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

[Ted Stern]

Chris Benham proposed IBIFA in May and June, 2010, on the election-methods mailing list:

http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-May/091807.html

http://election-methods.electorama.narkive.com/KdBxpweB/irrelevant-ballots-independent-fallback-approval-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.

IBIFA, simply stated, does the following:

• Find the highest rating R, for which there is at least one candidate X who is rated at or above level R on more ballots than any candidate is approved on ballots that rate X below R.
• If there is more than one such candidate X, elect the candidate X with the most ballots rating X at R or above.
• If no candidates satisfy the first criterion, for any approved rating R, elect the candidate with the highest approval over all ballots.

My modification is inserted with emphasis added.

• Find the highest rating R, for which there is at least one candidate X who is rated at or above level R on more ballots than any candidate is approved on ballots which rate X below R.
• If there is more than one such candidate X, then if there is at least one candidate Y who is rated above R on more ballots than the highest approved candidate on ballots that rate Y below R, elect the candidate Y with the most ballots rating Y above R.
• Otherwise, elect the candidate X with the most ballots rating X at R or above.
• If no candidates satisfy the first criterion, for any approved rating R, elect the candidate with the highest approval over all ballots.
  

I call this IBIFA variant "EXACT", because it uses an EXclusive Approval Comparison Threshold.  That is, the candidate compared to X is the one with maximum approval on ballots that exclude votes for X at some rating or above.  Like IBIFA, it is also cloneproof.

For EXACT, it is convenient to keep track of co-approval: the approval for candidates X[j] on a ballot containing candidate X[i] with rating k:

   for ballot in ballots:

     for candidate i on ballot with score k:

       if k approved:

         for candidate j on ballot with score m:

           if m approved:

             W[k,i,j] += 1

Note that W[k,i,i] is the total approval for candidate X[i] at rating k, and the total approval for candidate X[i] at rating k and higher is the sum of W[k,i,i] over all approved ratings k.

It should then be clear that the approval for any candidate j on a ballot that rates X[i] at R or higher is 

      Approval[j] - W[R,i,j] - W[R+1,i,j] ... - W[MaxScore,i,j]

The EXACT score for a candidate is tuple similar to Majority Judgment's "majority grade":

EXACT score for candidate X = (R, S, T)

where R is the rating at which X's votes at or above R are greater than the  highest approved candidate on ballots excluding X at R or above.;

If the number of ballots with X at rating R+1 and above is greater than those of the highest approved candidate on ballots excluding X at ratings R and above, then S = R+1, and T = votes for X at R+1 and above.

Otherwise, S = R and T = votes for X at R and above.

By sorting these tuples in descending order, one gets, as with Majority Judgment, an EXACT ranking for the candidates.

EXACT satisfies all the same properties as Majority Judgment, and in addition, is irrelevant-ballot-immune (IBI).  That is, a ballot containing approval only for non-contending candidates won't affect the results.

EXACT does require several N^2 arrays for summable storage, but note that no sorting of the ballots is required as with pairwise methods.

[William Waugh]

Is it Frohnmayer-balanced? That means, does every possible vote have an antivote? That means, a vote such that if submitted along with the original vote would cancel its effect on the outcome. That means, for all bags of votes B, the same candidates win, lose, and tie if B constitute all the votes in the election vs. with all the votes comprising B + {v} + {-v}, where v is any possible vote and -v is another vote, the antivote of v, and where + is bag union, and where {_} is bag singleton.

On Monday, December 25, 2017 at 6:43:36 PM UTC-5, Dodecatheon Meadia wrote https://groups.google.com/forum/?fromgroups#!topic/electionscience/VlSK2rgOblQ

[Ted Stern]

Hi William,

I believe that EXACT does indeed have an antivote for every vote.  It might depend on how approval is laid out in the score range.  It should be very similar to Majority Judgment, I would think.

It seems fairly straightforward that for this method, an approved score for candidate X is negated by a ballot disapproving X and approving candidates who were disapproved before. 

Therefore my intuition is that the antivote would be the complementary ballot.

If using a 4 slot ballot (scores 0 to 3), the complement of a ballot approving Alice at 3 and nobody else would be to approve everybody else at 3 and Alice at 0.

On a 4 slot ballot, however, there might be some issues with lower level approval complements.  I will consider that tonight.

Ted

On Tue, Dec 26, 2017 at 3:29 PM, William Waugh <2knuw...@snkmail.com> wrote:

Is it Frohnmayer-balanced? That means, does every possible vote have an antivote? That means, a vote such that if submitted along with the original vote would cancel its effect on the outcome. That means, for all bags of votes B, the same candidates win, lose, and tie if B constitute all the votes in the election vs. with all the votes comprising B + {v} + {-v}, where v is any possible vote and -v is another vote, the antivote of v, and where + is bag union, and where {_} is bag singleton.

On Monday, December 25, 2017 at 6:43:36 PM UTC-5, Dodecatheon Meadia wrote https://groups.google.com/forum/?fromgroups#!topic/electionscience/VlSK2rgOblQ

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[Ted Stern]

Hi Chris,

Your insertion is absolutely correct and does not change the method, but I think it is implied automatically.

If a candidate Y is rated above R on more ballots than the highest approved candidate on ballots that rate Y below R, then Y's above-R number of ballots is a lower bound for the number of ballots that rate Y at-or-above R.  So Y would always be "among those candidates".

Ted

On Wed, Feb 14, 2018 at 3:59 AM, Chris Benham <cben...@yahoo.com.au> wrote:

My modification is inserted with emphasis added.

• Find the highest rating R, for which there is at least one candidate X who is rated at or above level R on more ballots than any candidate is approved on ballots which rate X below R.
• If there is more than one such candidate X, then if there is at least one candidate Y who is rated above R on more ballots than the highest approved candidate on ballots that rate Y below R, elect the candidate Y with the most ballots rating Y above R.
• Otherwise, elect the candidate X with the most ballots rating X at R or above.
• If no candidates satisfy the first criterion, for any approved rating R, elect the candidate with the highest approval over all ballots.

To be more clear, shouldn't the second line read something like:

If there is more than one such candidate X, then among those candidates if there is at least one candidate Y who is rated above R on more ballots than the highest approved candidate on ballots that rate Y below R, elect the candidate Y with the most ballots rating Y above R.

?
 The method looks good, AFICT.

Chris Benham

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