 Amartya Sen (1970) extended Arrow’s framework by representing the preferences of individuals \(i\) as utility functions \(U_{i}\) that map the alternatives onto real numbers: \(U_{i}(x)\) is the utility that \(i\) obtains from \(x\).  A preference profile in Sen’s framework is a list \(\langle U_{1}, \ldots, U_{n}\rangle\) of utility functions, and a domain is a set of these. An aggregation function, now a social welfare functional, maps each profile in some domain onto a weak ordering of the alternatives.