Multivariate R\'enyi divergences characterise betting games with multiple lotteries

Abstract

We provide an operational interpretation of the multivariate R\'enyi divergence in terms of economic-theoretic tasks based on betting, risk aversion, and multiple lotteries. We show that the multivariate R\'enyi divergence $D_{\underline{\alpha}}(\vec{P}_X)$ of probability distributions $\vec{P}_X =(p^{(0)}_X,\dots,p^{(d)}_X)$ and real-valued orders $\underline{\alpha} = (\alpha_0, \dots, \alpha_d)$ quantifies the economic-theoretic value that a rational agent assigns to $d$ lotteries with odds $o^{(k)}_X \propto (p_X^{(k)})^{-1}$ ($k=1,\dots,d$) on a random event described by $p^{(0)}_X$. In particular, when the odds are fair and the rational agent maximises over all betting strategies, the economic-theoretic value (the isoelastic certainty equivalent) that the agent assigns to the lotteries is exactly given by $w^{\mathrm{ICE}}_{\underline{R}}=\exp[D_{\underline{\alpha}}(\vec{P}_X)]$, where $\underline{R}=(R_1,\dots,R_d)$ is a risk-aversion vector with $R_k = 1+\alpha_k/\alpha_0$ being the risk-aversion parameter for lottery $k$. Furthermore, we introduce a new conditional multivariate R\'enyi divergence that characterises a generalised scenario where the agent uses side information. We prove that this new quantity satisfies a data processing inequality which can be interpreted as the increment in the economic-theoretic value provided by side information; crucially, such a data processing inequality is a consequence of the agent's economic-theoretically consistent risk-averse attitude towards every lottery and vice versa. Finally, we apply these results to the resource theory of informative measurements in general probabilistic theories (GPTs). By establishing quantitative connections between information theory, physics, and economics, our framework provides a novel operational foundation for quantum state betting games with multiple lotteries in the realm of quantum resource theories.