IDEAS home Printed from
   My bibliography  Save this article

Mathematical Properties of the Banzhaf Power Index


  • Pradeep Dubey

    (The Rand Corporation, Main Street, Santa Monica, California 90406)

  • Lloyd S. Shapley

    (The Rand Corporation, Main Street, Santa Monica, California 90406)


The Banzhaf index of power in a voting situation depends on the number of ways in which each voter can effect a “swing” in the outcome. It is comparable---but not actually equivalent---to the better-known Shapley-Shubik index, which depends on the number of alignments or “orders of support” in which each voter is pivotal. This paper investigates some properties of the Banzhaf index, the main topics being its derivation from axioms and its behavior in weighted-voting models when the number of small voters tends to infinity. These matters have previously been studied from the Shapley-Shubik viewpoint, but the present work reveals some striking differences between the two indices. The paper also attempts to promote better communication and less duplication of mathematical effort by identifying and describing several other theories, formally equivalent to Banzhaf’s, that are found in fields ranging from sociology to electrical engineering. An extensive bibliography is provided.

Suggested Citation

  • Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
  • Handle: RePEc:inm:ormoor:v:4:y:1979:i:2:p:99-131

    Download full text from publisher

    File URL:
    Download Restriction: no


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ormoor:v:4:y:1979:i:2:p:99-131. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Mirko Janc) The email address of this maintainer does not seem to be valid anymore. Please ask Mirko Janc to update the entry or send us the correct email address. General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.