RESUMEN
Motivated by a horse betting problem, a new conditional Rényi divergence is introduced. It is compared with the conditional Rényi divergences that appear in the definitions of the dependence measures by Csiszár and Sibson, and the properties of all three are studied with emphasis on their behavior under data processing. In the same way that Csiszár's and Sibson's conditional divergence lead to the respective dependence measures, so does the new conditional divergence lead to the Lapidoth-Pfister mutual information. Moreover, the new conditional divergence is also related to the Arimoto-Rényi conditional entropy and to Arimoto's measure of dependence. In the second part of the paper, the horse betting problem is analyzed where, instead of Kelly's expected log-wealth criterion, a more general family of power-mean utility functions is considered. The key role in the analysis is played by the Rényi divergence, and in the setting where the gambler has access to side information, the new conditional Rényi divergence is key. The setting with side information also provides another operational meaning to the Lapidoth-Pfister mutual information. Finally, a universal strategy for independent and identically distributed races is presented that-without knowing the winning probabilities or the parameter of the utility function-asymptotically maximizes the gambler's utility function.
RESUMEN
Models of adaptive bet-hedging commonly adopt insights from Kelly's famous work on optimal gambling strategies and the financial value of information. In particular, such models seek evolutionary solutions that maximize long-term average growth rate of lineages, even in the face of highly stochastic growth trajectories. Here, we argue for extensive departures from the standard approach to better account for evolutionary contingencies. Crucially, we incorporate considerations of volatility minimization, motivated by interim extinction risk in finite populations, within a finite time horizon approach to growth maximization. We find that a game-theoretic competitive optimality approach best captures these additional constraints and derive the equilibria solutions under straightforward fitness payoff functions and extinction risks. We show that for both maximal growth and minimal time relative payoffs, the log-optimal strategy is a unique pure strategy symmetric equilibrium, invariant with evolutionary time horizon and robust to low extinction risks.