What is this subclass of the class of monotonic transformations?

Question

Let $u$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for greater than and equal to for all $x$ and $y$.

But I'm interested in a stronger condition. My question is, what is the set of all functions $v$ such that $v$ is not only a monotonic transformation of $u$, but also satisfies the condition $u(x) - u(y) < u(z) - u(w)$ if and only if $v(x)-v(y) < v(z) -v(w)$ and similarly for greater than and equal to for all $x,y,z,$ and $w$?

This set of functions certainly contains all positive affine transformations of $u$, i.e. all functions of the form $au+b$ where $a$ is positive. But are there any other functions in this set?

This question arose from my answer in Economics.SE here, where the context is that $u$ is a von Neumann-Morgernstern utility function, and the functions $v$ that satisfy the condition above are other possible utility functions a person could have.

Answer 1

This is a complete revision of a former answer.

The answer to you question is : no, in your case, there are no other functions satisfying the property you mention.

This is a corollary of Theorem 1 in Basu, Kaushik. "Cardinal utility, utilitarianism, and a class of invariance axioms in welfare analysis." Journal of mathematical Economics 12.3 (1983): 193-206. As it turns out, the continuity of $u$ does not matter. The only thing that matters is that the domain $\mathbb{R}$ of $u$ be somewhat "regular", see below.

The following is from Basu (1983)

Definition : The mapping $f:B\rightarrow \mathbb{R}$, where $B\subset \mathbb{R}$ is a positive affine transformation (PAT) if $\exists a,b \in \mathbb{R}$, with $b>0$, such that $\forall t \in B$ $$ f(t) = a + bt$$

Definition : The mapping $f:B\rightarrow \mathbb{R}$, where $B\subset \mathbb{R}$ is a first-difference preserving transformation (DPT) if $\forall t_1,t_2,t_3,t_4 \in B$ $$t_1 - t_2 \geq t_3 - t_4 \Leftrightarrow f(t_1) - f(t_2) \geq f(t_3) - f(t_4)$$

Theorem 1 : If $f: B\rightarrow \mathbb{R}$ and $B$ is dense in a connected subset of $\mathbb{R}$, then $f$ is a DPT if and only if it is a PAT.

To go from Theorem 1 to the desired corollary, simply notice that $\mathbb{R}$ is trivially dense in a connected subset of $\mathbb{R}$ (the subset being $\mathbb{R}$ itself).

In fact, this is noted by Basu himself in the following terms:

Assumption N (No restriction on transformation domain). $B = \mathbb{R}$.

Corollary 1.1. Given Assumption N, [...] the ability to compare first-differences of utility is equivalent to cardinality.