  We establish fundamental connections between utility theories of wealth from
the economic sciences and information-theoretic quantities. In particular, we
introduce operational tasks based on betting where both gambler and bookmaker
have access to side information, or betting tasks with double side information
for short. In order to characterise these operational tasks we introduce new
conditional R\'enyi divergences, and explore some of their properties.
Furthermore, we introduce an utility theory of wealth ratios, and operationally
interpret there the two-parameter $(q,r)$ generalised mutual information
measure recently introduced by V. M. Ili\'c and I. V. Djordjevi\'c; it
quantifies the advantage provided by side information in betting tasks for
utility theories of wealth ratios. Moreover, we show that the
Ili\'c-Djordjevi\'c conditional entropy satisfies a type of generalised chain
rule, which generalises that of Arimoto-R\'enyi. Finally, we address the
implications of these results on the quantum resource theories of informative
measurements and non-constant channels. Altogether, these results further help
strengthening the bridge between the theory of expected utility from the
economic sciences and Shannon's theory of information.
