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 ):
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Homomorphisms are usually defined as structure preserving mappings from one model to another. Representation theorems are taken to establish the existence of a homomorphism between a qualitative first-order structure endowed with some empirical relations and some sort of numerical first order structure. The classic example is in the measurement of hedonic utility using introspection. In that case, we describe the axiomatic conditions ensuring the existence of a mapping from a structure to the structure of reals with their standard ordering . Here is meant to be a (usually finite) set of alternatives or choices and the relation is meant to encode the introspectively accessible relation that something feels better than something else.
I have been thinking that this notion of a homomorphism was exactly the same as in model theory but it turns out there are some subtleties. In model theory, usually we say that if we have two structures and in the same signature (which is a set of constant symbols, relation symbols, and function symbols), then a homomorphism from , the domain of , to , the domain of is a function satisfying the following conditions:
- (i) For any constant symbol in , is
- (ii) For any -ary function symbol in and ,
- (iii:a) For any -ary relation symbol in and ,
The superscripts here indicate how the symbols are interpreted in the respective structures with objects or -tuples of objects. The conditions taken together are less demanding conditions than what is usually meant, it appears, in the theory of measurement. Here, we replace the third condition with
- (iii:b) For any -ary relation symbol in and ,
In model theory, a mapping satisfying (i), (ii), and (iii:b) is called a strong homomorphism. The condition ensures that if two objects, for example, are not -related in then the will remain unrelated by in the mapping to .
If we add the condition that , a strong homomorphism, is an injection then the map is usually called an embedding. Likewise if a strong homomorphism is a bijection then it is an isomorphism.