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.

Mathematical Properties of the Banzhaf Power Index

Universal banking is an alternative mechanism to a stock market for risk-sharing, for providing information for guiding investment, and for contesting corporate governance. In Germany, where the stock market has historically been small, banks hold equity stakes in firms and have proxy voting rights over other agents' shares. In addition, banks lend to firms and have representatives on corporate boards. If a banking relationship is a substitute for the stock market, then interaction with a bank should improve the performance of firms. But, if banks have private information about firms that they lend to and have monopolistic control over access to external capital markets, then bank interests may conflict with those of other equityholders, especially those whose shares are voted by the banks in proxy. We empirically investigate the influence of banks on the performance of German firms taking account of banks' equity holdings, the extent of banks' proxy voting rights, and the ownership structure of the firms' equity. We test for conflicts-of-interest in bank behavior and ask whether the relationship between banks and firms has changed between the 1970s and 1980s.

https://papers.ssrn.com/sol3/Delivery.cfm/nber_w5453.pdf?abstractid=1854831

Universal Banking and the Performance of German Firms

https://www.nber.org/system/files/working_papers/w5453/w5453.pdf

Universal banking and the performance of German firms

It is shown that the Banzhaf value can be obtained by differentiating the multilinear extension of a game at the midpoint of the unit cube. This gives us a composition theorem for the value of compound games. As an example, the values of the electoral college and presidential election “games” are approximated by the method of extensions.

Multilinear extensions and the banzhaf value

Homogeneity, independence, and power indices

/pdf/one-man-3-312-votes-a-mathematical-analysis-of-the-electoral-2fqv8mr14l.pdf

Banzhaf

Papers

One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College