Calculation of Banzhaf Voting Indices Utilizing Variable-Entered Karnaugh Maps
Ali Muhammad Ali Rushdi, Omar Mohammed Ba-Rukab
Journal of Advances in Mathematics and Computer Science · pp. 1–17 · Published 30 Jan 2017
10.9734/BJMCS/2017/31191Abstract
This paper is a tutorial exposition on how to translate concepts of voting systems to the Boolean domain, and consequently on how to use Boolean tools in the computation of a prominent index of voting powers, viz., the Banzhaf voting index. We discuss Boolean representations for yes-no voting systems, in general, and for weighted voting systems, in particular. Our main observation is that non-minimal winning coalitions are related to minimal ones via partial-order structures and also as particular subordinate loops that cover the all-1 cell in the Karnaugh map. We review the method of computing the total Banzhaf indices by the Conventional Karnaugh Map (CKM). Then we extend this method to handle larger problems via the Variable-Entered Karnaugh Map (VEKM). The map methods are demonstrated by two classical weighted voting systems.
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