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

Ted Stern dodecatheon at gmail.com
Wed Feb 14 11:46:42 PST 2018

```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 <cbenhamau at 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
>
>
> On 26/12/2017 10:13 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://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
>
> 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.
>
>
>
>