Mathematical Properties of the Banzhaf Power Index

Pradeep Kumar Dubey · Lloyd S. Shapley

681 citations0/0 files storedMathematics of Operations Research

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Game Theory and Voting Systems Social Power and Status Dynamics Opinion Dynamics and Social Influence

Abstract

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.

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References

1988A value for n -person games 2007The concept of power 1954A Method for Evaluating the Distribution of Power in a Committee System 1973An Introduction to Positive Political Theory 2015Values of Non-Atomic Games 1975Game Theory and Politics 1967New Methods of Thought and Procedure

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2000Determination of weights of interacting criteria from a reference set 1984Optimizing group judgmental accuracy in the presence of interdependencies 1988An axiomatization of the Banzhaf value 2002Chapter 4 Voting procedures 2018Deriving decision makers’ weights in group decision making: An overview of objective methods 2000Equivalent Representations of Set Functions 1997Ternary Voting Games 2001The Treaty of Nice and qualified majority voting

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