 Consider first an ordering over three regular options, e.g., the three holiday destinations Amsterdam, Bangkok and Cardiff, denoted \(A\), \(B\) and \(C\) respectively. Suppose your preference ordering is \(A\prec B \prec C\). This information suffices to ordinally represent your judgement; recall that any assignment of utilities is then acceptable as long as \(C\) gets a higher value than \(B\) which gets a higher value than \(A\). But perhaps we want to know more than can be inferred from such a utility function—we want to know how much \(C\) is preferred over \(B\), compared to how much \(B\) is preferred over \(A\). For instance, it may be that Bangkok is considered almost as desirable as Cardiff, but Amsterdam is a long way behind Bangkok, relatively speaking. Or else perhaps Bangkok is only marginally better than Amsterdam, compared to the extent to which Cardiff is better than Bangkok. This kind of information about the relative distance between options, in terms of strength of preference or desirability, is precisely what is given by an interval-valued utility function. The problem is how to ascertain this information.