Weighted voting systems: reliability versus rapidity

The weighted voting system (WVS) consists of n units that each 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 probability of making decisions 0 and 1 and by probability of abstaining for each input. The system fails if its output is not equal to its input. This paper shows that if the WVS consists of units that need different time to produce their outputs, the decision time of the entire system depends on the distribution of unit weights and on the value of τ. It shows also that a tradeoff exists between the system reliability and its rapidity. An algorithm that finds the system parameters maximizing its reliability under constraint imposed on the expected system decision time is suggested. Illustrative examples are presented.