Phase transitions in optimal betting strategies

Abstract

Kelly's criterion is a betting strategy that maximizes the long term growth rate, but which is known to be risky. Here, we find optimal betting strategies that gives the highest capital growth rate while keeping a certain low value of risky fluctuations. We then analyze the trade-off between the average and the fluctuations of the growth rate, in models of horse races, first for two horses then for an arbitrary number of horses, and for uncorrelated or correlated races. We find an analog of a phase transition with a coexistence between two optimal strategies, where one has risk and the other one does not. The above trade-off is also embodied in a general bound on the average growth rate, similar to thermodynamic uncertainty relations. We also prove mathematically the absence of other phase transitions between Kelly's point and the risk free strategy.

Universal constraints on selection strength in lineage trees

“…We note that eqs. (12) and (13) are similar to Fisher's relation, but with some differences: for instance, in the l.h.s of eqs. (12) and (13) the time derivative of mean fitness is replaced by the strength of selection for a given phenotypic trait.…”

“…Here, we derive universal constraints for the average value of a trait, and for its selection strength by exploiting a set of recent results known under the name of Thermodynamic Uncertainty Relations (TUR). These relations take the form of inequalities, which generalize fluctuation-response relations far from equilibrium [11], and which capture important trade-offs for thermodynamic and non-thermodynamic systems [12] as recently reviewed in [13]. Although our results are framed in the context of cell population in lineage trees, they apply more broadly to general stochastic processes defined on any branched tree.…”

Optimal spatial allocation of enzymes as an investment problem

“…For instance, as mentioned above, one could consider an objective function penalizing high variance as in Markovitz' problem [11]. One could also consider problems where the capital of the investor grows exponentially (iterative reinvestment) as in Kelly's problem [10] or more complex variants [48]. In each case, one needs to compute the marginal returns and the Hessian of the new objective function with respect to the resources invested in the different assets, and determine the optimal allocation of resources as the total budget gradually increases via the construction scheme.…”

Entropy, Economics, and Criticality

“…At a similar scale but for housing markets Crosato et al [ 10 , 11 ] have studied the criticality of city dynamics using maximum entropy techniques. At the individual agent level Dinis et al [ 12 ] studied phase transitions in optimal betting strategies using the Kelly criterion.…”