__Fractal Voting__

Copyright September 18, 2007

In this voting method, each voter
rates each candidate in the following manner. The most favored and least
favored candidates are identified. The most favored is given a rating of 1 and
the least favored 0. Now the voter proceeds to derive a rating for each of the
other candidates relative to most favored and least favored. Each of these will
receive a fractional rating between 0 and 1 that fits into one computer word.
For the sake of discussion we can assume a 32 bit computer word. The first bit
would determine whether the candidate was most favored (1) or least favored
(0). There could be more than one most favored or least favored candidate. The
second bit (2^{-1}) would determine if the candidate was less than or
greater then ˝ (0 or 1, respectively). Subsequent bits would determine further
subdivisions. The voting software would give the voter a choice of how many
subdivisions or slots to start with. For instance, if the voter chose to start
with 4 slots, he or she would be presented with a screen with a line subdivided
into 4 segments and a list of candidates. The voter could then drag each
candidate from the list onto one of the 4 slots or subdividions. Then each
candidate would be represented by 3 bits. Likewise, to separate the candidates
into 8 groups would take 4 bits etc. The voter could potentially separate the
candidates into 2^{30} groups which would be far more than even the
most astute voter would probably need to distinguish among the candidates. The
voter need not be aware of the numbers that are attached to each slot. The
candidates in any particular slot would be preferred to those in the slot to
the left and indifferent among the candidates in the same slot. Following is an
example in which a voter has specified 4 slots. Candidate J is most preferred
and candidate K is least preferred. H is preferred to I which is preferred to
A, B, C, D, E and F. B, C and E are indifferent and are preferred to A, D and F
which are indiffferent to each other.

The voter only need distinguish among the candidates to his own individual sensitivity level. For instance, if he or she prefers only to distinguish between good and bad candidates, this would only take 2 slots or 2 bits (1 bit is needed just to inducate “most preferred” or “least prefered.”) Now say the voter has distinguished among the candidates so that all slots except one contain 1 candidate. The remaining slot contains 3 candidates and the voter wishes to distinguish among these. He or she need only right click on that slot and a menu comes up giving him or her the choice to subdivide that slot. He or she then chooses how many slots to subdivide that slot into. For instance, the voter may think that he or she needs 8 slots to distinguish among the 3 candidates properly. After choosing 8 from the submenu, the screen refreshes to show a line divided into 8 segments with the 3 candidates listed on the side. Again the voter drags these candidates onto the proper slot. Now he can choose to view the entire line-up of candidates. Note that he does not need to subdivide each of the original slots. For instance, in the example above he could choose to subdivide the second slot and distinguish among E, C and B, but remain indifferent among A, F and D in the first slot. He or she need only go into as much detail as he or she prefers, and then only on as many line segments or slots as he or she prefers. Since expanding any line segment brings up similar choices as were made on the previous screen, we call this method fractal voting. From Wikipedia the definition of fractal is the following:

a **fractal** is "a rough or fragmented geometric shape that can be
subdivided in parts, each of which is (at least approximately) a reduced-size
copy of the whole".

The definition
corresponds exactly to this method of voting in that, when a voter wishes to
distinguish among candidates, he or she need only click on that line segment or
slot corresponding to those candidates to bring up a screen again with a line
subdivided into segments that represents a closer up examination of the
original slot.

This method of
voting has several advantages over Range Voting:

1)
Every
vote is a “full strength” vote. In Range Voting it’s possible for a voter to
submit a “partial strength” vote if he doesn’t pin his most favored candidate
to the maximum range vote and his least favorite candidate to the minimum range
vote.

2) The voter need not be concerned with
assigning numbers to candidates. The software does that for him. He or she only
need be concerned with ranking the candidates and indicating intensity of
ranking. The method is completely visual and easy to implement via touch screen
or mouse clicking.

3) The voter can choose his or her own
sensitivity level. For instance, if he or she wants to vote “approval style,”
he or she need only separate the candidates into 2 slots. So the arbitrariness
of choosing what range to use (whether to use 1-10 or 1-100, for instance) is
eliminated.

4) The voter can go into detail selectively
in those parts of the overall ranking that concern him or her while doing a
rough ranking in other parts of the overall scale. In range the voter needs to
consider each candidate over the whole scale.

5) The binary ranking submitted by each voter
is a logical consequence of the voting method, but the voter need not be
concerned with numbers at all. He or she need only be concerned with a visual
on-screen representation of the rankings.

Each voter will submit one binary computer word that contains a ranking for each candidate which in turn is a number between 0 and 1. A voter can abstain from voting on any candidate; the result is the same as giving that candidate a 0 or “least prefered” status. To choose the winner, one need only tote up the scores for each candidate over all voters. The figure of merit, which can be considered voter satisfaction or social utility, is the same as the winning score. This can then be divided by the voting population to get a normalized figure of merit or social utility between 0 and 1.

We can compare fractal voting with other methods because the
voter’s submitted ballots under fractal allows us to derive the votes that
would be submitted under any other voting method. For instance, the “most
favored” candidate under fractal would be the Plurality vote. If there are
multiple “most favored” candidates, a random selection among them would be
a suitable way of deciding the Plurality vote since Plurality allows only one
candidate to be voted for. With Approval the threshold would have to be
assumed. A threshold of 0.5 is reasonable since the voter has ranked everyone
above that threshold closer to “most favored” and vice versa. It should also be
easy to derive the Range votes, and they should correspond very closely to
Fractal. Fractal allows more individual
sensitivity than range because the Range limits set the sensitivity for each
voter whereas with Fractal the individual determines the selectivity. In each
case the votes would have to be normalized to the 0-1 range. They are already
normalized for Plurality and Approval and are easily normalized for range. Then
the voting methods can be compared by writing a computer program that first
chooses one of the possible profiles for each of the voters and conducts a
theoretical election. Over a large number of such elections, it would be
possible to statistically compare the performance of the various voting
methods.