 A social welfare functional (SWFL), also denoted \(F\), is a function that assigns to each profile \(\langle W_1, W_2 , \ldots ,W_n\rangle\) (in some domain of admissible profiles) a social preference relation \(R = F(W_1, W_2 , \ldots ,W_n)\) on \(X\), with the familiar interpretation. Again, when \(F\) is clear from the context, we write \(R\) for the social preference relation corresponding to \(\langle W_1, W_2 , \ldots ,W_n\rangle\). The output of a SWFL is similar to that of a preference aggregation rule (again, we do not build the completeness or transitivity of \(R\) into the  definition[9]),  but its input is richer.