 A grade profile is a list \(\langle G_{1}, \ldots, G_{n}\rangle\) of grade functions, one for each of the people \(1, \ldots, n\). Notions of a multiprofile domain and a social welfare function \(f\) are defined as in Arrow’s framework, putting grade profiles instead of Arrow’s profiles of weak orders (the output of a social welfare function is not a social grading of the alternatives but a social ordering of them, just in Arrow’s original framework). The conditions of Arrow’s theorem are reformulated accordingly; for instance, the reformulated domain assumption is: 