Bet-hedging via Kelly betting in a limited environment leads to logistic growth in the Game of Fitness

In finance and population dynamics, it is well known that achieving the optimal logarithmic growth rate in an uncertain environment is accomplished by a bet-hedging strategy known as Kelly betting. Such bets assume exponential growth with an unlimited upper bound on population size and a known success probability to determine the bet size. However, practical systems are constrained by finite resources and space, in which case populations instead follow logistic growth with a carrying capacity, defined as the maximum population size that the environment can sustain. In this paper, we introduce a population-dependent Kelly betting strategy that explicitly incorporates environmental limits and show analytically that it converges to the logistic growth function when a carrying capacity is present. To illustrate this, we propose a spatially explicit simulation model, called the Game of Fitness, which models growth as the outcome of local reproductive and dormancy decisions in a finite environment. Within the Game of Fitness, the success probability is not assumed a priori, but instead is predicted from the evolving spatial interactions of the population. This model reveals how departures from idealised Kelly assumptions arise when spatial and informational constraints are taken into account. In particular, the results reveal the effects of spatial reproduction mobility and environmental information availability, showing that the former is more dominant than the latter. These findings demonstrate how Kelly-based growth is modified in realistic, constrained environments, providing insight into biological populations and structurally analogous growth processes in economic and financial systems.