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Forest Simmons fsimmons at
Fri Jun 15 07:50:57 PDT 2001

On Thu, 14 Jun 2001, Blake Cretney wrote:

> On Thu, 14 Jun 2001 14:47:10 -0700 (PDT)
> Forest Simmons <fsimmons at> wrote:
> > In this regard it should also be noted that someone's "voting power"
> > defined to be the probability that their vote will be pivotal,
> varies from
> > state to state, and that contrary to popular opinion, the Electoral
> > College system that converts states into voting blocs favors the
> voters in
> > large states more than in small states.
> > 
> > This is because even though the smaller states have super
> proportional
> > representation, in many cases it is not enough to compensate for the
> added
> > voting power that comes from being a member of a large bloc. 
> > 
> > A hypothetical example will make this abundantly clear.  Suppose
> that
> > Transfersylvania has three provinces P1, P2, and P3, with respective
> > populations of 1 million, 2 million, and 9 million voting citizens.
> > 
> > Suppose that in the Electoral College each province gets a member
> for each
> > million of its citizens plus two extra, like the system used in the
> U.S.A.
> > 
> > So the numbers of electoral votes for the respective provinces are
> 3, 4,
> > and 11. 
> Let's imagine a country with an electoral college, and three
> provinces.  In this country, the electoral college grants no extra
> electoral votes to the smaller provinces above proportionality.
> The province populations are 48%, 48% and 4%.  The give 48, 48, and 4
> votes respectively.  So, winning any two provinces wins the election. 
> A candidate is as concerned about winning the small province as either
> of the larger ones.  So, the small province is disproportionately
> influential.  I don't think that the advantage for large states in
> your example can be generalized.

That's why I said "in many cases" instead of saying "in all cases" :-)

However, in typical cases with many voting blocs of randomly distributed
sizes, the voting power is supposed to be approximately proportional to
the square of the size of the bloc. [Though I haven't verified this by
calculation or simulation.]

Whence the "square root rule of thumb" that you might have heard of
relative to representation in weighted voting systems.

In particular, in the U.S.A. the calculations have been done for the power
of each state in the electoral college.

A table on page 74 of the fourth edition of "Excursions in Modern
Mathematics" by Tannenbaum and Arnold shows that every state smaller than
Ohio has a smaller percent of power than its percent of electoral votes,
while each of the three largest states has a greater percent of power than
its percent of electoral votes. 

The other three states in between Ohio and Texas (Illinois, Pennsylvania,
and Florida) have power roughly in proportion to their number of electoral
votes, depending on whether you use the Banzhaf or the Shapely-Shubik
power calculation.

These two calculations differ in the assumed probability distribution.

Remember that voting power is defined in terms of the probability of
having a pivotal vote. That calculation depends on the underlying
probability distribution.

Anyway, my main point stands unscathed: variable voting calculations based
only on percentage of the population represented, and not taking actual
voting power into account ... such calculations are pretended fine tuning
when the coarse tuning hasn't been completed. It's like trying to decide
what the seventh decimal place should be when the second decimal place is
still in question.


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