A generalized median voting method elects the alternative that minimizes the total distance to the ballots. But how do we gauge the distance from an alternative to a ballot in the Universal Domain context? One piece of the puzzle is that the position of a ballot B in issue space is most simply represented by the position of the most favored alternative on that ballot Y=f(B). So we need a metric d(X,Y), for the distances between the various possible positions of the respective alternatives (X), and the fixed positions (Y) of the ballot favorites. The simplest metric I can think of in the Universal Domain context for the distance from a moving (i.e. adjusting towards minimality) alternative X to a fixed alternative Y, is the number of ballots on which Y is preferred over X. This makes sense, because as X moves directly away from stationary Y, the number of ballots on which Y is preferred over X can only increase. Put these pieces of the puzzle together and we can model the total distance from X to the ballots as the sum .. S(X)=Sum(over Y) of d(X, Y)*f(Y), where d(X,Y) is the number of ballots on which Y outranks X, and f(Y) is the percentage of ballots on which Y is the favorite alternative. So the purest median method I can come up with in the UD context is to elect argmin S(X). If I am not mistaken, this method satisfies the FBC. If you prefer that the winner be uncovered, you can trade in the FBC for Landau efficiency by attaching a Landau afterburner: While argmin S(X) is covered, eliminate X, and replace it with the remaining alternative closest to X in its value of S among the alternatives that cover X. -Forest

We should call this method MEPO for .Min Expected Pairwise Opposition in comparison with Min Max Pairwise Opposition MMPO. The main defect of MMPO is Plurality failure, which cannot afflict MEPO as long as E is the Random Favorite Lottery Expectation, i.e. the Benchmark Lottery Expectation. By the way, when Kevin first posted about MMPO, he based it on the same geometry that I used to describe MEPO: As X moves further from Y, the number of ballots that prefer Y over X increases. -Forest On Wed, Oct 12, 2022, 11:29 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > A generalized median voting method elects the alternative that minimizes > the total distance to the ballots. But how do we gauge the distance from an > alternative to a ballot in the Universal Domain context? > > One piece of the puzzle is that the position of a ballot B in issue space > is most simply represented by the position of the most favored alternative > on that ballot Y=f(B). > > So we need a metric d(X,Y), for the distances between the various possible > positions of the respective alternatives (X), and the fixed positions (Y) > of the ballot favorites. > > The simplest metric I can think of in the Universal Domain context for the > distance from a moving (i.e. adjusting towards minimality) alternative X to > a fixed alternative Y, is the number of ballots on which Y is preferred > over X. > > This makes sense, because as X moves directly away from stationary Y, the > number of ballots on which Y is preferred over X can only increase. > > Put these pieces of the puzzle together and we can model the total > distance from X to the ballots as the sum .. > > S(X)=Sum(over Y) of d(X, Y)*f(Y), > > where d(X,Y) is the number of ballots on which Y outranks X, and f(Y) is > the percentage of ballots on which Y is the favorite alternative. > > So the purest median method I can come up with in the UD context is to > elect argmin S(X). > > If I am not mistaken, this method satisfies the FBC. > > If you prefer that the winner be uncovered, you can trade in the FBC for > Landau efficiency by attaching a Landau afterburner: > > While argmin S(X) is covered, eliminate X, and replace it with the > remaining alternative closest to X in its value of S among the alternatives > that cover X. > > -Forest >

If we normalize S(X) by converting numbers of ballots to percentages, then S(X) becomes the probability that a random voter would prefer the favorite on a randomly drawn ballot, to X. On Thu, Oct 13, 2022, 9:14 AM Forest Simmons <forest.simmons21@gmail.com> wrote: > We should call this method MEPO for .Min Expected Pairwise Opposition in > comparison with Min Max Pairwise Opposition MMPO. > > The main defect of MMPO is Plurality failure, which cannot afflict MEPO as > long as E is the Random Favorite Lottery Expectation, i.e. the Benchmark > Lottery Expectation. > > By the way, when Kevin first posted about MMPO, he based it on the same > geometry that I used to describe MEPO: > As X moves further from Y, the number of ballots that prefer Y over X > increases. > > -Forest > > On Wed, Oct 12, 2022, 11:29 PM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> A generalized median voting method elects the alternative that minimizes >> the total distance to the ballots. But how do we gauge the distance from an >> alternative to a ballot in the Universal Domain context? >> >> One piece of the puzzle is that the position of a ballot B in issue >> space is most simply represented by the position of the most favored >> alternative on that ballot Y=f(B). >> >> So we need a metric d(X,Y), for the distances between the various >> possible positions of the respective alternatives (X), and the fixed >> positions (Y) of the ballot favorites. >> >> The simplest metric I can think of in the Universal Domain context for >> the distance from a moving (i.e. adjusting towards minimality) alternative >> X to a fixed alternative Y, is the number of ballots on which Y is >> preferred over X. >> >> This makes sense, because as X moves directly away from stationary Y, the >> number of ballots on which Y is preferred over X can only increase. >> >> Put these pieces of the puzzle together and we can model the total >> distance from X to the ballots as the sum .. >> >> S(X)=Sum(over Y) of d(X, Y)*f(Y), >> >> where d(X,Y) is the number of ballots on which Y outranks X, and f(Y) is >> the percentage of ballots on which Y is the favorite alternative. >> >> So the purest median method I can come up with in the UD context is to >> elect argmin S(X). >> >> If I am not mistaken, this method satisfies the FBC. >> >> If you prefer that the winner be uncovered, you can trade in the FBC for >> Landau efficiency by attaching a Landau afterburner: >> >> While argmin S(X) is covered, eliminate X, and replace it with the >> remaining alternative closest to X in its value of S among the alternatives >> that cover X. >> >> -Forest >> >

Let's look at a Chicken example: 49 C 26 A>B 25 B (sincere B>C) The pairwise matrix M is [[0, 26, 26] [25,0,51 ][49,49,0]] The random ballot favorite row vector V is [[26, 25, 49]] The matrix product VM is [[25^2+49^2,26^2+49^2,26^2+25×51]] Evidently the third component of this vector is the smallest so argmin S(X) is candidate C.... the B faction Chicken ploy failed. Now, let's look at ... 40 A 10 A=C 10 B=C 40 B We have M equal to [[0,50,40] [50,0,40] [10,10,0]] The random ballot favorite row vector is [[45,45,10]] The matrix product VM is the row vector [[45*50+10^2,45*50+10^2,45^2+45^2]] Evidently, argmin S(X) is {A,B}, so A and B are tied for the win. Our method MEPO seems to respect Plurality, unlike MMPO. More and more, I like the idea of MEPO with a Landau afterburner. Stitching on the afterburner leaves a slight seam in the rare cases it needs to be applied to achieve Landau efficiency. [None of our examples have needed it, so far.] -Forest On Thu, Oct 13, 2022, 9:14 AM Forest Simmons <forest.simmons21@gmail.com> wrote: > We should call this method MEPO for .Min Expected Pairwise Opposition in > comparison with Min Max Pairwise Opposition MMPO. > > The main defect of MMPO is Plurality failure, which cannot afflict MEPO as > long as E is the Random Favorite Lottery Expectation, i.e. the Benchmark > Lottery Expectation. > > By the way, when Kevin first posted about MMPO, he based it on the same > geometry that I used to describe MEPO: > As X moves further from Y, the number of ballots that prefer Y over X > increases. > > -Forest > > On Wed, Oct 12, 2022, 11:29 PM Forest Simmons <forest.simmons21@gmail.com> > wrote: > >> A generalized median voting method elects the alternative that minimizes >> the total distance to the ballots. But how do we gauge the distance from an >> alternative to a ballot in the Universal Domain context? >> >> One piece of the puzzle is that the position of a ballot B in issue >> space is most simply represented by the position of the most favored >> alternative on that ballot Y=f(B). >> >> So we need a metric d(X,Y), for the distances between the various >> possible positions of the respective alternatives (X), and the fixed >> positions (Y) of the ballot favorites. >> >> The simplest metric I can think of in the Universal Domain context for >> the distance from a moving (i.e. adjusting towards minimality) alternative >> X to a fixed alternative Y, is the number of ballots on which Y is >> preferred over X. >> >> This makes sense, because as X moves directly away from stationary Y, the >> number of ballots on which Y is preferred over X can only increase. >> >> Put these pieces of the puzzle together and we can model the total >> distance from X to the ballots as the sum .. >> >> S(X)=Sum(over Y) of d(X, Y)*f(Y), >> >> where d(X,Y) is the number of ballots on which Y outranks X, and f(Y) is >> the percentage of ballots on which Y is the favorite alternative. >> >> So the purest median method I can come up with in the UD context is to >> elect argmin S(X). >> >> If I am not mistaken, this method satisfies the FBC. >> >> If you prefer that the winner be uncovered, you can trade in the FBC for >> Landau efficiency by attaching a Landau afterburner: >> >> While argmin S(X) is covered, eliminate X, and replace it with the >> remaining alternative closest to X in its value of S among the alternatives >> that cover X. >> >> -Forest >> >