Translation of Weighted Voting Concepts to the Boolean Domain: The Case of the Banzhaf Index Ali

This chapter 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.