[EM] PAV versus List PR

[Forest Simmons]

Lets see how PAV stacks up against list PR.

Suppose there are K seats to be filled in a multiwinner election, and that
there are M single minded factions with members in proportion n1:n2: ...
:nM .  Suppose further that the sum n1 + n2 + ... + nM is exactly K, the
number of seats to be filled.  This is the ideal situation for list PR.
If every member of each faction votes for the faction's list (and none
other) then the votes for the respective factions will be in exact
proportion to the membership of the factions, so the respective factions
will have  n1, n2, ... nM representatives, in precise proportion to their
respective sizes.

Now how does PAV stack up here?

Suppose that on an Approval Ballot each faction member approves the party
line and none other.

Consider a combination C of candidates having exactly n1, n2, ... nM
members from the respective factions.  Let's see if this combination can
be improved upon from the point of view of PAV.

Remember PAV gives a score of (1 + ... + 1/n1)*n1 + (1 + ... + 1/n2)*n2 +
... + ...  + (1 + ... + 1/nM)*nM to this combination C .

Suppose, for example, that we remove the last layer of representation from
the third faction and award it to the seventh faction. Then the net change
in the PAV score would be  n7/(1+n7) - n3/n3, which reduces to  -1/(1+n7)
,  a negative change.

The more layers that are relocated, the more negative the effect on the
score.

So the List PR winners are precisely the PAV winners.

The advantage of PAV comes in when the factions are not so precisely
defined.  There is a serendipitous effect when preferences overlap, so
that super proportional representation can be achieved. 

Remember that Michael Welford pointed out that different apportionment
rules (when precise proportional representation would require a fraction
of a representative) correspond to slightly different DRLR (discounted
redundancy layers of representation) sequences for PR. The harmonic
sequence is the one that most favors the minorities. For r between zero
and one, any constant multiple of any sequence of the form 1/(1+r),
1/(2+r), ... 1/(k+r), ... would also yield proportional representation in
the precise case.  The closer r is to one, the more the big guys are
favored when fractional candidates are indicated by the arithmetic.

It turns out that the sum 1/2 + ... + 1/(k+1) , which corresponds to the
case when r=1, is an underestimate of the natural log of (k+1), and the
sum 1 + ... +1/k is an over estimate of the same quantity, so the log of
(k+1) yields a compromise between the conservative and liberal
apportioning of "fractional representatives." 

In Bart's example, under this compromise, the small faction with 34% or
less of the voting population would not be represented in a two seat
government, because the compromise PAV score 66%*ln(2+1) + 34%*ln(0+1) is
greater than 66%*ln(1+1) + 34%*ln(1+1) .  However, if the smaller faction
had 37% or more of the voting population, then they would be represented
with one of the two seats because 63%*ln(2+1) + 37%*ln(0+1) is smaller
than the sum 63%*ln(1+1) + 37%*ln(1+1) . 

Note that the sum of the logs is the log of the product, so maximizing
this compromise PAV score is equivalent to maximizing the product (1+k1) *
(1+k2) * (1+k3) ... (1+kN) where N is the number of voters, and kj is the
number of approvals by the j_th voter (in the combination being scored). 

To adjust this compromise PAV to higher resolution ballots like rankings
and ratings (which do have their legitimate uses in various contexts) just
let kj be the total number of those resolution units supported by the j_th
voter in the combination being scored.  So if the resolution unit is .05,
and the j_th voter supports candidates (within the combination being
scored)  in the amounts .30, .25, .40, and zeros for all the rest, then kj
would be 6+5+8=19 and the factor (1+kj) would be 20. 


I hope this helps elucidates PAV and its relationship to apportionment and
List PR. 

I won't have time to expound the axiomatic derivation of PAV in the near
future. I'm swamped at work, at home, at church, etc.

I will try to write a few more elucidating notes on the subject as time
will permit.

In particular, suppose that f and g are increasing continuous real valued
functions, and that f and g are composition inverses of each other, so
that f(g(x)) = x = g(f(x)) .  Suppose that we want to achieve
representation in proportion f(n1):f(n2): ... :f(nM), as opposed to simple
direct proportion.  Then DRLR based on the sequence 1/g(1), 1/g(2), ...
1/g(k), ...  will do the job. 

The demonstration mimics that for ordinary PAV, which is the case where f
and g are both the identity function.

In this case, the ideal M is f(n1) + ... + f(nM) and we consider the score
(1/g(1) + ... + 1/g(f(n1)))*n1 + ... + (1/g(1) + ... + 1/g(f(nM))*nM
, corresponding to the kind of proportion that we want.

Removing a layer from the first faction and awarding it to the last would
give a net change in the score of nM/g(1+f(nM)) - n1/g(f(n1)), which can
be written in the form g(f(nM))/g(1+f(nM)) - 1 , since g and f are
composition inverses. 

To see that this difference is negative, notice that the denominator
g(1+f(nM)) is greater than the numerator g(f(nM)), since g is an
increasing function and  1+f(nM) is greater than f(nM).

Similarly, any other alteration from the required proportion would
decrease the generalized PAV score.  QED.

Functions analogous to logarithms for these generalized proportions can be
obtained by integrating 1/g(x) with respect to x, and these methods can be
adapted to higher resolution settings just as well (with or without the
log analogs.)

Well, there's a lot of exploring to be done in these fields.  I hope
someone with more time on his/her hands than I have gets interested enough
to take the ball and run with it.

Forest