Fundamental connections between utility theories of wealth and information theory

Abstract: 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.