 We thus see that an interval-valued utility measure over options can be constructed by introducing lottery options. As the name suggests, the interval-valued utility measure conveys information about the relative sizes of the intervals between the options according to some desirability scale. That is, the utilities are unique after we have fixed the starting point of our measurement and the unit scale of desirability. In the above example, we could have, for instance, assigned a utility value of 1 to \(A\) and 5 to \(C\), in which case we would have had to assign a utility value of 4 to \(B\), since 4 is 3/4 of the way between 1 and 5. In other words, once we have assigned utility values to \(A\) and \(C\), the utility of \(L'\) and thus \(B\) has been determined. Let us call this second utility function \(u'\). It is related to our original function as follows: \(u'=4\cdot u +1\). This relationship always holds between two such functions: If \(u\) is an interval-valued utility function that represents a preference ordering, \(\preceq\), and \(u'\) is another utility function that also represents this same preference ordering, then there are constants \(a\) and \(b\), where \(a\) must be positive, such that \(u'=a\cdot u + b\). This is to say that interval-valued utility functions are unique only up to positive linear transformation.