University of Arkansas, Fayetteville
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Abstract

On the whole, political power can he very difficult to quantify. A person may be powerful due to his or her personal charm, wealth, fame, credibility, or influential connections. Political bodies do not account for these qualities when creating voting procedures; they only assign voting rules to specific positions. For example, most would say that in the United States government that a Senator is more powerful than a Representative, but less powerful than the President, without knowing any way to quantify or verify those differences. Since the 1950's, mathematicians and political scientists have attempted to create mathematical models that partially describe an individual's power as a voting member of a committee, board, or legislative body. These models have resulted in four major "power indexes" that describe the percentage of a body's total power held by each individual member. The four most prominent power indexes are the Shapley-Shubik, Banzhaf, Johnston, and Deegan-Packel, each of which uses a different theory to calculate the probability that an individual's vote will decide whether a proposal passes or fails. The research in this paper develops formulas to calculate the four-power indexes for legislatures that are unicameral, bicameral, unicameral with committees, and bicameral with committees. These formulas have several variables (up to ten) and have many (up to several thousand) terms for typical sizes of state legislative chambers. Using Mathematica computer software the four power indices are computed for various legislative configurations and the indices' behavior are studied. Then these methods are applied to the Arkansas State Government by calculating the power indexes of the Governor, Senate, House, House Committee members, and Senate Committee members. By examining the theories behind the four power indexes and available historical evidence, the paper concludes by analyzing which indexes, if my, provide the best model for the political power structure of the Arkansas State Government.

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