Asymmetric weighted voting systems

The weighted voting system (WVS) studied consists of n units, each of which provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. 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 non-abstaining units. Otherwise, system output is 0. The system input is either 0 or 1. Every unit is characterized by the probability of making decisions 0 and 1 and by probability of abstaining for each input. The system fails if the system output is not equal to its input. In this paper, an asymmetric WVS is suggested in which each voting unit has two weights. The first weight is applied when the unitʹs decision is 0 and the second weight is applied when the unitʹs decision is 1. The asymmetry of unit weights allows the WVS designer to take advantage of the knowledge of statistical asymmetry of voting units (asymmetric probabilities of making correct decisions with respect to the input). The paper presents an algorithm for asymmetric WVS reliability evaluation. This algorithm is based on using a universal generating function technique. For a system consisting of voting units with given reliability characteristics, one can maximize the entire system reliability by choosing proper unit weights and threshold value. An algorithm is suggested which finds the optimal unit weights and the threshold. A genetic algorithm is used as the optimization tool. An example is presented in which the superiority of asymmetric WVS over regular symmetric one is demonstrated.