Let $A_n$ be the set of pairs $(p, q)$ of probability distributions on $\{1, \ldots, n\}$ such that $q_i = 0 \implies p_i = 0$. (This is exactly the condition needed to guarantee that $D(p\|q) < \infty$.)
Theorem Let $(I: A_n \to [0, \infty))_{n \geq 1}$ be a sequence of functions. Then $I$ is a scalar multiple of relative information if and only if:
- the functions $I$ are measurable;
- the functions $I$ are permutation-invariant (that is, $I(p\sigma\|q\sigma) = I(p\|q)$ for all $\sigma \in S_n$);
- $I(p\|p) = 0$ for all $p$;
- we have $$ \begin{align*} & I\bigl(tp_1, (1 - t)p_2, p_3, \ldots, p_n \,\|\, uq_1, (1 - u)q_2, q_3, \ldots, q_n)\bigr)\\ & = I(p\|q) + p_1 I\bigl((t, 1 - t)\,\|\,(u, 1 - u)\bigr) \end{align*} $$ for all distributions $p, q$ and all $t, u \in [0, 1]$ such that $((t, 1 - t), (u, 1 - u)) \in A_2$.
I don't know who first stated or proved this theorem. I found it myself (inspired by John and Tobias's paper), and wrote about it here. But it seems unlikely that it's new. It could have been found any time since the 1950s, and probably has been. The literature search is made difficult by the fact that relative information is studied in multiple disciplines (mathematics, physics, engineering, statistics, ...) and goes by many names.
Edit (months later) Something very similar to this result was implicitly proved by Kannappan and Ng in 1972. I just wrote up a shorter proof as arXiv:1712.04903, and the history is discussed in Remark 2.7 there. This remark includes a summary of the theorem of Hobson that John originally asked about.