[EM] Instant Approval Runoff or Instant Freedom Voting

[Forest Simmons]

Tom Ruen has recently reminded us of the possibilities of Approval Runoffs
or "Freedom Voting" in committee meetings and similar situations.

For the record I would like to explain how this type of runoff can be
simulated instantly from Dyadic Approval ballots as easily (and
accurately) as IRV simulates simple minded plurality runoff from
preference ballots. 

Recall that Dyadic Approval ballots express an hierarchy of approval in
one of two equivalent ways.

For example a voted ballot could look like this:

A > B >> C,D > E,F,G >>>  H > I,J >> K > L,M,N

Or the same preferences could be expressed like this:

111 A
110 B
101 C,D
100 E,F,G
011 H
010 I,J
001 K
000 L,M,N

We interpret either ballot as follows: A through G are approved, while the
rest are not approved. And if all of the approved candidates dropped out,
H through J would be approved, and the rest not approved. If all of the
candidates except C through G dropped out or were eliminated, then C and D
would be approved, but not the others. Etc.

Our instant version of Freedom Voting will be a faithful model of the
method that it simulates to the extent that the following Weak
Independence of Irrelevant Alternatives (WIIA) condition holds true.

If a candidate is eliminated from an Approval ballot, that ballot is still
valid for the remaining candidates UNLESS the eliminated candidate was the
last one in her category (i.e. last approved or last rejected).

Here's how to do the runoff given the Dyadic Approval Ballots:

1. Convert all of the ballots to the second form, with all of the approval
code labels left justified. (Some ballots may use more levels of hierarchy
than others.)

2. Simulate an Approval election using only the leading (left most) bit of
each approval code label.

3. Eliminate the candidate with the lowest approval score.

4. Cross out that candidate from each ballot, and her code label if she
was the last one with that label.

5. If the change in a ballot results in all of the leading (most
significant) bits being the same, then delete all of the leading bits.
Repeat if necessary until at least two labels have different leading
bits.

6. Simulate another Approval election using only the leading bits.

7. Remove the least approved candidate, and adjust the ballots as before.

8. When a ballot reaches the stage where all of the bits of all of the
labels are crossed out, that ballot may be archived.

9. Repeat until one candidate is left.

In Tom Ruen's version of the Approval Runoff half of the candidates are
eliminated each time, instead of just one.  If the Instant Approval Runoff
ballots are being hand processed, that option would save some time,
otherwise there is no need for it.

I believe that this method would pick the same winner as Dyadic Approval
most of the time.

The main advantages of Dyadic Approval over Instant Freedom Voting are ...

(1) Dyadic Approval is a matrix method, so the ballots can be scored once
and for all in matrix form, and the matrices added to get one matrix
containing all of the needed information.

(2) Dyadic Approval results in an ordered list of the candidates like the
one Hugo Harth was requesting.

(3) Dyadic Approval doesn't require the WIIA (Weak Independence of
Irrelevant Alternatives) assumption that Instant Freedom Voting
requires.

On the other hand, Instant Freedom Voting may be easier to understand for
the average voter.

By the way, I believe that WIIA models reality faithfully to the extent
that the voters are rational, consistent, and careful in their choices.
Basically it says that we can infer from Dyadic Approval ballots what the
voters' approval choices would be in an approval runoff.

Methods that use preference ballots to infer voter choices in other kinds
of runoffs must either make similar assumptions or admit that their
simulation is only approximate. 

Whether or not an instant simulation faithfully models some repeated
voting episode method (dare I say runoff?) the resulting instant method
may have merit in its own right.

I hope you like this one.

Any questions?

Forest