters ... and putting in > things helpful to voters that technical readers take for granted. > > That's plenty for now. Next time a tournament friendly version that is > efficiently precinct summable ... or why not now? > > While more than one candidate remains ... eliminate > Friends(PL(MinMaxPS,MaxMinPS)) > EndWhile > Elect the remaining candidate. > > MinMaxPS is the candidate whose Maximal Pairwise Support is minimal. > > MaxMinPS is the candidate whose Minimal Pairwise support is maximal. > > -Forest > > On Sat, Mar 18, 2023, 7:02 PM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> >> >> At each stage, among the remaining candidates let MaxTop and MaxBot, >> respectively, be the candidates with the largest Top and Bottom counts ***. >> >> Let COACC be the candidate with the greater of these two max counts. >> >> Eliminate every candidate X that is outranked by the COACC candidate on >> a majority of those ballots that show any preference between them .... >> between X and the COACC, that is. >> >> Also eliminate COACC itself if its Bottom Count is greater than its Top >> Count ... otherwise keep it. >> >> ***[At any stage a candidate's Top count is the number of ballots on >> which it is not outranked by any remaining candidate. Similarly, its Bottom >> count is the number of ballots on which it outranks no remaining candidate. >> These Top and Bottom counts are supposed to be updated (by vote transfers >> from eliminated candidates to those remaining) between all elimination >> stages.] >> >> Example >> >> 48 C >> 28 A>B >> 24 B >> >> MaxTop and MaxBot are C and A, respectively, with counts of 48 and 72. >> >> Since 72 is larger than 48, the obvious approval cutoff COACC is A, and >> because A's bottom count is greater than its top count, A itself will be >> eliminated along with the candidate B, that is disapproved because it is >> outranked by A=SOACC on a majority (28 to 24) of the ballots that show any >> preference ... leaving C as the winner. >> >> Note that the cutoff candidate SOACC will always be someone that should >> obviously be approved (most Top) or else someone obviously disapproved >> (most bottom). Of these two obvious choices, the more obvious of the two, >> the one with the greater max count, is the SOACC cutoff. >> >> This completely resolves the two most difficult approval questions ... >> where to place the cutoff between approved and disapproved, and whether or >> not to approve candidates ranked exactly on the cutoff boundary. The voters >> don't have to worry about it at all unless they want to over-ride the SOACC >> cutoff for some personal reason. >> >> Anybody find any significant fault with this method compared to their >> previous favorite? >> >> Easy or hard to understand? >> >> Likely or unlikely to elect the "best" candidate? >> >> Easy to sell or hard to sell? >> >> Best selling point? >> >> Biggest drawback? >> >> Thanks! >> >> -Forest >> >> >> >> >> >> >> >>