If an ordinal-scaled utility function is defined via strictly increasing transformation, how can it represent a case of indifference?

Question

Problem: According to Wulf Gaertner’s (2009, p. 13) A Primer in Social Choice Theory, any strictly increasing transformation of an individual’s ordinal utility function is informationally equivalent. He says that:

“The only thing which matters in terms of available information is whether, say, commodity bundle $a$ is preferred to or indifferent to or dispreferred to commodity bundle $b$ or, expressed in terms of utility indices, whether $a$ has a higher or equally high or lower utility index than $b$.”

This makes sense, however, when I looked up the definition of a `strictly increasing function/transformation’ I became confused regarding how this would allow one to represent cases of indifference as well as cases of strict preference.

Ken Binmore’s (1983, p. 109) Mathematical Analysis: A Straightforward Approach formally defines an increasing and strictly increasing function as:

Increasing function: $\forall x,y \in X: (x > y) \Longrightarrow ( \ f(x) \geq f(y) \ )$

Strictly increasing function: $\forall x,y \in X: (x > y) \Longrightarrow ( \ f(x) > f(y) \ )$

So a strictly increasing transformation in utility terms–––for individual $i$–––would (I think) be defined formally as:

Strictly increasing transformation: $\forall x,y \in X: ( \ u_i(x) > u_i(y) \ ) \Longrightarrow ( \ f(u_i(x)) > f(u_i(y)) \ )$

But, since this is defined solely in terms of strict inequality ($ \ u_i(x) > u_i(y) \ $) I’m confused as to how this can allow for cases where an individual is indifferent between two alternatives: $xI_iy = ( \ u_i(x) = u_i(y) \ )$. Or cases of three or more alternatives where $i$ strictly prefers some and is indifference regarding others: e.g. $ ( \ u_i(x) = u_i(y) \ ) > u_i(z) $. Wouldn’t these be ordering cases, where the orderings are not characterised by strict inequality of utilities?

Duplication information: I am aware that a question very similar to this has been asked before by Eric '3ToedSloth' (Ordinal utility and monotonic transformations). Eric '3ToedSloth'’s question was focused on the definitions of both ordinal-scaled utility functions and monotone transformations. He also mentioned the difficulty of representing indifference. The reason that I have asked this question is that the accepted answer (by 201p) was focused on answering the question by showing that increasing (/weakly monotone) transformations destroy ordering information, rather than the issue which I am most confused about: how ordinal utility functions (defined in terms of strictly increasing transformations) can represent indifference. (There is a chance that 201p’s answer did cover this, but I am too stupid to be able to see it). Ideally I would have posed this as a clarificatory comment, but I lack the reputation level to post one. I am happy for my question to be closed if the community thinks it is too similar :).

Answer 1

I think you are most confused about what is meant by “defined in terms of strictly increasing transformations” or being “informationally equivalent.

You should think it in the following way, a utility function represents someone’s preferences if the function gives a higher/lower/equal value from bundle a than from bundle b if that person prefers more/less/the same bundle a than bundle b. Suppose that such function is $u(x)$, then you can ask, is there a unique function with such properties (I.e. that represents the person’s preferences) and the answer is NO. Any strictly increasing monotonic transformation of $u(x)$, say $f(u(x))$ is also a function describing the preferences of the person. Hence we say that these two are valid representations, and are equivalent (or informationally equivalent).

Note that whenever you are comparing two bundles you can use any one of the possible representations, and the “strictly increasing monotonic transformation” part is just to check if two functions are equivalent representations of preferences.