2023/06/22 by Ali Muhammad Ali Rushdi, Rushdi, Ali Muhammad, Muhammad A. Rushdi +1
Economics, Econometrics and Finance · #Computer Science and Game Theory (cs.GT) #FOS: Computer and information sciences #Game Theory and Voting Systems
paper · pdf · doi:10.48550/arxiv.2306.13684
openalex publication_date 2023/06/22 · openalex created_date 2023/06/28 · openalex updated_date 2026/07/28
We explore the switching-algebraic computation of the Banzhaf indices for general and monotone or unrestricted systems. This computation is achieved via (a) two Boolean-quotient formulas that are valid when the voting system is not necessarily monotone (e.g., when coalition formation is restricted), (b) four Boolean differencing formulas and six Boolean-quotient formulas that are applicable when the decision switching function is a positively polarized unate one. We also provide switching-algebraic formulas for certain Banzhaf-related indices, including the power-to-initiate index (PII), and the power-to-prevent index (PPI), as well as satisfaction indices. Moreover, we briefly address other Banzhaf-related indices, including the Strict Power Index (SPI) and the Public Good Index (PGI). We illustrate the various indices formulas by way of four examples of voting systems, each considered first as an unrestricted monotone system and then subjected to a restriction on the formation of a coalition between two particular voters.