Kristofer, This elimination method helps to explain the "unreasonable effectiveness" of what we have been calling Max X>Y. While there remains no unbeaten candidate among the uneliminated... "vindicate" the victor X and eliminate the loser Y in the strongest paireise defeat. Vindication of X is executed by raising X to the level of Y on any ballot that (misguidedly) ranked Y above X, before eliminating Y. We say "misguidedly" because X was already ranked above or equal to Y on the greatest majority of ballots. So vindication is a new kind of vote transfer that applies when the strongest (remaining) defeat brings about the elimination. In the case of three candidates, that kind of vote transfer will be sufficient to turn X into a majority winner ... and in general, will likely change the strongest victor into an undefeated candidate. Example 49 C 26 A>B 25 B If we measure defeat strength by Winning Votes + Losing Truncations, we see that the defeat of A by C is strongest: WV+LT = 49+74 = 173. Vindication updates the ballot profile as 49 C 26 C>B 25 B So C now defeats B 75 to 25! Try this out on your favorite ballot profiles, and see if you like it. fws

A slightly simpler version for everyday citizens: When there is no candidate with more than half of the first rank votes ... 1.Find the candidate X that has the strongest head-to-head victory over any other candidate Y. This means (i) that X outranks Y on more ballots than Y outranks X, and (ii) that the number of ballots on which X outranks Y, plus the number of ballots on which Y out ranks no other candidate, is larger than the corresponding sum for any other pair of candidates. 2. Eliminate Y after replacing all of Y's first rank ballot appearances with X. 3. After these two steps if there is still no candidate with more than half of the first place votes, repeat steps (1) and (2) among the uneliminated candidates, until only one candidate remains. Note that in each elimination stage the votes of the most strongly defeated loser are transferred to the winner of that maximal strength defeat. We call this kind of vote transfer, "vindication" (of the winner). fws On Fri, Sep 8, 2023, 10:56 AM Forest Simmons <forest.simmons21@gmail.com> wrote: > Kristofer, > > This elimination method helps to explain the "unreasonable effectiveness" > of what we have been calling Max X>Y. > > While there remains no unbeaten candidate among the uneliminated... > "vindicate" the victor X and eliminate the loser Y in the strongest > pairwise defeat. > > Vindication of X is executed by raising X to the level of Y on any ballot > that (misguidedly) ranked Y above X, before eliminating Y. > > We say "misguidedly" because X was already ranked above or equal to Y on > the greatest majority of ballots. > > So vindication is a new kind of vote transfer that applies when the > strongest (remaining) defeat brings about the elimination. > > In the case of three candidates, that kind of vote transfer will be > sufficient to turn X into a majority winner ... and in general, will likely > change the strongest victor into an undefeated candidate. > > Example > > 49 C > 26 A>B > 25 B > > If we measure defeat strength by Winning Votes + Losing Truncations, we > see that the defeat of A by C is strongest: WV+LT = 49+74 = 123. > > Vindication updates the ballot profile as > > 49 C > 26 C>B > 25 B > > So C now defeats B 75 to 25! > > Try this out on your favorite ballot profiles, and see if you like it. > > fws >

Example of "Vindication": 11 A>B 15 C>A 7 B>C A>B by 26-7 is the strongest win, so B is eliminated and the 7 B>C votes become A>C votes and A wins over C 18-11. If the 7 B>C voters changed their votes to C>B (Favorite Betrayal), C wins. If the 7 B>C voters don't vote at all C wins (participation failure). If the 15 C>A voters vote C>B instead (Burial), C wins. Sounds fishy to me, though there's probably something I'm missing. There usually is. Jim Faran ________________________________ From: Election-Methods <election-methods-bounces@lists.electorama.com> on behalf of Forest Simmons <forest.simmons21@gmail.com> Sent: Friday, September 8, 2023 8:17 PM To: Kristofer Munsterhjelm <km_elmet@t-online.de>; EM <Election-methods@lists.electorama.com> Subject: Re: [EM] Vindication A slightly simpler version for everyday citizens: When there is no candidate with more than half of the first rank votes ... 1.Find the candidate X that has the strongest head-to-head victory over any other candidate Y. This means (i) that X outranks Y on more ballots than Y outranks X, and (ii) that the number of ballots on which X outranks Y, plus the number of ballots on which Y out ranks no other candidate, is larger than the corresponding sum for any other pair of candidates. 2. Eliminate Y after replacing all of Y's first rank ballot appearances with X. 3. After these two steps if there is still no candidate with more than half of the first place votes, repeat steps (1) and (2) among the uneliminated candidates, until only one candidate remains. Note that in each elimination stage the votes of the most strongly defeated loser are transferred to the winner of that maximal strength defeat. We call this kind of vote transfer, "vindication" (of the winner). fws On Fri, Sep 8, 2023, 10:56 AM Forest Simmons <forest.simmons21@gmail.com<mailto:forest.simmons21@gmail.com>> wrote: Kristofer, This elimination method helps to explain the "unreasonable effectiveness" of what we have been calling Max X>Y. While there remains no unbeaten candidate among the uneliminated... "vindicate" the victor X and eliminate the loser Y in the strongest pairwise defeat. Vindication of X is executed by raising X to the level of Y on any ballot that (misguidedly) ranked Y above X, before eliminating Y. We say "misguidedly" because X was already ranked above or equal to Y on the greatest majority of ballots. So vindication is a new kind of vote transfer that applies when the strongest (remaining) defeat brings about the elimination. In the case of three candidates, that kind of vote transfer will be sufficient to turn X into a majority winner ... and in general, will likely change the strongest victor into an undefeated candidate. Example 49 C 26 A>B 25 B If we measure defeat strength by Winning Votes + Losing Truncations, we see that the defeat of A by C is strongest: WV+LT = 49+74 = 123. Vindication updates the ballot profile as 49 C 26 C>B 25 B So C now defeats B 75 to 25! Try this out on your favorite ballot profiles, and see if you like it. fws

James, Thanks for contributing and exploring your great example! Here's the way I look at it: When there are only three factions, if voters are sincere and well informed about each others' preferences, the two factions that are farthest apart will (most likely) be "anti-favorites" of each other. And the other faction favorite will be ranked top by itself ... so it will not be the anti-favorite of any of the three factions. In your illuminating example every candidate is an anti-favorite of some faction, so most likely not all factions are sincere. In fact, the most likely cause of the cycle was the burial of a weak centrist (B in this case) by the strongest of the other factions (C in this example) ... that is, the one most confident about benefiting from the cycle creation. In deed all of the classical Condorcet methods (MinMax, Ranked Pairs, Schulze's Beatpath CSSD, Jobst's River, etc) break the cycle at the weakest link, eliminating B. Nanson, Baldwin, and IRV also start by eliminating B, thereby bolstering the confidence of the B nuriers. And without the vindication transfer of votes to the candidate A that defeated the eliminated candidate pairwise, all of these methods reward the burying faction by electing C. Vindication may seem vindictive, but it's the retaliation needed to punish the buriers by making their sneaky ploy backfire! Thanks Again. fws On Fri, Sep 8, 2023, 8:47 PM James Faran <jjfaran@buffalo.edu> wrote: > Example of "Vindication": > > 11 A>B > 15 C>A > 7 B>C > > A>B by 26-7 is the strongest win, so B is eliminated and the 7 B>C votes > become A>C votes and A wins over C 18-11. > > If the 7 B>C voters changed their votes to C>B (Favorite Betrayal), C wins. > If the 7 B>C voters don't vote at all C wins (participation failure). > If the 15 C>A voters vote C>B instead (Burial), C wins. > > Sounds fishy to me, though there's probably something I'm missing. There > usually is. > Jim Faran > ------------------------------ > *From:* Election-Methods <election-methods-bounces@lists.electorama.com> > on behalf of Forest Simmons <forest.simmons21@gmail.com> > *Sent:* Friday, September 8, 2023 8:17 PM > *To:* Kristofer Munsterhjelm <km_elmet@t-online.de>; EM < > Election-methods@lists.electorama.com> > *Subject:* Re: [EM] Vindication > > A slightly simpler version for everyday citizens: > > When there is no candidate with more than half of the first rank votes ... > > 1.Find the candidate X that has the strongest head-to-head victory over > any other candidate Y. This means (i) that X outranks Y on more ballots > than Y outranks X, and (ii) that the number of ballots on which X outranks > Y, plus the number of ballots on which Y out ranks no other candidate, is > larger than the corresponding sum for any other pair of candidates. > > 2. Eliminate Y after replacing all of Y's first rank ballot appearances > with X. > > 3. After these two steps if there is still no candidate with more than > half of the first place votes, repeat steps (1) and (2) among the > uneliminated candidates, until only one candidate remains. > > Note that in each elimination stage the votes of the most strongly > defeated loser are transferred to the winner of that maximal strength > defeat. We call this kind of vote transfer, "vindication" (of the winner). > > fws > > > > On Fri, Sep 8, 2023, 10:56 AM Forest Simmons <forest.simmons21@gmail.com> > wrote: > > Kristofer, > > This elimination method helps to explain the "unreasonable effectiveness" > of what we have been calling Max X>Y. > > While there remains no unbeaten candidate among the uneliminated... > "vindicate" the victor X and eliminate the loser Y in the strongest > pairwise defeat. > > Vindication of X is executed by raising X to the level of Y on any ballot > that (misguidedly) ranked Y above X, before eliminating Y. > > We say "misguidedly" because X was already ranked above or equal to Y on > the greatest majority of ballots. > > So vindication is a new kind of vote transfer that applies when the > strongest (remaining) defeat brings about the elimination. > > In the case of three candidates, that kind of vote transfer will be > sufficient to turn X into a majority winner ... and in general, will likely > change the strongest victor into an undefeated candidate. > > Example > > 49 C > 26 A>B > 25 B > > If we measure defeat strength by Winning Votes + Losing Truncations, we > see that the defeat of A by C is strongest: WV+LT = 49+74 = 123. > > Vindication updates the ballot profile as > > 49 C > 26 C>B > 25 B > > So C now defeats B 75 to 25! > > Try this out on your favorite ballot profiles, and see if you like it. > > fws > >