Ted Stern

2017-12-25 23:43:35 UTC

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.

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.