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

Ali Muhammad Ali Rushdi; Omar Mohammed Ba-Rukab

Ali Muhammad Ali Rushdi Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80204, Jeddah, 21589, Kingdom of Saudi Arabia.
Omar Mohammed Ba-Rukab Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, P.O.Box 344, Rabigh 21911, Saudi Arabia.

Keywords: Voting system, Banzhaf index, coalition, Conventional Karnaugh map, Variable-Entered Karnaugh Map

Abstract

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.