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Multivariate Rényi divergences characterize betting games with multiple lotteries

Andrés F. Ducuara1,2,3,4,*, Erkka Haapasalo5,†, and Ryo Takakura6,7,‡

  • *Contact author: andres.dg@ntu.edu.sg
  • †Contact author: cqteth@nus.edu.sg
  • ‡Contact author: takakura.ryo.qiqb@osaka-u.ac.jp
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APS Open Sci. 1, 000138 – Published 16 September, 2026

DOI: https://doi.org/10.1103/h8zk-pl1p

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. Specifically, we show that the multivariate Rényi divergence Dα̲(P⃗X) of probability distributions P⃗X=(pX(0),...,pX(d)) and real-valued orders α̲=(α0,...,αd) quantifies the economic-theoretic value that a rational agent assigns to d lotteries with odds oX(k)∝(pX(k))−1 (k=1,...,d) on a random event described by pX(0). In particular, when the odds are fair and the rational agent is allowed to maximize over all betting strategies, the economic-theoretic value (precisely, the isoelastic certainty equivalent) that the agent assigns to the lotteries is exactly given by wR̲ICE=exp[Dα̲(P⃗X)], where R̲=(R1,...,Rd) is a risk-aversion vector with Rk=1+αk/α0 being the risk-aversion parameter associated with lottery k. Furthermore, we introduce a conditional multivariate Rényi divergence that characterizes a generalized scenario where the rational agent is allowed to have access to side information. We prove that this quantity satisfies a data processing inequality that can be interpreted as the increment in the economic-theoretic value provided by side information; crucially, we show that such a data processing inequality is a consequence of the agent's economic-theoretically consistent risk-averse attitude toward every lottery and vice versa. Moreover, we demonstrate the applicability of these results to the resource theory of informative measurements within the operational framework of general probabilistic theories encompassing quantum theory. By establishing quantitative connections between information theory, physical theories, and theories from economics, our framework provides an operational foundation for quantum state betting games with multiple lotteries in the realm of quantum resource theories.

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APS Open Sci. 1, 000138– Published 16 September, 2026
Vol. 1
  • Received 8 February 2026
  • Accepted 26 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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