[EM] Public Proposal Verbiage

[Forest Simmons]

Preface:

In a March 2004 Scientific American article  entitled, "The Fairest Vote of
All," Partha Dasgupta and Eric Maskin (now a Nobel Laureate) argued
persuasively for their conception of a "True Majority Winner" of a single
winner election based on ranked choice ballots.

Taking for granted the Majority Criterion that mandates electing the
candidate that outranks all of the other candidates on more than half of
the ballots (when there is such a candidate), they propose that when there
is no such candidate, when possible they at least keep this less demanding
but crucial property of a Majority top ranked candidate: such a candidate
outranks any competitor on more ballots than not.

Why not say, "on more than half of the ballots" instead of "more ballots
than not"?

Because voters are not required to rank all of the candidates. Indeed, some
voters may simply "bullet vote" for their favorite, while leaving the other
candidates unranked or "truncated."

The dictionary definition of "majority" is flexible enough to include this
usage of "more than not," so Dasgupta and Maskin's "True Majority Winner"
terminology is perfectly acceptable to Webster, Cambridge, OED, etc.

Their Scientific American article briefly alluded to the rare public
election possibility where a ballot set might yield neither a "more than
half" first place majority winner nor a (less demanding) True Majority
Winner.

It was not the purpose of their article to prescribe a course of action to
cover that rare case, since they were not making a proposal for a specific
election method to be adopted and written into law for some specific
democratic electorate.

Their purpose was to expound and publicize to the broader scientific
community and other interested citizens a principle that has been respected
among social choice thinkers at least since the time of Ramón Llull of
twelfth century Spain.

We now pick up where they left off with a proposal for how to decide the
winner in the case of no True Majority Winner (TMW).

For ease of reference we repeat (my wording of) the Dasgupta/Maskin
definition of True Majority Winner, namely a candidate that outranks every
competitor on more ballots than not.

Also, "bullet ballot" ... a ballot that truncates after its top choice.

We also need the concept of a "ballot superset:" In the current context it
is a ballot set augmented with a number of bullet ballots to gauge how far
away a ballot set is from having a True Majority Winner.

Our idea is to complete the quest for a True Majority Winner by augmenting
the given ballot set with the bare minimum of bullet ballots to ensure the
existence of a True Majority Winner for the augmented ballot set. In other
words, we elect the candidate closest to being a TMW when there is no TMW.

So here it is:

If the submitted set of marked ballots does not have a True Majority Winner
(i.e. a candidate that outranks each opponent on more ballots than not),
then elect the True Majority Winner of the smallest ballot superset that
does have a True Majority Winner.

The above description completely and decisively defines the winner without
recommending one procedure over another for tallying the submitted ballots.

There are many possible counting procedures, (some more efficient than
others) but any that require multiple passes through the ballot set (as do
elimination methods like Instant Runoff) are inefficient, hence to be
avoided.

One efficient procedure is to immediately (at the precinct level) summarize
each ballot in the form of a table with K rows and K columns, where K is
the number of candidates.  The i_th entry in the j_th row of the table is a
one or zero depending on whether or not candidate j outranks candidate i on
the ballot being tabulated.

Once a ballot is converted to this K by K tabular format it can be added in
to the precinct total. In turn the precinct totals are added together at
some central location to arrive at a grand total table T.

Apparently the i_th entry in the j_th row of table T is the number of
ballots on which candidate j outranks candidate i.

Similarly, the i_th entry in the j_th column of table T is the number of
ballots on which candidate i outranks candidate j.

Therefore, when we subtract the corresponding elements of the j_th column
from the j_th row we get a new table D in which the i_th entry of the j_th
row is the difference between the number of ballots on which j out ranks i
and the number of ballots on which i outranks j.

If this difference is positive, then candidate  j outranks candidate i  on
more ballots than not.

Therefore, if the j_th row of D has all positive differences, then
candidate j is the True Majority Candidate.

If there is no such row j with all positive entries, find the row j that
needs the least multiple of the "bullet row" added to it in order to wipe
out all of its (row j's) negative entries.

A bullet ballot row consists entirely of ones: [1, 1, ..., 1].

This row j identifies the True Majority Winner of the ballot set that has
been augmented with the minimum number of bullet ballots to achieve a TMW
... in ther words the candidate closest to being a TMW of the original
ballot set.

Don't worry about the details of this tally procedure  .. that's for
trained election officials to learn. But do take note that a methodical
method involving mostly copying and adding of table entries (derived from
ranked ballots) with a subtraction of table columns from corresponding
table rows is all there is before determining how many bullet ballot rows
are need to wipe out all of the negatives from one row ... just methodical
use of arithmetic ... for someone equipped with an adding machine to worry
about.

Questions?

Suggestions for improved exposition?

Gripes?

Thanks!

-Forest
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