[EM] EXACT, a Majority Judgment-like IBIFA variant w/FBC and IBI

[Ted Stern]

See inserted reply below:

On Sun, Feb 4, 2018 at 6:53 AM, Kristofer Munsterhjelm <km_elmet at t-online.de
> wrote:

> On 12/26/2017 12:43 AM, Ted Stern wrote:
>
>> Chris Benham proposed IBIFA in May and June, 2010, on the
>> election-methods mailing list:
>>
>> http://lists.electorama.com/pipermail/election-methods-elect
>> orama.com/2010-May/091807.html
>> <http://lists.electorama.com/pipermail/election-methods-elec
>> torama.com/2010-May/091807.html>
>>
>> http://election-methods.electorama.narkive.com/KdBxpweB/irre
>> levant-ballots-independent-fallback-approval-ibifa
>> <http://election-methods.electorama.narkive.com/KdBxpweB/irr
>> elevant-ballots-independent-fallback-approval-ibifa>
>>
>> http://wiki.electorama.com/wiki/IBIFA
>> <http://wiki.electorama.com/wiki/IBIFA>
>>
>> IBIFA is, as originally stated, a "Bucklin-like method meeting Favorite
>> Betrayal and Irrelevant Ballots."  Its key principle is to compare the
>> ballots voting for a candidate at-or-above a particular rating to the
>> most-approved candidate on the complementary ballots.  When the former
>> exceeds the latter, a meaningful threshold has been crossed, unlike the
>> arbitrary 50% threshold of median rating methods.  This is what enables
>> IBIFA to yield the same result if irrelevant ballots are added or
>> dropped.  By construction, IBIFA is cloneproof.
>>
>> With this in mind, I realized that a minor modification of IBIFA would
>> make it more like Majority Judgment, reducing later-harm and improving
>> Condorcet consistency (though not completely), while satisfying the same
>> criteria as MJ.
>>
>
> Do you think it's possible to generalize that strategy to handle
> multiwinner elections as well? I can't see any obvious ways, but it would
> be nice if one could make multiwinner methods with implicit thresholds as
> well, since the vast majority uses explicit thresholds (usually the Droop
> quota).


Single-winner rated methods can be adapted to an ER-Bucklin-Droop
multiwinner method.  Just choose each seat using the single-winner method
and weighted ballots.

Then reweight the ballots by finding the score at-and-above which at least
one Droop quota of voters approves the seat winner, and reweight those
ballots to remove one Droop quota.

The difference from straight ER-Bucklin multiwinner is that the
quota-finding score is *not* the one used to determine the seat winner.


>
>
> EXACT does require several N^2 arrays for summable storage, but note
>> that no sorting of the ballots is required as with pairwise methods.
>>
>
> Pairwise methods don't strictly require that ballots are sorted. E.g. the
> following will produce a Condorcet matrix (if there's no truncation or
> equal rank):
>
> for i = 1 to num ballots
>         for j = 1 to num candidates
>                 c1 = candidate ranked at jth place on the ith ballot
>
>                 for k = j+1 to num candidates
>                         c2 = candidate ranked at kth place on the ith
> ballot
>                         matrix[c1 beats c2] += weight of ballot i
>
> It's a bit of a nitpick, though :-)
>

I apologize for my earlier assertion.  When I tried coding the pairwise
array myself, using the numpy package in Python, I found a very compact way
to do so, much along those lines.  Using the matrix optimized numpy
package, imported as "np",

        for ballot, w in zip(ballots,weight):
            for v in range(1,maxscore):           # Assuming all rated
candidates are approved
                A  += w * np.multiply.outer(np.where(ballot==v,1,0),
                                            np.where(ballot<=v,1,0))


In other words, the matrix update to the pairwise array is, for each rating
v, the outer product of (a) a vector with ones where candidates have score
v, with (b) another vector with ones where candidates have scores less than
or equal (or less than) v.
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