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

TL;DR

This paper characterizes the multivariate Re9nyi divergence through betting games, linking it to economic value, risk aversion, and side information, with applications to quantum resource theories.

Contribution

It introduces an operational interpretation of multivariate Re9nyi divergence in economic betting scenarios and extends it with a new conditional divergence satisfying a data processing inequality.

Findings

Quantifies economic value of lotteries using Re9nyi divergence

Introduces a conditional divergence with side information

Connects information theory, physics, and economics in quantum resource theories

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.