A few weeks ago, I was having a chat with Todd and some others in the office and it was in the conversational mix that cardinal utility had the property of preserving “intervals.” It was occasionally also mentioned that such utility representations were closed under “linear transformations.” I was confused by the discussion and at first I didn’t know why. On my walk home that day, I remembered I had heard those sorts of claims before. I typically think of a linear transformations as any mapping from a vector space to a vector space , both over a field , with the following properties of linearity:
if is in the field then for , ;
if , then .
For example, the equation is a linear transformation from the vector space of the set of reals back into itself. So, . When we speak of the algebra on in one dimension, is the underlying set for the vector space as well as the field.
Note that has the first property; suppose for example that and . Then
It also has the second property; for example, let and ; then
But clearly, this linear transformation does not preserve intervals:
I didn’t think I could be wrong about my understanding of the conventional use of the term “linear.” I thought maybe what people mean instead of “linear” in this context is that cardinal utility was closed under the class affine transformations. That is, the class containing all transformations of the following form (where is a scalar value in the field of and ):
The class of affine transformations is a proper superclass of the class of linear transformations. So the class of affine transformations does not always preserve intervals since is an affine transformation with set to the zero vector. It is easy to see that the class of affine transformations does not preserve ratios, for example let . If we let and , we have that:
To make matters somewhat worse, it appears that the form of is quite correctly called the form of a linear equation. However, some linear equations are not linear transformations because they fail to have the property of additivity. To wit:
Perhaps this subtle confusion between linear equations and linear transformations leads to a terminological confusion which makes people say that cardinal utility is closed under positive linear transformations, as in the final section of this Wikipedia article on cardinal utility (where cardinal utility is construed as von Neummann-Morgenstern utility).
Both our example transformations, and preserve the order of the reals. That is, for , we have that
Note that this property, positive monotonicity, is a consequence of the fact that and . One compelling feature of positive monotone transformations is that they preserve an existing order. However if a kind of utility only preserves an underlying order, it is typically regarded as ordinal utility in contrast to cardinal utility.
The class of translations does preserve intervals. These are the class of transformations of the following form (for ):
The effect of this transformation is to move a vector space in some direction (to the “left” or “right” by some value . For example let . Then if and ,
But if a utility measurement is only closed under translations, then that closure condition is tantamount to the view that there is something absolute about scale. It is not disputed that von Neumann-Morgenstern utility is closed under more than simply translations.
I think all the while the problem is that it is simply cumbersome in conversation to say which property is preserved by the affine transformations. It is neither ratios nor intervals but rather ratios of intervals. To see this, consider the following proposition.
Claim. Let and let be affine. Then
Continuing we have that
The following table summarizes what I think the correct picture should be:
|Difference Scale||Translation||Order, Intervals,
If all of the above is correct then it should be appear that it is technically incorrect to ever say that cardinal utility has the invariance property of preserving intervals even though we may call such utilities, perplexingly, an interval scale. Furthermore, while a cardinal utility like a vNM utility is closed under positive linear transformations, its invariance class is actually more properly understood as the class of positive affine transformations.