Optimal growth in an uncertain environment

Bet-hedging in biological phenomena is a well recognized process that can manifest itself as phenotypic switching: an individual has to choose the best strategy to adapt to a stochastic environment in order to maximize the growth rate of its own population. Fluctuations can lead to the extinction of the population itself, no matter how high the growth rate is. The same model can be applied to Kelly' s horse races studies, where the gambler has to choose the best strategy to maximize his payoff. We propose to add a definition of "risk" to Kelly's horse races model in order to map the phenotypic switching model into Kelly's one, taking into account the "bankruptcy" probability, that we want to minimize. We end up considering the volatility as a measure of risk, as well as the probability of extinction itself and comparing the different definitions. Results show a similarity between the two notions. Furthermore, we analyse the model with a game-theoretic approach, to see if there is any possible reduction, under some particular conditions, of the set of strategies into a smaller one, called the "essential part of the game". We found a general way to map an optimization problem to a game-theoretic one, showing how reduction leads to the same results.