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.


  1. 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 304-346.

  2. 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 92-96.

  3. Values of Large Games VI: Evaluating the Electoral College Exactly, Irwin Mann and Lloyd S. Shapley. RAND Corporation Memo RM-3158, 1962.

  4. Multi-Member Electoral Districts---Do They Violate the "One Man, One Vote" Principle?, John F. Banzhaf III. Yale Law Journal, Vol. 75, No. 8, July 1966, pgs 1309-1338.

  5. Weighted Voting Doesn't Work: A Mathematical Analysis, John F. Banzhaf III. Rutgers Law Review, Vol. 19, No. 2, Winter 1965, pgs 317-343.

  6. Mathematical Recreations: Election Fever in Blockvotia. Ian Stewart, Scientific American, July 1995, pgs 88-89.

  7. 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. 787--792.

  8. Weighted Voting: A Mathematical Analysis for Instrumental Judgments, W. Riker and L. Shapley. RAND Corporation Memo P-3318, 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.

  1. 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 1-9.

  2. 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 290-307.

  3. Mathematical Properties of the Banzhaf Power Index, Pradeep Dubey and Lloyd S. Shapley. Mathematics of Operations Research, Vol. 4, No. 2, May 1979, pgs 99-131.

Here are three more recent works. I haven't read them (yet). Note that two of them are available on the web.

  1. 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.

  2. 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.

  3. Weighted Banzhaf Values, Tadeusz Radzik, Andrzei S. Nowak, and Theo S.H. Driessen. Mathematical Methods of Operations Research, Vol. 45, No. 1, 1997, pgs 109-118.

Here are two mathmetics texts that discuss the Banzhaf Power Index.

  1. 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 college-level 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 Shapley-Shubik 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.

  2. 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.

  3. 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
Last update: 21 Feb 2000