The fundamental theorem of natural selection in optimization and games

Abstract

The Fisher's fundamental theorem of natural selection (FTNS) is a matter of longstanding debate in the community of mathematical biologists. Many researchers proposed different clarifications and mathematical reconstructions of the Fisher's original statement. The present study is motivated by our opinion that the controversy can be resolved by examining the Fisher's statement within the framework of two mathematical theories that are inspired by the Darwinian formalism: evolutionary game theory (EGT) and evolutionary optimization (EO). We present four rigorous formulations (some of them previously reported) of FTNS in four different setups that come from EGT and EO. Our study demonstrates that FTNS in its original form is correct only in certain setups. In order to be recognized as a universal law, the Fisher's statement should be: (a) clarified and completed and (b) relaxed by replacing the words "is equal to" with "does not exceed". Moreover, the real meaning of FTNS can be best understood from the information-geometric point of view. Such an approach shows that FTNS imposes an upper geometric bound on information flows in evolutionary systems. In this light, FTNS appears to be a statement about the intrinsic time scale of an evolutionary system. This leads to a novel insight: FTNS is an analogue of the time-energy uncertainty relation in physics. This further emphasizes a close relation with results on speed limits in stochastic thermodynamics.

Keywords: Evolutionary games; Fisher speed; Information-geometric optimization; Kullback–Leibler divergence; Uncertainty principle.