 To formalize this, let \(S\) be the set of all available grades (‘\(S\)’ for grading scale), where \(S\) is non-empty and linearly ordered from highest to lowest. For instance, \(S\) could be \(\{\)A, B, C, D, E\(\}\) or \(\{ \text{Distinction,}\) \(\text{Merit,}\) \(\text{Pass,}\) \(\text{Fail}\}.\)  Each individual \(i\)’s assignment of grades to the alternatives in \(X\) is a function \(G_i\) that assigns to each element \(x \in X\) a grade \(G_i (x)\) from the set \(S\). A combination of grade assignments across the individuals in \(N\), \(\langle G_1, G_2 , \ldots ,G_n\rangle\), is called a profile of grade assignments. A grade aggregation rule is a function \(f\) that assigns to each profile of grade assignments \(\langle G_1, G_2 , \ldots ,G_n\rangle\) (in some domain of admissible ones) a collective grade assignment \(G = f(G_1, G_2 , \ldots ,G_n)\), where, for each \(x \in X, G(x)\) is the collective grade for \(x\).