[EM] Electrical Network Resistance Model (was Ideas for distance Schulze tie-breaking)

[Forest Simmons]

Here is a highly flawed first attempt at decloning the "electometer"
network:

Unlike the straightforward case of decloning the beatpath resistance
method, the electrical network resistance model allows for parallel paths
of current, which introduces another source of clone dependence.
To compensate for that we need to include another factor in the resistance
diodes.

 Recall that every pair of candidates (j, k) is connected by two diodes (or
possibly one equivalent composite circuit element, but let's go with two
dodes for clarity):

The diode for current from node j to k has a rectifier that enforces that
direction in series with a resistor whose resistance is jointly
proportional with three factors ... (1) the number of ballots on which
candidate k is designated favorite, (2) the number of of ballots on which
candidate k is ranked ahead of candidate j, and (3) the number of ballots
on which j is designated as "anti-favorite."

It is this last factor of resistance that allows for candidate j to be
cloned in the context of current flowing out of j into k, just as the first
factor allows for k to be cloned in the context of current flowing into k
from j.

If j and k, respectively, are replaced by clone sets of cardinality #j and
#k, respectively, then the directional diode from j to k will be replaced
by #j*#k diodes whose combined resistance will be the same as the single
resistance described above for the diode from j to k.

Unfortunately, this last claim of combined resistance cannot be correct
because only in series (as opposed to parallel) is resistance additive.

Does anybody see how to rescue this approach?

Perhaps through conductance, which is additive in parallel ...

-Forest

El mar., 1 de mar. de 2022 1:38 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> I wrote this in a hurry ... the basic method should be called
> MinMaxLeastTotalResistanceBeatPath.
>
> The total resistance TR of a beatpath BP is the sum of its losing votes.
>
> For each pair of candidates (j,k) let TLR(j,k) be the total resistance of
> the beatpath of least resistance from j to k. Elect the candidate j that
> minimizes the maximum value of TLR(j,k) as k varies over the other
> candidates.
>
> To get the de-cloned version, weight each losing vote by the number of
> ballots on which the pairwise loser is designated favorite.
>
> Any electrical engineer should be able to design an "electo-meter" network
> that has, for each pair of candidates (j,k), a diode from node j to node k
> that allows current to move from j to k with resistance jointly
> proportional to the number of ballots on which k is ranked ahead of j and
> the number of ballots on which k is designated favorite.
>
> Elect the candidate j that minimizes the maximum network resistance from
> node j to node k as k varies over the other nodes. [Put the source probe of
> a directional ohm meter at j while varying the sink probe over the other
> nodes (k).]
>
> One of ypu engineers correct me if my EE terminology isn't quite right.
>
> -Forest
>
> El mar., 1 de mar. de 2022 9:25 a. m., Forest Simmons <
> forest.simmons21 at gmail.com> escribió:
>
>> Define the total resistance of the losers in a beatpath as the sum of the
>> losing votes. Unfortunately, minimization of this total beatpath resistance
>> is a clone dependent method.
>>
>> However, the method can be decloned.
>>
>> Let L be the vector of losing votes along the beatpath. Let F be the
>> vector of corresponding favorite votes, i.e. F_k is the number of ballots
>> on which candidate k is designated as favorite.
>>
>> The decloned resistance of the beatpath is the dot product of L and F.
>>
>> Minimizing this dot product over beatpaths is a clone free method that
>> could be called de-cloned Min Resistance BeatPath (dcMinRBP).
>>
>> Use this method to break ties.
>>
>> Better yet, use dcMinRBP as the main method, and break ties with Schulze.
>>
>> El dom., 27 de feb. de 2022 6:05 a. m., Kristofer Munsterhjelm <
>> km_elmet at t-online.de> escribió:
>>
>>> Like minmax, Schulze sometimes has a lot of ties, particularly for
>>> elections with few voters. But perhaps this way could work to break ties
>>> in a consistent way:
>>>
>>> Let p[A, B] be the strength of the strongest beatpath going from A to B.
>>>
>>> Now do ext-Minmax on the p matrix. This is minmax (i.e. A's score is A's
>>> weakest pairwise contest vs B), but ties are broken by looking at second
>>> weakest pairwise contests, third weakest, etc.
>>>
>>> Since there always exists a candidate A who is the "Condorcet winner"
>>> according to p, and this is the same as the Schulze winner, and both
>>> Minmax and ext-Minmax pass Condorcet, this method should agree with
>>> Schulze when there are no ties.
>>>
>>> But do we lose any criteria from this? I don't know; it's just a thought
>>> that occurred to me as a way to make ext-Minmax more Schulze-like.
>>> Possibly there exist situations where there's a tie according to
>>> Schulze, and this breaks the tie in a way that (say) depends on clones;
>>> but I'm not sure how to construct such an example.
>>>
>>> Another possible drawback is that the social order may suffer, because
>>> minmax doesn't pass Condorcet loser. So it might be that the social
>>> ranking is A>B>C>D by Schulze, and this is strict (A has 3 strong
>>> beatpaths, B has 2, C has 1, D has none), but D's weakest beatpath is
>>> stronger than C's; and then the ext-Minmax modification returns a
>>> different social order even though there are no ties at any point.
>>>
>>> This might point to the idea being not entirely defensible. One could,
>>> of course, only break the ties that exist by ext-Minmax, but that feels
>>> rather like a hack.
>>>
>>> Perhaps it would be doable to augment Floyd-Warshall to maximize the
>>> leximin of the beatpath instead of just its minimum - among the paths
>>> with the strongest weakest link, find the one with the strongest
>>> second-weakest link, etc. Then comparing the full vectors of defeats
>>> along the beatpath thus recovered could resolve ties in a way more
>>> consistent with the Schulze method itself.
>>>
>>> I'm still only working by intuition, though; I haven't rigorously
>>> checked any of these ideas.
>>>
>>> -km
>>> ----
>>> Election-Methods mailing list - see https://electorama.com/em for list
>>> info
>>>
>>
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