Multivariate Rényi divergences characterise betting games with multiple lotteries

TL;DR

The paper provides an operational interpretation of the multivariate Rényi divergence in economic tasks involving betting on multiple lotteries under risk aversion, showing that the isoelastic certainty equivalent coherently encodes $D_{\underline{\alpha}}(\vec{P}_X)$ when odds are fair. It introduces a conditional version $D_{\underline{\alpha},\beta}$ with a data-processing inequality, capturing the value added by side information in betting scenarios. Extending to general probabilistic theories, it connects state betting, measurements, and resource theories, establishing a GPT-wide resource monotone for measurement informativeness based on mv Rényi divergences. The framework blends information theory, physics, and economics, yielding a quantitative foundation for multi-lottery quantum-state betting and informative measurements within quantum resource theories.

Abstract

We provide an operational interpretation of the multivariate Rényi divergence in terms of economic-theoretic tasks based on betting, risk aversion, and multiple lotteries. We show that the multivariate Rényi divergence $D_{\underlineα}(\vec{P}_X)$ of probability distributions $\vec{P}_X =(p^{(0)}_X,\dots,p^{(d)}_X)$ and real-valued orders $\underlineα = (α_0, \dots, α_d)$ 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 $w^{\mathrm{ICE}}_{\underline{R}}=\exp[D_{\underlineα}(\vec{P}_X)]$, where $\underline{R}=(R_1,\dots,R_d)$ is a risk-aversion vector with $R_k = 1+α_k/α_0$ being the risk-aversion parameter for lottery $k$. Furthermore, we introduce a new conditional multivariate Rényi 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.

Figures (8)

• Figure 1: Summary of the current work. We study three major themes in colored boxes and discuss their quantitative connections represented by gray lines. Technical details will be found in the underlined section attached with each topic.
• Figure 2: The area of allowed parameter vectors $\underline{\alpha}=(\alpha_0,\alpha_1,\alpha_2)\in\mathbb R^3$ for the multivariate Rényi divergence $D_{\underline{\alpha}}$ in the case $d=2$. The area is confined onto the affine plane $\sum_{k=0}^2\alpha_k=1$ in $\mathbb R^3$. The special points fixing this plane are $e_0=(1,0,0)$, $e_1=(0,1,0)$, and $e_2=(0,0,1)$.
• Figure 3: Illustration of a single-lottery betting game. A random event (horse race) is described by a random variable $X$ and distributed according to a probability mass function $p_X$. A bookmaker proposes an odds function $o_X$, and a gambler considers placing a betting strategy $b_X$. The gambler is assumed to have a risk-averse character represented by a utility function $u_{R}$, with $R$ the risk-averse parameter. Before the gambler fully committing to playing the game, a friend proposes to pay the gambler a fixed amount, say $w$, to persuade him to walk away from the betting game. Will the gambler take the friend's offer over gambling on the lottery? The solution to this decision problem is given by the isoelastic certainty equivalent $w^{\rm ICE}_{R} \equiv w^{\rm ICE}(p_{X}, o_X, b_{X}, u_{R})$ of the betting game as it represents the economic-theoretic value that the gambler assigns to the lottery in question. Explicitly, the gambler will choose to gamble whenever $w^{\rm ICE}_{R} > w$, and will take the friend's offer otherwise.
• Figure 4: Schematic representation of a betting game extended to a multi-lottery framework. A random event (horse race) is described by a random variable $X$ distributed according to a probability mass function $p_X^{(0)}$. In this setting, a group of $d$ bookmakers propose $d$ odds functions $\vec{O}_X=(o_X^{(1)},\ldots,o_X^{(d)})$, and a gambler places $d$ betting strategies $\vec{B}_X=(b_X^{(1)},\ldots,b_X^{(d)})$. Accordingly, the gambler's risk profile is characterised by a risk-aversion vector $\underline{R} = (R_1,\ldots,R_d)\in\mathbb{R}_{\ge 0}^d$ within the utility function $u_{\underline{R}}$, where each $R_k$ is the risk-aversion parameter associated with the $k$th lottery. To persuade the gambler to withdraw from the betting, a fixed offer of $w$ per lottery is presented. The decision to accept this offer or play depends on the isoelastic certainty equivalent $w^{\mathrm{ICE}}_{\underline{R}} \equiv w^{\mathrm{ICE}}(p_{X}^{(0)},\vec{O}_X,\vec{B}_{X},u_{\underline{R}})$, which quantifies the gambler's valuation of the lotteries. Explicitly, the gambler chooses to play (rejecting the persuasion) whenever $w^{\mathrm{ICE}}_{\underline{R}} > w$.
• Figure 5: Multi-lottery betting games with side information. Unlike standard multi-lottery betting games described in \ref{['fig:fig2']}, the random variable $X$ is now correlated to another random variable $G$ and jointly distributed according to a joint probability mass function $p_{XG}^{(0)}$. The gambler places $d$ betting strategies $\vec{B}_{X|G}=(b_{X|G}^{(1)}, \ldots, b_{X|G}^{(d)})$ having access to the side information $G$. The economic-theoretic value of the lotteries is given by $w^{\rm ICE}_{\underline{R}} \equiv w^{\rm ICE}(p_{XG}^{(0)},\vec{O}_X,\vec{B}_{X|G},u_{\underline{R}})$.

Theorems & Definitions (16)

• Definition 1: Unconditional multivariate Rényi divergence farooq2024Verhagen_et_al_2024
• Definition 2: Conditional multivariate Rényi divergence
• Proposition 1: Data processing inequality with respect to the main system
• Proposition 2: Data processing inequality with respect to the conditioning system
• Corollary 1: Data processing inequality
• Corollary 2: Optimal betting strategies
• Corollary 3: The unconditional multivariate Rényi divergence characterises multi-lottery betting, optimal betting strategies, and fair odds
• Corollary 4: Optimal betting strategies
• Corollary 5: Conditional multivariate Rényi divergence characterises multi-lottery betting games, access to side information, optimal betting strategies, and fair odds
• Corollary 6: Economic-theoretic interpretation of the data processing inequality