 By assigning real numbers to alternatives, welfare profiles contain a lot of information over and above the profiles of orderings on \(X\) they induce. Many different assignments of numbers to alternatives can give rise to the same orderings. But we may not consider all this information meaningful. Some of it could be an artifact of the numerical representation. For example, the difference between the profile \(\langle W_1, W_2 , \ldots ,W_n\rangle\) and its scaled-up version \(\langle 10\cdot W_1, 10 \cdot W_2,\ldots , 10\cdot W_n\rangle\), where everything is the same in proportional terms, could be like the difference between length measurements in centimeters and in inches. The two profiles might be seen as alternative representations of the exact same information, just on different scales.