What started as Chain Climbing has evolved into a Banks efficient take-down Elimination protocol.... whenever you eliminate an alternative, let it take-down with it any alternative it defeats ... an expedient for conferring Banks efficiency on elimination methods. [A method is Banks efficient iff it is the head of a maximal transitive beatpath ... each path member defeats all subsequent path members, not only its immediate successor.] The heuristic for this policy is that once you have gone to all of the trouble to figure out who is so bad or weak to deserve Elimination, you should get your money's worth by taking down with it all of the candidates that are even weaker or worse ... at least in the sense of being directly defeated by it.. If at every step there was a candidate defeated by every remaining alternatives, eliminating that one by itself would be an ideal IIA step. If we are eliminating from the bottom of a list, we can get the "defeated by all remaining" in line for elimination by one pass of a pairwise sink sort. Even if there is no universally defeated alternative among the remaining alternatives, then the sink sort loser should still be one of the usual suspects to eliminate anyway ... assuming the candidates starting near the bottom of the list were already relatively unpromising according to whatever criterion goverened the list formation in the first place. Here's a proposed method: "Sink Loser Take-Down" Lacking a pairwise undefeated candidate ... 0. List the candidates in the Universal Tie Breaking Order with the most promising candidates at the top of the list L. [This UTBOrder is a highly decisive natural extension of the Implicit Approval Order as highlighted recently in the EM Thread entitled "General Purpose Universal Domain Tie Breaking Order."] 1. If only one uneliminated candidate remains on the list, elect that candidate, ... else ... 2. ... update the list by sink sorting it pairwise. 3. Let "pivot" be the candidate ending up at the bottom of this updated list. 3. Update the list again by eliminating the pivot along with every candidate defeated by it. 4. Repeat steps 1 through 3 until a winner appears in step 1. To make this more efficient, in step 2 take advantage of the fact that it only takes one pass of sink sort to find the pivot alternative. Here's the one pass procedure: Initialize the candidate at the top of the list as "pivot". While there exists some candidate X immediately below "pivot," swap them if X defeats "pivot", else let X be the new "pivot." EndWhile. Here is a bonus sincerity check: Do a final runoff between the winner (so far) and the last pivot, using a fresh set of ballots dedicated to this binary contest. Since this runoff is a secret ballot final binary choice, there is no incentive for insincerity on these runoff ballots. Furthermore, in the likely case that the lack of an undefeated pairwise winner was caused by burial of a sincere Condorcet Winner, this step will almost surely restore that sincere CW. What do you think? -fws

On 5/4/23 22:48, Forest Simmons wrote: > What started as Chain Climbing has evolved into a Banks efficient > take-down Elimination protocol.... whenever you eliminate an > alternative, let it take-down with it any alternative it defeats ... an > expedient for conferring Banks efficiency on elimination methods. A thought about this: Could we create a method that works in this way but also produces the same result as Smith//Range for a three-way contest? Suppose we have an ABCA cycle and everybody but that cycle has been eliminated. Then there are six possible Range orders: ABC, ACB, BAC, BCA, CAB, and CBA, where ABC means that A has the highest score, B is next, and C is last. If C is last, then C is eliminated and takes A with him. B wins. If B is last, then B is eliminated and takes C with him. A wins. If A is last, then A is eliminated and takes B with him. C wins. So this only produces the "proper" outcome if the Range order is ACB, BAC or CBA. The other half of the time, it'll be incorrect. So for maintaining the VSE of a high-VSE method, it might not be a good idea. It might be useful for strategy, though. An interesting question is if take-down elimination Benham loses any strategy resistance. With IRV being as chaotic as it is, we could hardly lose any VSE :-) And another thought: wanting Banks might be overkill -- at least for current versions of the "homotopy method". Although on the hand, if we do get large honest Condorcet cycles, that shows there's genuine competition. So who knows what could happen after Condorcet has been in use for a long time? -km

On Fri, May 5, 2023, 1:09 PM Kristofer Munsterhjelm <km_elmet@t-online.de> wrote: > On 5/4/23 22:48, Forest Simmons wrote: > > What started as Chain Climbing has evolved into a Banks efficient > > take-down Elimination protocol.... whenever you eliminate an > > alternative, let it take-down with it any alternative it defeats ... an > > expedient for conferring Banks efficiency on elimination methods. > > A thought about this: Could we create a method that works in this way > but also produces the same result as Smith//Range for a three-way contest? > > Suppose we have an ABCA cycle and everybody but that cycle has been > eliminated. Then there are six possible Range orders: ABC, ACB, BAC, > BCA, CAB, and CBA, where ABC means that A has the highest score, B is > next, and C is last. > > If C is last, then C is eliminated and takes A with him. B wins. > If B is last, then B is eliminated and takes C with him. A wins. > If A is last, then A is eliminated and takes B with him. C wins. > > So this only produces the "proper" outcome if the Range order is ACB, > BAC or CBA. The other half of the time, it'll be incorrect. So for > maintaining the VSE of a high-VSE method, it might not be a good idea. > > It might be useful for strategy, though. An interesting question is if > take-down elimination Benham loses any strategy resistance. With IRV > being as chaotic as it is, we could hardly lose any VSE :-) > > And another thought: wanting Banks might be overkill It's not Banks per se that seems important to me ... it just turns out that this Banks efficient protocol is a very inexpensive, simple way to make any elimination method Landau efficient ... which is important for immunity against pairwise domination complaints ... such cheap insurance it would be irresponsible to pass it up! -- at least for > current versions of the "homotopy method". Although on the hand, if we > do get large honest Condorcet cycles, that shows there's genuine > competition. So who knows what could happen after Condorcet has been in > use for a long time? > > -km >