[EM] Discounted Redundant Layers of Representation

[Forest Simmons]

More thoughts on PR through AV:

If you believe (like I do) that AV is superior to IRV, then it could be
plausible to you (as it is for me) that a PR system based on AV has a good
chance of being superior to one (like STV) that merely extends IRV.

As I mentioned in an earlier posting I'm really proposing a whole class of
multiwinner voting methods, which we could call DRLR for Discounted
Redundant Layers of Representation.  Every sequence of decreasing positive
weights for "redundant layers" defines a member of this class.

Besides the harmonic sequence 1, 1/2, ... 1/k, ... which (up to a constant
multiple) is the unique member of DRLR that yields PR, I proposed (the
more radical) square of the harmonic sequence which would tend to give
like minded segments of the electorate voting power (as opposed to mere
representation)in proportion to their size. 

A noteworthy sub family of DRLR consists of methods given by geometric
sequences such as 1, q, ... q^k, ... .  If  q is smaller than the
reciprocal of the number of voters, this method awards the election to
that coalition that represents the greatest number of voters with at least
one layer of representation (as indicated by their approval), and breaks
ties by giving it to the coalition that yields the greatest number of
ballots with two layers of representation, etc.

With this extreme method you would probably want to have a side condition
that eliminated candidates with less than the fraction 1/(n+1) of the vote
(where n is the number of positions to be filled by the election) or else
restrict the voting power of winners with few votes, etc.

Now back to general q in this subclass of DRLR given by geometric
sequences:  If B is an approval ballot in a multiwinner election with n
positions to fill, and k=k(B,C) is the number of candidates approved by B
among those in coalition C, then this ballot contributes 1 + q + ... +
1/q^(k-1), which can also be written as (1-q^k)/(1-q), to the coalition C. 

Multiplying all such expressions by the positive constant (1-q) preserves
the order, so that coalition wins which maximizes the sum over k of 1-q^k,
or equivalently, the one that minimizes the sum of all terms of the form
q^k.  In other words, to find a winning coalition minimize over C the sum
over B of the expression q^k(B,C) . 

In the context of protective clothing we can give a probability
interpretation to this result: if q is the probability that "something
bad" can get through one of the layers of protection, then (assuming
independence) q^k is the probability that it will get all of the way
through and damage part B of the body when C is the chosen ensemble of
protective gear.

So we are minimizing the expected number of body parts that will be
allowed to suffer.

In the team representing a school in the math olympics, we are
minimizing the expected number of problems that will baffle all members of
the team.

In the political context, q might be the probability that a representative
will neglect your interests given that you approved him.  Then we are
minimizing the expected number of voters that are left without anyone
defending their interests.

Of course it is very hard to estimate what q should be in any particular
case. Nevertheless, this analysis does yield some insight. 


Now back to the PR case:  The sum 1 + 1/2 + ... + 1/k can be represented
as the integral with respect to q from zero to one of the expression 1 + q
+ ... + q^(k-1).  As we have seen the integrand can be simplified to
(1-q^k)/(1-q) .  This time we can not get rid of the divisor (1-q) because
q is a variable under the integral sign. 

There are two advantages of using this compact form.

First, we can still give a probability interpretation to the result.  This
time we don't have to wonder which q to use, since all q's between zero
and one are blended together with emphasis on the q's close to one. (This
is the effect of the divisor 1-q . ) Roughly interpreted, this method is
based on a skeptical estimate of how much good you can expect out of the
representatives you vote for. 

Second,the expression (1-q^k)/(1-q) still has a mathematical value even
when k is not a whole number.  So we can easily extend our DRLR methods to
Cardinal Rating ballots.  In that case just define k(B,C) to be the sum of
B's levels of approval for members of C. 

Furthermore, every Ordinal Ranking can be converted into an appropriate CR
by even spacing, i.e. by normalizing the Borda Count.  Therefore, all of
the methods of DRLR (including the essentially unique PR member of the
family) can be applied to the Borda Count, as well.

Next time, if I don't get side tracked again, I'll show how we are led
inexorably to DRLR methods in general and the harmonic sequence in
particular if we want to use Approval Voted Ballots as our basis for
Proportional Representation.

In summary, we have reason to hope that PR through Approval Voting will
yield a practical, interesting, and valuable addition to the known
methods.  Of course, we should be cautious until extensive simulation and
test cases have been examined. The little I have had time to do has not
been disappointing.

Forest