[EM] A Decisive, Landau Efficient, Generalized Median, Single Winner Method

[Forest Simmons]

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 at 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 at 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...
>>
>>
>>
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