The median X of a finite set of distinct points arranged along a straight line segment will always minimize the sum of distances from it to the other points. [If the points are not distinct, a weighted sum does the job.] Consequently one way to generalize the concept of "median" in a general metric space is by minimization of (weighted) sums of distances. Thus, the Kemeny-Young method chooses the "finish order" that minimizes its sum of distances to the ballots, i.e to their respective rank orders. In this context, the distance from a ballot order to a potential finish order is their Kendall-tau distance, the total number of basic order reversals necessary to convert one order into the other. There are two unnecessary difficulties associated with Kemeny-Young: (1) The number of finish orders that need to be checked grows exponentially with the number of candidates, even when all we need is the winner of a single winner election. (2) The method is clone dependent ... a fatal flaw in the context of electoral politics. The basic spoiler problem that sparked election method reform in the first place was a failure of clone independence. Even IRV with all of its other problems, is clone independent. The method we propose is both clone independent and computationally efficient. The key innovation is that we gauge the distance from ballot B to a potential winner X by the cost of the least expensive beatpath from X to the candidate f(B) that is favored above all others on ballot B. I'm going to break here to let this idea sink in a little before filling in the few remaining details. To be continued...

We now finish our description of the method introduced previously, by defining the cost of a beatpath from candidate X to candidate Y as the number of steps in the path plus epsilon times the losing vote total along the path. Any positive infinitesimal will do for epsilon, because all sufficiently small standard positive values will give the same result as determining cost primarily by the number of steps, and secondarily by the losing vote sum. Note that if X is in the Landau set, there will always be a a beatpath of two or fewer steps to Y. On the other hand, if X is not in Landau, then for some Y, no beatpath from X to Y will have fewer than three steps. It follows that this method is Landau efficient. Questions? Thanks! Forest On Fri, Oct 7, 2022, 6:54 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > The median X of a finite set of distinct points arranged along a straight > line segment will always minimize the sum of distances from it to the other > points. [If the points are not distinct, a weighted sum does the job.] > > Consequently one way to generalize the concept of "median" in a general > metric space is by minimization of (weighted) sums of distances. > > Thus, the Kemeny-Young method chooses the "finish order" that minimizes > its sum of distances to the ballots, i.e to their respective rank orders. > > In this context, the distance from a ballot order to a potential finish > order is their Kendall-tau distance, the total number of basic order > reversals necessary to convert one order into the other. > > There are two unnecessary difficulties associated with Kemeny-Young: > (1) The number of finish orders that need to be checked grows > exponentially with the number of candidates, even when all we need is the > winner of a single winner election. > (2) The method is clone dependent ... a fatal flaw in the context of > electoral politics. The basic spoiler problem that sparked election method > reform in the first place was a failure of clone independence. Even IRV > with all of its other problems, is clone independent. > > The method we propose is both clone independent and computationally > efficient. > > The key innovation is that we gauge the distance from ballot B to a > potential winner X by the cost of the least expensive beatpath from X to > the candidate f(B) that is favored above all others on ballot B. > > I'm going to break here to let this idea sink in a little before filling > in the few remaining details. > > To be continued... > > >

It may be necessary for monotonicity compliance to replace "total" with "max" in the phrase "the losing vote total along the path" On Fri, Oct 7, 2022, 7:28 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > We now finish our description of the method introduced previously, by > defining the cost of a beatpath from candidate X to candidate Y as the > number of steps in the path plus epsilon times the losing vote total along > the path. > > Any positive infinitesimal will do for epsilon, because all sufficiently > small standard positive values will give the same result as determining > cost primarily by the number of steps, and secondarily by the losing vote > sum. > > Note that if X is in the Landau set, there will always be a a beatpath of > two or fewer steps to Y. > > On the other hand, if X is not in Landau, then for some Y, no beatpath > from X to Y will have fewer than three steps. > > It follows that this method is Landau efficient. > > Questions? > > Thanks! > > Forest > > > On Fri, Oct 7, 2022, 6:54 PM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> The median X of a finite set of distinct points arranged along a straight >> line segment will always minimize the sum of distances from it to the other >> points. [If the points are not distinct, a weighted sum does the job.] >> >> Consequently one way to generalize the concept of "median" in a general >> metric space is by minimization of (weighted) sums of distances. >> >> Thus, the Kemeny-Young method chooses the "finish order" that minimizes >> its sum of distances to the ballots, i.e to their respective rank orders. >> >> In this context, the distance from a ballot order to a potential finish >> order is their Kendall-tau distance, the total number of basic order >> reversals necessary to convert one order into the other. >> >> There are two unnecessary difficulties associated with Kemeny-Young: >> (1) The number of finish orders that need to be checked grows >> exponentially with the number of candidates, even when all we need is the >> winner of a single winner election. >> (2) The method is clone dependent ... a fatal flaw in the context of >> electoral politics. The basic spoiler problem that sparked election method >> reform in the first place was a failure of clone independence. Even IRV >> with all of its other problems, is clone independent. >> >> The method we propose is both clone independent and computationally >> efficient. >> >> The key innovation is that we gauge the distance from ballot B to a >> potential winner X by the cost of the least expensive beatpath from X to >> the candidate f(B) that is favored above all others on ballot B. >> >> I'm going to break here to let this idea sink in a little before filling >> in the few remaining details. >> >> To be continued... >> >> >>

It is now obvious to me that the simplest, adequate measure of beatpath cost, and the one that completes the "friendliness" measure in our longstanding foA-fpC thread, is simply the number of steps in the beatpath. The fewer the number of steps in the beatpath from X to Y, the friendlier Y is to X. Time to summarize the method that has culminated in this thread. I'm tempted to call it the Munsterhjelm Venzke Heitzig or MVH method after the principal contributors of the key ideas. Many other EM List members from (Demorep to Lung) have contributed in ways too numerous to enumerate. I can truly say that anybody who made regular contributions to the EM List in the form of questions, criticisms, or clarifications over any period of a year or more has contributed to my understanding of voting methods ... for which I am very grateful. Special mention to Benham, Ossipoff, Rob LeGrand, Andrew Jennings, Joe Weinstein, Bart Ingles, etc. for tolerating my newbie efforts. HVM: Elect the candidate X with the smallest total of beatpath costs from X to the ballot favorites. In other words, for each ballot B, and each candidate X, let Cost(X,fpB) be the minimum number of steps in a beatpath from X to fpB, the first place favorite of ballot B. Note that if X=fpB, then the beatpath from X to F(B has zero steps, so the cost is zero. Let SumfpB(X) be the sum over all ballots B of Cost(X,fpB). Elect argmin SumfpB(X) It is not hard to show that this simple, decisive, Universal Domain election method is clone independent, Landau efficient, and monotone. If these claims hold up to close scrutiny, and MVH truly does generalize fpA-fpC, then why not propose it for public elections? -Forest On Fri, Oct 7, 2022, 6:54 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > The median X of a finite set of distinct points arranged along a straight > line segment will always minimize the sum of distances from it to the other > points. [If the points are not distinct, a weighted sum does the job.] > > Consequently one way to generalize the concept of "median" in a general > metric space is by minimization of (weighted) sums of distances. > > Thus, the Kemeny-Young method chooses the "finish order" that minimizes > its sum of distances to the ballots, i.e to their respective rank orders. > > In this context, the distance from a ballot order to a potential finish > order is their Kendall-tau distance, the total number of basic order > reversals necessary to convert one order into the other. > > There are two unnecessary difficulties associated with Kemeny-Young: > (1) The number of finish orders that need to be checked grows > exponentially with the number of candidates, even when all we need is the > winner of a single winner election. > (2) The method is clone dependent ... a fatal flaw in the context of > electoral politics. The basic spoiler problem that sparked election method > reform in the first place was a failure of clone independence. Even IRV > with all of its other problems, is clone independent. > > The method we propose is both clone independent and computationally > efficient. > > The key innovation is that we gauge the distance from ballot B to a > potential winner X by the cost of the least expensive beatpath from X to > the candidate f(B) that is favored above all others on ballot B. > > I'm going to break here to let this idea sink in a little before filling > in the few remaining details. > > To be continued... > > >

A slight correction is needed: the simple version, is Smith Efficient, but not quite Landau efficient. For that we need a tweak to the cost of a beatpath that is a concave down increasing function of k: c(k,epsilon)=(1-epsilon^k)/(1-epsilon), in other symbols ... Sum over j from zero to k of epsilon^j, where epsilon is some fixed, positive infinitesimal. On Sun, Oct 9, 2022, 3:45 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > It is now obvious to me that the simplest, adequate measure of beatpath > cost, and the one that completes the "friendliness" measure in our > longstanding foA-fpC thread, is simply the number of steps in the > beatpath. The fewer the number of steps in the beatpath from X to Y, the > friendlier Y is to X. > > Time to summarize the method that has culminated in this thread. I'm > tempted to call it the Munsterhjelm Venzke Heitzig or MVH method after the > principal contributors of the key ideas. Many other EM List members from > (Demorep to Lung) have contributed in ways too numerous to enumerate. I > can truly say that anybody who made regular contributions to the EM List in > the form of questions, criticisms, or clarifications over any period of a > year or more has contributed to my understanding of voting methods ... for > which I am very grateful. Special mention to Benham, Ossipoff, Rob LeGrand, > Andrew Jennings, Joe Weinstein, Bart Ingles, etc. for tolerating my newbie > efforts. > > HVM: Elect the candidate X with the smallest total of beatpath costs from > X to the ballot favorites. > > In other words, for each ballot B, and each candidate X, let Cost(X,fpB) > be the minimum number of steps in a beatpath from X to fpB, the first place > favorite of ballot B. Note that if X=fpB, then the beatpath from X to F(B > has zero steps, so the cost is zero. > Let SumfpB(X) be the sum over all ballots B of Cost(X,fpB). > Elect argmin SumfpB(X) > > It is not hard to show that this simple, decisive, Universal Domain > election method is clone independent, Landau efficient, and monotone. > > If these claims hold up to close scrutiny, and MVH truly does generalize > fpA-fpC, then why not propose it for public elections? > > -Forest > > On Fri, Oct 7, 2022, 6:54 PM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> The median X of a finite set of distinct points arranged along a straight >> line segment will always minimize the sum of distances from it to the other >> points. [If the points are not distinct, a weighted sum does the job.] >> >> Consequently one way to generalize the concept of "median" in a general >> metric space is by minimization of (weighted) sums of distances. >> >> Thus, the Kemeny-Young method chooses the "finish order" that minimizes >> its sum of distances to the ballots, i.e to their respective rank orders. >> >> In this context, the distance from a ballot order to a potential finish >> order is their Kendall-tau distance, the total number of basic order >> reversals necessary to convert one order into the other. >> >> There are two unnecessary difficulties associated with Kemeny-Young: >> (1) The number of finish orders that need to be checked grows >> exponentially with the number of candidates, even when all we need is the >> winner of a single winner election. >> (2) The method is clone dependent ... a fatal flaw in the context of >> electoral politics. The basic spoiler problem that sparked election method >> reform in the first place was a failure of clone independence. Even IRV >> with all of its other problems, is clone independent. >> >> The method we propose is both clone independent and computationally >> efficient. >> >> The key innovation is that we gauge the distance from ballot B to a >> potential winner X by the cost of the least expensive beatpath from X to >> the candidate f(B) that is favored above all others on ballot B. >> >> I'm going to break here to let this idea sink in a little before filling >> in the few remaining details. >> >> To be continued... >> >> >>