[EM] Strong Reverse Symmetry Clone Proof Borda

[Richard Lung]

It's gratifying to be praised for helping something you don't yourself 
comprehend. Forest has that good nature.

It may be worth mentioning that the binomial stv use of an election 
count and an exclusion count may both benefit a candidate. If the 
candidate is popular enough to gain an election quota, but not too 
unpopular to gain an exclusion quota,  then a poor exclusion count can 
consolidate a good election count, for the same candidate..

Unlike classical Borda count, but in iine with the Gregory count, later 
preferences do not count against former preferences. When there is a 
conflict with a candidate both popular (in the election count) and 
unpopular (in the exclusion count) it is a reflection of real 
differences in opinion between different factions of the electorate.

Unlike traditional stv, the Gregory count, via an extension of Meek 
method keep-value counting to all candidates, not just elected 
candidates, is repeated on the voters preferences, in reverse, as an 
unpreference vote. This necessitates the counting of abstentions, to 
ascertain the balance of liking and disliking of the candidates, by the 
voters.

This in turn means all the preferences are counted, which amounts to a 
complete dimension of choice counted --  not merely an uncertain 
fraction of a dimension counted.

Real elections with binomial stv should show how the entropy of the 
voters usual exponential decline in preferences -- or unpreferences -- 
works or fails to work for decisive results, in practise.

Sorry, if this explanation was too cryptic.

Regards,

Richard Lung.





To give credit where credit is due, I want to thank Richard Lung for 
getting us thinking along the inclusion/exclusion axis corresponding to 
near-first/near-last ballot ranks ... That contributed to my confidence 
in the lottery/anti-lottery idea as a means of de-cloning Borda, 
Copeland. and Kemeny-Young, which has panned out so beautifully.

Thanks, Richard!!!

El vie., 14 de ene. de 2022 8:18 a. m., Forest Simmons 
<forest.simmons21 at gmail.com> escribió:

    Two equivalent ways of decloning Borda ...  both relying on a
    combination of the benchmark and anti-benchmark lotteries BML and ABML.

    First method:

    Ballot B contributes to candidate X's score the sum of BML(j)
    [summed over candidates j ranked below X] minus the sum of ABML(k)
    [summed over candidates k ranked above X].

    Think of it this way ... the importance of candidate X to the voter
    of ballot B is the estimated strength of the candidates she likes
    less than X, minus the estimated weakness of the candidates she
    likes better:

    [If the ones you like better are weak and the ones you dislike in
    comparison are strong, you should support X.]

    [Also this is a natural DSV kind of way to specify an approval cutoff]

    Method 2.

    The score for X is the difference in the BML expectation of X's
    pairwise matrix row and the ABML expectation of X's pairwise matrix
    column.

    The former is a source of pride for X ...the weighted average
    pairwise scores of X over the candidates, where the weights are
    estimates of the strengths of those candidates.

    The latter expectation is a source of chagrin for X ... the weighted
    average of the scores against X by the other candidates, where the
    weights are estimates of the weakness of those candidates... if you
    are soundly trounced by a weak candidate, you should feel chagrin,
    if not shame!

    This gives another idea for a kind of MinMax/MaxMin method:

    Let M(X) be the Max value of the product of ABML(k)*P(k,X), where
    P(k, X) is the percentage of ballots on which k is ranked ahead of X.

    Similarly, let m(X) be the min value of P(X, j)*BML(j), where P(X,j)
    is the percentage of ballots ranking X above j.

    Elect argmin(M(X)-m(X)).

    [Maybe this is another key to fpC-fpA]

    Note the strong symmetry of this method, also.

    All in all, I'm pretty excited about the usefulness of the
    anti-benchmark lottery in conjunction with the benchmark lottery.

    If BML(X) is an estimate of X's viability, then ABML(X) is an
    estimate of X's weakness.

    Of course, the BML is not the only clone-free lottery ...Jobst's
    MaxParC, the Martin Harper Lottery, the Random Approval Lottery, the
    Nash Lottery and its "Ultimate Lottery" generalization ...all of
    these and others have their corresponding anti-lottery counterparts
    ... just reverse the ballots to get them.

    Why didn't we think of this before?  It was staring us right in the
    face!

    What other obvious ideas are still invisible to us?

    -Forest


    El vie., 14 de ene. de 2022 6:24 a. m., Forest Simmons
    <forest.simmons21 at gmail.com> escribió:

        Kristofer,

        I have a feeling you might like this method:

        For simplicity assume complete rankings.

        For each candidate X, let fp(X) and lp(X), respectively, be the
        total number of first place votes and last place votes,
        respectively, of the candidates that are pairwise defeated by X,
        and pairwise defeat X, respectively.

        Elect argmax(fp(X)-lp(X))

        What do you think?

        -Forest




        El vie., 14 de ene. de 2022 2:11 a. m., Kristofer Munsterhjelm
        <km_elmet at t-online.de> escribió:

            On 14.01.2022 11:01, Kristofer Munsterhjelm wrote:
             > On 14.01.2022 04:05, Forest Simmons wrote:
             >> I've always admired the basic idea of Kemeny-Young,
            namely a cost metric
             >> for conversion of one ranking into another ... simply
            the number of
             >> transpositions (elementary swaps) required.
             >>
             >> The deal-breaker draw-back of the Kemeny cost metric is
            that it is clone
             >> dependent ... if it were independent of clones the "
            cost" of
             >> transposing AB to BA would be the same as the net cost
            of transposing
             >> the rank order of respective clones of A and B:
             >>
             >> the order a1a2a3b1b2b3
             >>  to the order b3b2b1a3a2a1
             >
             > Of course, I have to restate here that Ranked Pairs is
            such a method:
             > the metric is just leximax. (ordering a is better than
            ordering b if the
             > greatest pairwise victory consistent with a is higher
            than the greatest
             > pairwise victory consistent with b, with tiebreaks for
            second greatest,
             > third greatest, etc.) And it's cloneproof.

            Here's a thought: let parameterized p-Kemeny be: take the lp
            norm of the
            vector of all pairwise victories consistent with the output
            ordering.
            Break ties by gradually lowering p, i.e. l(p-epsilon),
            further ties by
            l(p-2 epsilon), etc.

            Then Kemeny is l_1 and RP is the limit as p approaches
            infinity, l_inf.
            I think? Are any other values of p interesting?

            -km
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