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 :).