 Different voting methods require different types of information from the voters as input. The input requested from the voters are called ballots. One standard example of a ballot is a ranking of the set of candidates. Formally, a ranking of \(X\) is a relation \(P\) on \(X\), where \(Y\mathrel{P} Z\) means that “\(Y\) is ranked above \(Z\),” satisfying three constraints: (1) \(P\) is complete: any two distinct candidates are ranked (for all candidates \(Y\) and \(Z\), if \(Y\ne Z\), then either \(Y\mathrel{P} Z\) or \(Z\mathrel{P} Y\)); (2) \(P\) is transitive: if a candidate \(Y\) is ranked above a candidate \(W\) and \(W\) is ranked above a   candidate \(Z\), then \(Y\) is ranked above \(Z\) (for all candidates \(Y, Z\), and \(W\), if \(Y\mathrel{P} W\) and \(W\mathrel{P} Z\), then \(Y\mathrel{P} Z\)); and (3) \(P\) is irreflexive: no candidate is ranked above itself (there is no candidate \(Y\) such that \(Y\mathrel{P} Y\)). For example, suppose that there are three candidates \(X =\{A, B, C\}\). Then, the six possible rankings of \(X\) are listed in the following table: 