55. Weighted Voting Systems

Voting is an ancient method for a group such as a meeting or an electorate to make a collective decision or express an opinion after some discussions, deliberations, or election campaigns. Participants who give an option or choose a candidate are called voters; therefore, a simplest voting system consists certain number of qualified voters and candidates. The most common application in weighted voting systems is, for example, the US Electoral College, where the number of electoral votes for each state is based upon its population. This chapter states the modeling of threshold weighted voting systems and the dynamic analysis of two terms including indecisive effect and known/unknown inputs. The system reliability models presented in the chapter are based on the assumptions that the system operates as two types of input values (0, 1) and three types of output values (0, 1, x) with three types of errors, and the components are unequally weighted and subject to three failure-modes (stuck-at-0, stuck-at-1, stuck-at-x). For any weighted voting system, a decision rule is required although different rules may result different system performance in terms of the system reliability. For instance, the current decision rule of US Electoral College is who wins the election obtaining at least 270 electoral votes, which results four former US presidents won the elections with less national populate votes than their opponents in history.

In general, the weighted voting system (WVS) consists of n units assigned with individual weights, each of which provides a binary decision (0 or 1) or abstains (x) from voting. A generic decision rule can be defined as the system output is 1 if the cumulative weight of all 1-opting units is at least a prespecified threshold τ of the cumulative weight of all nonabstaining units. If the indecisive effect is considered, weights of abstaining units can be added in the decision rule such as the system output is 1 if the cumulative weight of all 1-opting units is at least a prespecified threshold τ of the sum of all nonabstaining units and prespecified indecisive parameter θ of all abstaining units. The system fails if the generated output is not equal to its original input. Recent research results indicate that, under specified assumptions, multiple approaches can be used to quantify the reliability of the weighted voting system. This chapter demonstrates the development of decision rules and the evolution of approaches of generating reliability function. Some related works are addressed to provide a full picture of WVS, and some future works are proposed to attract attentions.