Banzhaf power index: relevant references
This is a list of relevant references. I have read the first two, glanced at
the fourth and fifth, and read the sixth and the two chapters cited in the
seventh. I have not seen the others, but I found the references in my
recursive search, so I have repeated them. If you are interested, scan
Banzhaf [1], the Mann and Shapley article [3] and in [7], and the Shapley and Shubik
article in [7] for the relevant mathematics and the prose
definitions of power. You might also skim through Sickels [2] for his objections. Those four references are the ones I
found most interesting and useful. I do, however, summarize them in my
discussion on the Banzhaf
power index web page.
I do not recommend reading Stewart's article
since he gets it quite wrong; I was very confused
by what he did, to the point of creating an incorrect implementation that was
available on these pages for a very long time. You can read it for general
ideas about half of the method (though Stewart presents his summary as being
of the whole method), but honestly, I think it will be more confusing than
elucidating.
References
 One Man, 3,312 Votes: A Mathematical Analysis of the
Electoral College, John F. Banzhaf III. Villanova Law Review,
Vol. 13, No. 2, Winter 1968, pgs 304346.
 The Power Index and the Electoral College: A
Challenge to Banzhaf's Analysis, Robert J. Sickels. Villanova Law
Review, Vol. 14, No. 1, Fall 1968, pgs 9296.
 Values of Large Games VI: Evaluating the Electoral
College Exactly, Irwin Mann and Lloyd S. Shapley. RAND
Corporation Memo RM3158, 1962.
 MultiMember Electoral DistrictsDo They Violate
the "One Man, One Vote" Principle?, John F. Banzhaf III. Yale
Law Journal, Vol. 75, No. 8, July 1966, pgs 13091338.
 Weighted Voting Doesn't Work: A Mathematical
Analysis, John F. Banzhaf III. Rutgers Law Review,
Vol. 19, No. 2, Winter 1965, pgs 317343.
 Mathematical Recreations: Election Fever in
Blockvotia. Ian Stewart, Scientific American, July 1995,
pgs 8889.
 Game Theory and Related Approaches to Social Behaviours:
Selections, edited by Martin Shubik. R. E. Krieger, 1975.
In particular, see Chapter 10, The a priori voting strength of the
electoral college, by Irwin Mann and Lloyd S. Shapley, which (it says)
largely repeats [3] and another RAND Corp. memo.
Also see Chapter 9, A method for evaluating the distribution of power in a
committee system, by Lloyd S. Shapley and Martin Shubik. That article is
reprinted from American Political Science Review, Vol. 48, 1954,
pgs. 787792.
 Weighted Voting: A Mathematical Analysis for Instrumental
Judgments, W. Riker and L. Shapley. RAND Corporation Memo P3318, 1966.
Here are some more references. The first two of these introduce themselves
as being basically reproductions of the works at the RAND Corporation (such
as reference [3] above, but that's not what I found
reprinted  yet, anyway), so they probably should be considered as having
dates in the early 1960s.
 Values of Large Games, I: A Limit Theorem, N. Z. Shapiro and
L.S. Shapley. Mathematics of Operations Research, Vol. 3, No. 1,
February 1978, pgs 19.
 Values of Large Games, II: Oceanic Games, J.W. Milnor and
L.S. Shapley. Mathematics of Operations Research, Vol. 3, No. 4,
November 1978, pgs 290307.
 Mathematical Properties of the Banzhaf Power Index, Pradeep Dubey
and Lloyd S. Shapley. Mathematics of Operations Research, Vol. 4,
No. 2, May 1979, pgs 99131.
Here are three more recent works. I haven't read them (yet). Note that two
of them are available on the web.
 A method for
evaluating the distribution of power in policy games: strategic power in the
European Union, Bernard Steunenberg, Dieter Schmidtchen, and Christian
Koboldt. 1997 Annual Meeting of the American Political Science
Association, August 1997.
 Values
for Multialternative Games and Multilinear Extensions, Rie Ono. Working
Paper No.166. Toyama University, October 1996. Prof. Ono has a lot of good
links on her research page on Politics and
Elections.
 Weighted Banzhaf Values, Tadeusz Radzik, Andrzei S. Nowak, and
Theo S.H. Driessen. Mathematical Methods of Operations Research,
Vol. 45, No. 1, 1997, pgs 109118.
Here are two mathmetics texts that discuss the Banzhaf Power Index.
 For All Practical Purposes: Introduction to Contemporary
Mathematics, (fourth edition), by Joseph Malkevitch et al., published by
W. H. Freeman and Company, New York, 1997.
I read this and it looked like a good introduction at a basic level for anyone
with collegelevel math experience and good high school students. It doesn't
go into how you could compute it for a large system, however. The whole text
was apparently used for a course in the University of Georgia Math Department
called Mathematics
of Decision Making. One interesting note in this text is that it says
that the Banzhaf Power Index was independently developed by James Coleman at
RAND Corporation. It goes on to say that the Banzhaf index was used
more in court cases than the ShapleyShubik index, and theorizes that this is
because John Banzhaf is a lawyer. I would guess that this probably also
explains why his name is on it and why it gets a little more press.
 Excursions in Modern Mathematics (second edition), by Tannebaum
and Arnold is used as a text for several courses. I haven't been able to come
up with a copy of this text, however.
 Finally, Temple University math
professor John Allen Paulos
(author of Innumeracy: Mathematical Illiteracy and Its Consequences
among others) discussed the Banzhaf Power Index in the first
essay included in his book A Mathematician Reads the Newspaper, which
prompted Richard Bernstein of the New York Times to write the following.

In his new book, the mathematician John Allen Paulos continues his witty
crusade against mathematical illiteracy ...... Mr. Paulos's little essay
explaining the Banzhaf power index and how it relates to Lani Guinier's ideas
about empowering minorities is itself worth the price of the book.


High praise, indeed. I read the essay, and it was entertaining and made a
good point. But read it for yourself.
Go back to Banzhaf power
index main page.
Mark Livingston
Dept. of Computer Science
Univ. of North Carolina
livingst@cs.unc.edu
Last update: 21 Feb 2000