# [EM] Two means of proportionality

Kristofer Munsterhjelm km_elmet at t-online.de
Sun Feb 2 03:23:28 PST 2014

```I think I see the difference between the proportionality divisor methods
try to achieve and the proportionality that largest remainder methods
try to achieve.

The largest remainder methods define their proportionality as quota-wise
proportionality: a given group should at least get its quota's worth of
seats, no matter what.

On the other hand, the divisor methods (at least Webster/Sainte-Laguë)
try to attain an ideal reflection of a certain probability distribution
whose parameters are given by the votes. For instance, in Sainte-Laguë,
this distribution is approximately a multinomial distribution, where the
probability for a party being drawn is equal to its proportion of first
place votes. In a sense, these produce the "most fair random result",
like what you would get if you drew at random, but if you could exclude
those that happened to be too extreme due to bad luck.

I say "approximately" becauses, the outcome produced by Webster
optimizes the chi-squared metric (Sainte-Laguë index), which AFAIK only
asymptotically gives the most likely rendering of the multinomial
distribution.

(This also points to a relation between lotteries and highest-average
methods.)

-

So if we were to make a ranked ballot version of Sainte-Lague in a less
hackish way than my CPO-SL v2, it should retain that logic of trying to
ideally represent some kind of distribution.

For Condorcet, that gives two possible strategies:
a. Find some kind of distribution/model that can solve the sliver
problem, then use this as a base method for a CPO-style "exclude all not
in either matchup" method to fix the problem of later votes being
concealed by earlier ones.
b. Find a distribution that, when we ideally represent it, gives a
Condorcet outcome in the single-winner case and a multinomial
distribution when everybody plumps for a single party.

The first may be easier, while the second would be more axiomatic (i.e.
more "elegant" in being from first principles). I'm not a good enough
statistician to really do either.

A related way of looking at it might be to consider whether the problems
that we get when applying Sainte-Lague to small seat sizes can be fixed
by trying to accurately represent something else than fractions of first
preferences, or whether no matter what strategy we pick, we'll have to
use another method to deal with the quantization to a small number of seats.

If it can be fixed by accurately representing something else, alone,
then that suggests approach b. If it can't, then that suggests approach a.

```

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