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

[Chris Benham]

> 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-electorama.com/2010-May/091807.html 
> <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://election-methods.electorama.narkive.com/KdBxpweB/irrelevant-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.
>
> 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.




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