If the intention is to express Arrow’s independence condition this is a mistake because \(I^*\), though similar in appearance to I, has a different content. I says that whenever everybody’s preferences concerning a pair of options are the same in one profile as they are in another, the social preference must also be the same at the two profiles, as far as this pair is concerned. This is not what \(I^*\) says because the embedded antecedent ‘for all \(i: xR_{i}y\) if and only if \(xR^*_{i}y\)’ is satisfied not only when everybody’s preferences among \(x\) and \(y\) are the same in \(\langle R_{i}\rangle\) as they are in \(\langle R_{i}^*\rangle\), but in other instances as well.  For example, suppose that in \(\langle R_{i}\rangle\) everybody is indifferent between some social state \(T\) and another state \(S\) (in which case for all \(i\), both \(T R_{i} S\) and \(S R_{i} T\)), while in \(\langle R_{i}^*\rangle\) everybody strictly prefers \(T\) to \(S\) (for all \(i\), \(T R^*_{i} S\) but not \(S R^*_{i} T\)). Then the antecedent ‘for all \(i: T R_{i} S\) if and only if \(T R^*_{i} S\)’ is satisfied, although individual preferences among \(T\) and \(S\) are not the same in the two profiles. \(I^*\) sometimes constrains \(f\) though \(I\) does not and it is a more demanding condition.