Independence of Irrelevant Alternatives (choice version): For all environments \(S\) within \(X\), and all profiles \(\langle R_{i}\rangle\) and \(\langle R^*_{i}\rangle\) in the domain of \(f\), if \(\langle R_{i}\rangle|S = \langle R^*_{i}\rangle|S\), then \(C(S)= C^*(S)\).