From: egl1@cunixf.cc.columbia.edu (Elizabeth G. Levy)
Subject: Voter Influence
Reply-To: egl1@cunixf.cc.columbia.edu (Elizabeth G. Levy)
Organization: Columbia University
Date: Sun, 26 Jul 1992 16:33:05 GMT

The following is a reply to the question of whether it would be better to
vote in California or Hawaii.  Someone contended that one's vote doesn't
count unless one's state is pivotal to winning the electoral college, and
one's vote is pivotal to winning that state.  By this analysis, one's vote
is worthless.

I argue that the vote is worth something, though not a whole lot.  We seem
to have a difference in our definition on the influence of a voter.  The
forementioned poster essentially argues that a voter has influence if and
only if that voter decides the outcome of the election.  This is true in a
very concrete sense.  I argue that each voter has a certain amount of
influence because of the "rules of the game", by the way the electoral
college is set up.  This is an abstract approach, and is based on
cooperative game theory.

Let's consider the following situation.  Three towns A, B and C have
populations of 5000, 4000 and 3000, respectively.  Suppose they each send
a representative to a county council, on which each have one vote a piece.
By these rules, each town has an equal amount of influence.  This is
obvious, but I want to formulate this fact in a convoluted way (ah, the
joys of mathematics!): except for an unanimous vote, each "winning
coalition" of towns (a winning coalition is one in which that coalition is
able to pass a piece of legislation. In this case, any majority coalition)
becomes a losing coalition when one of the towns changes its vote.  For
instance, towns A and B vote yes to a piece of legislation. Since this is
a two town majority, it is a winning coalition.  If, however, town A votes
no, the coalition of B and only B voting yes is a losing coalition.  We
define the power of a town (the influence) by the number of coalitions for
which that town is pivotal (the number of coalitions that are winning, but
would lose if the town changes its vote).  Each town then has the same
amount of power by this definition, since each town is pivotal to two
coalitions.  The winning coalitions are:

	ABC	no town is pivotal
	AB	A is pivotal, and B is pivotal
	A C	A is pivotal, and C is pivotal
	 BC	B is pivotal, and C is pivotal

Suppose that town A argues that it should have more influence, because it
has a greater population than either town.  The state creates a
"proportional vote" system, where town A has 5 votes, B has 4 votes, and C
has 3 votes.  Does this change the situation?  The answer is no.  A
majority of votes is required to pass a piece of legislation, so town A
still needs one of the others to be on a winning coaltion, etc.  By
looking at which coalitions are winning, and which towns are pivotal to
these coalitions, we can see that the "rules of the game" have not
changed.  Again, this is an abstract way of examining the power of a
player in a given game.  This method does not consider what happens when a
certain vote is taken, but how much power is confered by the structure of
the game.  The original poster looked at a concrete election, which would
be a single play of the game.  I was attempting to consider the structure
of the electoral college, and how much influence a voter would have
through that.  We do not know how a vote will fall, so we must take this
abstract approach.

So, the way to examine the influence of state in the electoral college is
to look at the number of coalitions to which that state is pivotal. The
number of electoral votes just kinda obfuscates the matter.

We originally considered California and Hawaii.  California, having more
electoral votes, will most likely be pivotal in more coalitions.  I don't
really want to figure out how many: "the electoral collge game" would have
2^50 total coalitions. :-)

OK.  For each given state, we must consider the subgame of voters.  We
must consider which coalitions of voters will win the state, and if a
voter is pivotal to such coalitions.  This is actually easier, since we
are looking at simple majorities.  Each voter is symmetric in the sense
that the votes have equal weight.  Therefore each voter has equal power. I
think we can then divide the power of a state obtained by playing "the
electoral college game" by the number of registered voters in that state.

Residents of Hawaii, which has less power than California because it is
less pivotal, may actually have greater individual influence than a CA
voter.  Their power, though less, isn't split among 31 million residents
(I don't know how many are registered).  Remember that the number of
electoral votes includes senators, so is skewed slightly towards smaller
states.

In any case, I'm not sure if Hawaii voters is more influential than
California voters.  I would have to write up the game, etc.  I do,
however, recall seeing this analysis.  Someone's master's thesis, or
somesuch.  I think books on cooperative game theory might contain snippets
of this. (one can do the same sort of thing with, say, the Security
Council of the UN.  I think the 5 permanent members hold collectively 93%
of the power through the voting rules alone.  I've seen an analysis of the
American legislative system, though I don't remember the details.  In any
case, winning coalitions would have to have 51% of the House, and 51% of
the Senate, and the President, or else have 67% of the House, and 67% of
the Senate, etc.  The President does have a considerable amount of power
purely by the rules, though I don't recall the exact numbers).