Weighted voting systems

This paper deals with reliability and cost evaluation of weighted dynamic-threshold voting-systems. The particular voting system studied consists of n units that each provide a binary decision (0 or 1) or abstain from voting (x). The system output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all nonabstaining units. Otherwise the system output is 0. Systems of this type are encountered in many areas ranging from target to pattern recognition, safety monitoring and human organization systems. These results for a voting system with a particular structure can easily be modified for other voting systems as well. A general mathematical model of this system is presented. However, its immense combinatorial complexity (even for small-scale and trivial systems) makes the model virtually inapplicable for real systems. By placing two restrictions on the generality of the model parameters, the model can be greatly simplified, which in turn makes the model applicable for evaluating systems of any size. The imposed restrictions are minor, and are fulfilled in most practical situations, so that the derivations in this paper are helpful for the design and analysis of practical, weighted dynamic-threshold voting-systems