A resource theory of gambling
Maite Arcos, Renato Renner, Jonathan Oppenheim
TL;DR
This work reframes gambling as an adversarial resource theory to quantify informational advantage, extending Kelly’s criterion from its traditional asymptotic form to finite and single-shot regimes via a risk–reward framework governed by Rényi divergences. The authors connect information-theoretic divergences with CRRA utilities, showing how risk aversion selects optimal strategies and recovers familiar utility-based results, while simultaneously enabling a distributed, side-information generalization that yields asymptotic Nash equilibria. By embedding gambling in a resource-theoretic structure with explicit allowed operations, free states, and monotones, they map wealth growth to a tangible information-processing resource (communication) and lay out a clear path to quantum generalizations. The framework unifies economic and information-theoretic perspectives and opens avenues for applications in thermodynamics and quantum information through Adversarial Quantum Resource Theories.
Abstract
Betting games provide a natural setting to capture how information yields strategic advantage. The Kelly criterion for betting, long a cornerstone of portfolio theory and information theory, admits an interpretation in the limit of infinitely many repeated bets. We extend Kelly's seminal result into the single-shot and finite-betting regimes, recasting it as a resource theory of adversarial information. This allows one to quantify what it means for the gambler to have more information than the odds-maker. Given a target rate of return, after a finite number of bets, we compute the optimal strategy which maximises the probability of successfully reaching the target, revealing a risk-reward trade-off characterised by a hierarchy of Rényi divergences between the true distribution and the odds. The optimal strategies in the one-shot regime coincide with strategies maximizing expected utility, and minimising hypothesis testing errors, thereby bridging economic and information-theoretic viewpoints. We then generalize this framework to a distributed side-information game, in which multiple players observe correlated signals about an unknown state. Recasting gambling as an adversarial resource theory provides a unifying lens that connects economic and information-theoretic perspectives, and allows for generalisation to the quantum domain, where quantum side-information and entanglement play analogous roles.