A new long-term measure of sustainable growth under uncertainty

The trade-off between short-term success and long-term sustainability is a common subject of great importance both in the biological evolution of organisms and in the economic activities of human beings. In evolutionary biology, bet-hedging theories have described it as the trade-off between the (within-generation) arithmetic mean fitness and the (between-generation) geometric mean fitness of a genotype. Accordingly, bet-hedging strategies observed in various organisms are regarded as optimizing the geometric mean fitness. To increase the geometric mean fitness signifies to suppress the between-generation variance in the mean fitness. Thus, this view is consistent with mean-variance portfolio analysis in which the standard deviation of a portfolio is regarded as a measure of risk. In the present study, we provide yet another measure of long-term sustainability, which is based on minimization of the probability of extinction/bankruptcy that randomly varying population/asset size after a long time becomes less than a certain small value. We present results for representative examples to show that the present criterion gives a qualitatively similar but quantitatively different prediction from the traditional ones. In particular, we emphasize that maximizing survival probability (i.e. minimizing extinction probability) is equivalent neither to maximizing geometric mean fitness nor to minimizing variance in mean fitness, while these three are consistently related to each other.