Elimination methods generally proceed by eliminating the "worst" remaining candidate at each step. "Worst" is defined in various ways ... the simpler the better, all else being equal. No gauge of "worst" is infallible ... especially when defined in one sound bite. Robert's dictum, can be thought of as a corrective... all else being equal, do not elect a pairwise beaten candidate ... especially if she is considered to be "worse" than the other remaining candidates. Think of BTR IRV. Here "worst" means fewest transferred top votes ... an appealing criterion ... but not infallible. The tentative judgment based on "fewest top votes" is not a reliable standard of worst when center squeeze is a real possibility. Here's a suggestion for incorporating Robert's dictum in conjunction with any notion of "worst" ... whether fewest top votes, most bottom, least pairwise support, most pairwise opposition, or any thing else: While there remains at least one un-eliminated candidate ... eliminate the "worst" remaining one ... AFTER eliminating any (and every) candidate pairwise defeated by it. EndWhile [Then elect the last candidate to be eliminated.] Rationale: If there exists a candidate X pairwise defeated by the candidate Y considered to be "worst" by some tentative criterion, then evidently a majority of the participating voters consider X to be even worse than Y ... not withstanding the tentative judgment of of Y being "worst." In other words, whatever the criterion for "worst" may be ... that judgment is only tentative until confirmed by a democratic majority of the participating voters ... hence it can and should be overridden when the voters (according to their ranked preference ballots) prefer keeping Y over X. Any elimination method following this template will be Condorcet efficient. Beyond that ... absent a Condorcet Winner, it will still elect an uncovered candidate ... important insurance against loser complaints ... insurance that no extant public methods offer. So fill in the template wih your favorite standard for "worst" ... least GPA, most disapproval, worst majority judgment, fewest total yards gained, fewest free throws completed, fewest technical errors, etc... use your imagination ... but remember, for public election proposals ... the simpler the better ..."defeat by the strongest majority of participating voters" might barely pass. In general, elimination methods (like IRV) fail monotonicity ... except a few them when they are based on a fixed, monotonically generated agenda. For example, when "worse" means worst according to a monotonically generated agenda, this method (like Sequential Pairwise Elimination) is monotone ... otherwise probably not. -Forest

If, at any elimination stage there remains a candidate X that is defeated pairwise by all of the other remaining candidates, this method eliminates it (as it should) at that stage ... even if X does not satisfy the user supplied definition of "worst", because whichever candidate is deemed worst will sweep X out before its own elimination ... "after every candidate it defeats (including X) is eliminated." Richard the VoteFair guy (among others) has proposed inserting this Condorcet Loser elimination step into IRV. This method does it seamlessly. Some other comments in line below ... On Tue, Feb 21, 2023, 1:28 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > Elimination methods generally proceed by eliminating the "worst" remaining > candidate at each step. > > "Worst" is defined in various ways ... the simpler the better, all else > being equal. > > No gauge of "worst" is infallible ... especially when defined in one sound > bite. > > Robert's dictum, can be thought of as a corrective... all else being > equal, do not elect a pairwise beaten candidate ... especially if she is > considered to be "worse" than the other remaining candidates. > > Think of BTR IRV. Here "worst" means fewest transferred top votes ... an > appealing criterion ... but not infallible. > > The tentative judgment based on "fewest top votes" is not a reliable > standard of worst when center squeeze is a real possibility. > > Here's a suggestion for incorporating Robert's dictum in conjunction with > any notion of "worst" ... whether fewest top votes, most bottom, least > pairwise support, most pairwise opposition, or any thing else: > > While there remains at least one un-eliminated candidate ... > eliminate the "worst" remaining one ... > AFTER > eliminating any (and every) candidate pairwise defeated by it. > EndWhile > > [Then elect the last candidate to be eliminated.] > > Rationale: If there exists a candidate X pairwise defeated by the > candidate Y considered to be "worst" by some tentative criterion, then > evidently a majority of the participating voters consider X to be even > worse than Y ... not withstanding the tentative judgment of Y being "worst." > > In other words, whatever the criterion for "worst" may be ... that > judgment is only tentative until confirmed by a democratic majority of the > participating voters ... hence it can and should be overridden when the > voters (according to their ranked preference ballots) prefer keeping Y over > X. > They will always prefer keeping Y over X when X is a "Condorcet Loser", assuming sincere ballots. > > Any elimination method following this template will be Condorcet > efficient. Beyond that ... absent a Condorcet Winner, it will still elect > an uncovered candidate ... important insurance against loser complaints ... > insurance that no extant public methods offer. > > So fill in the template wih your favorite standard for "worst" ... least > GPA, most disapproval, worst majority judgment, fewest total yards gained, > fewest free throws completed, fewest technical errors, etc... use your > imagination ... but remember, for public election proposals ... the simpler > the better ..."defeat by the strongest majority of participating voters" > might barely pass. > > In general, elimination methods (like IRV) > I should have included Baldwin, Nanson, Coombs, etc ... not just pick on IRV. fail monotonicity ... except a few them when they are based on a fixed, > monotonically generated agenda. > > For example, when "worse" means worst according to a monotonically > generated agenda, this method (like Sequential Pairwise Elimination) is > monotone ... otherwise probably not. > Another case where this method is monotone is when "worst" means "most unranked." That is so simple that it really fits the bill as a simple Condorcet completion proposal: Absent a Condorcet Winner ... eliminate the least democratically acceptable candidates in stages until there is an un-defeated candidate to be elected among those remaining. In this context "least democratically acceptable" refers to the candidates deemed worthy of ranking by the fewest voters, as well as any candidates too weak to defeat them (if there are any that weak). > -Forest > > > >