 Independence: For any two profiles \(\langle Pr_1, Pr_2,\ldots, Pr_n\rangle\) and \(\langle Pr^*_1, Pr^*_2, \ldots ,Pr^*_n\rangle\) in the domain of \(F\) and any \(p \in X\), if for all \(i \in N\) \(Pr_i(p) = Pr^*_i(p)\), then \(Pr(p) = Pr^*(p)\).