[EM] Definitions for Comparing Voting Power

[Alex Small]

I haven't really had the chance to do anything with these, but these
definitions seem reasonable for analyzing any electoral college where some
members have more voting power than others:


Definition:  An electoral college is a set EC of electors e_i with votes
v_i.

Note:  All sets are implicitly subsets of EC unless otherwise stated.

Definition:  A set M is a majority if the number of votes held by electors
in M is a majority of the number of votes held by all electors in EC.

Definition:  A set A has no voting power if there exists no set S such that
S is not a majority but S U A is a majority.

Definition:  Two sets A and B have equal voting power if

1)	for any set S such that if S is not a majority but S U A is a
majority then S U B is also a majority
and
2)	for any set T such that T is not a majority but S U B is a majority
then S U A is also a majority.

Definition:  A set A has more power than B if there exists a set S such
that S and S U B are not majorities but S U A is a majority.

Definition:  The elements of a set A have n/m times the power of the
individual elements of a set B if

1)  each of the n elements of B have equal power
2)  each of the m elements of A have equal power
3)  A and B have equal power.

Questions that might be interesting:

1)  Can one identify necessary or sufficient conditions so that relative
voting power is just proportional to the number of votes?  (i.e. are
conditions such that the state of CA has 55/3 times the power of RI?)
2)  What is the effect of Senatorial votes?  If conditions are not such
that CA has 55/3 times the power of RI, what would happen if CA had 53
votes and RI had 1 vote?

Reasons for asking these questions:
1)  My understanding is that the power of a state in the Banzhaf index is
calculated by Monte-Carlo simulations, determining the probability that a
given state will tip the outcome.  I wonder how those numbers would compare
to an axiomatic approach.
2)  Even if the axiomatic approach is not tractable for something as large
as the US EC, it might be useful for analyzing smaller bodies.  The
original 13 states might be a more tractable problem.  One could then ask
whether the states got what they bargained for at the Constitutional
Convention.


--
"Frodo gave his finger for you."