From: egl1@cunixf.cc.columbia.edu (Elizabeth G. Levy)
Subject: Re: Voter Influence
Reply-To: egl1@cunixf.cc.columbia.edu (Elizabeth G. Levy)
Organization: Columbia University
Date: Mon, 27 Jul 1992 15:46:08 GMT

In article <oeQjU5K00h5yAjBtUm@andrew.cmu.edu> Conrad Poelman <cp2w+@andrew.cmu.edu> writes:
>Elizabeth gives a fairly clear explanation of her voter influence
>measurement technique, in which a town's/state's influence is given by
>the number (should it be fraction?) of coalitions in which that group is
>"pivotal", i.e. essentially coalitions in which that group's vote can
>make the result go either way.
>
>But my original question was really only with her last step:
>Excerpts from netnews.alt.politics.elections: 26-Jul-92 Voter Influence
>Elizabeth G. Levy@cunixf (5444)
>
>> 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).
>How do you do this "splitting" among 31 million voters?  In your first
>post, you implied this was to be done simply by dividing the influence
>of the state by the number of voters.  This seems inconsistent with your
>previous method - why not do the "coalitions" thing again within the
>state, to determine when you as an individual voter are a member of a
>pivotal winning coalition?  A voter is a member of a "pivotal winning
>coalition" within his/her state only if the election is determined by a
>single vote.

Actually, I am implicitly considering pivotal coalitions of voters.
Voters, by the rules of the game, are symmetric, and have equal
influence.  I can then divide the state's influence by the number of
voters.  If you want to look at the 2^31,000,000 coalitions, you can,
but you will get the same results.  More convolutedly :-), the
measurement of power I'm using is called the Banzhaf index, and, if
the players are symmetric in the sense that "changing the names of the
players" (a permutation) does not affect the game, the index will
yield an equal measurement of influence for all players.

What I'm considering is how the "rules of the game" determine the
influence of an individual player.  I am NOT considering concrete
situations.

>
>So if you compute voter influence within states this way, you are
>essentially doing exactly what I proposed - a voter's "influence" is the
>probability that his/her single vote will determine the outcome of the
>election.  Then you can choose which state to vote in so as to maximize
>this probability.

Actually, come to think of it, you're right.  The Banzhaf index
_is_ a measurement of the probability of being pivotal.  The index
is usually presented in a normalized form, where player power is
considered relative to other players, not as raw probabilities.

>However, given that that probability is virtually
>zero no matter what you do, I don't see why this strategy makes much
>sense.  In fact, I suspect that there never has been and never will be
>an election swayed by a single vote, so why compute a voter's influence
>based on the probability of this happenning?

As said, I'm examining an abstract measurement of power, as determined
by the rules of the game.  I agree that there will never be a national
election determined by a single vote (though, I believe, the 1960
election may have turned on a few thousand votes in Chicago.  Billy
Joel could have sung "RMN, blown away, what else do I have to say?"
though the euphony would have been somewhat lessened), but this says
nothing on how the rules affect voter power.

And, yes, the actual numbers concerning the power of individual
voters is vanishingly small, at best.  A state's voters may
weild twice the influence of another state's, but having 2/20,000,000
instead of 1/20,000,000 is no great consolation.  Better to play
the lottery, I suppose.  As said, these uses of cooperative game
theory are restricted to the thesis of a bored grad student.

Actually, the Banzhaf index did play a part when Canada considered
a new constitution.  The players, in this case, were Canadian provinces,
and the measurement had something to do with how much influence each
would have if they adopted a new system of representation.  I don't
remember the details, but that was a moment of fame for an obscure
branch of game theory.


>
>Another problem with any of these measures is that they don't take any
>real-world knowledge into accout.  Although they don't explicitly state
>it, if you try to interpret the results in terms of "probability" you
>are really assuming that each person's vote is essentially random. 
>Also, this ignores the fact that 30-50% of the voters in any state vote
>along straight party lines, so their votes really aren't in question,
>and Washington D.C. and Arkansas will vote for Clinton no matter what.

The take into account the political parties, one may come up
with a more convoluted coalition formation.  One may take loyal party
members as one player wielding X votes in the game.  As said, what
I've presented have more to do with electoral college rules than
reality.  I was a pure math person in a past life, after all.

>
>So any system which only looks at the behavior of a single voter is
>meaningless, because a single voter will never effect the outcome of the
>election.  Better voting influence measures would look at whole groups
>of voters and determine how their votes will influence the outcome of
>the election.  Perhaps a better rule of voting would be along rule
>utilitarian lines - "Act in the manner which, if followed by everyone,
>would result in the best outcome".  I think this translates fairly
>directly into "vote for the person you think is best, regardless of what
>you think their chances of winning are."

Actually, this is closer to what I'm going to do than any game theoretic
considerations.


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