# [EM] Binary Clone Tree method (was reverse Llull)

Forest Simmons fsimmons at pcc.edu
Thu Mar 26 17:12:37 PDT 2020

```We need a better name: "Natural Coalitions?"

Here is a Proportional Representation multiwinner method based on the
binary tree structure of candidate cluster/coalition/clone families,
whatever we want to call it:

1. Construct a natural metric on candidate/issue space. We can devote an
entire thread to the interesting possibilities. The field is wide open.

2. Use the metric to do a cluster anslysis on the candidates with output in
the form of a binary tree.

3. The tree diagram is published.

4. Voters submit ballots that indicate only their favorite or tied for
favorite candidates.

5. The tree diagram is annotated with the favorite support numbers, with
"tied for favorite" support counted fractionally.

6. Assuming K winners, multiply each support number by K over the total
number of voters. That way the total support will be K, so that after  vote
transfers are complete each candidate will have one whole unit of support.

7. Start at the root node of the tree, and call the branch with the
greatest total support B1, and the other branch B2.

8. If the support S1 of B1 is greater than its number of leaves L1, assign
each of its leaves full unit support and transfer the remainder (S1 - L1)
to the other branch B2 by multiplying each of the B2 support numbers by a
common factor alpha that will ensure that total combined support remains
constant at the value K, the number of candidates.

9. If S1 is not greater than L1, then round S1 to the nearest whole number
S1' and multiply each numerical support in B1 by the factor S1'/S1. Then
multiply each support number in B2 by a factor that makes the total support
for B2 candidates add up to (K - S1').

10. Recursively apply this procedure to the sub trees B1 and B2 so that in
the end exactly K leaves are left with positive support, namely one unit
each.

That' it.

It is useful in view of the fractional support possibilities inherent in
this procedure, to think of total support being represented by K
milliliters or cubic centimeters of some incompressible fluid like liquid
water.

Since the fluid is incompressible, multiplication by alpha (for example) is
merely an intermediate purely mathematical book keeping step that implies
the necessary fluid transfers to the various leaves of the tree.

In the case of K = 1 the method reduces to the zero info single winner
strategy mentioned in my two previous messages.

Who has a good idea for a name for this method?

On Tuesday, March 24, 2020, Forest Simmons <fsimmons at pcc.edu> wrote:

> So the method is well adapted to zero info and perfect info voting and
> everything in between.
>
> But how about the case of gross disinformation that plagues US politics as
> seen in the current Democratic primary?
>
> Disinformation can be thought of as negative information, closer to the
> zero end than the positive end of the above spectrum. So I recommend using
> the zero info strategy in that case.
>
>
>
> On Tuesday, March 24, 2020, Forest Simmons <fsimmons at pcc.edu> wrote:
>
>> This binary clone tree based method, like Reverse Llull, was designed
>> with perfect preference info in mind, but this method, unlike Reverse
>> Llull, is well adapted to zero info voting, as well.
>>
>> Here is a near optimal zero info strategy: always vote for the branch