Natural Selection as an Inhibitor of Genetic Diversity: Multiplicative Weights Updates Algorithm and a Conjecture of Haploid Genetics

In a recent series of papers a surprisingly strong connection was discovered between standard evolutionary models of natural selection and Multiplicative Weights Updates Algorithm, a ubiquitous model of online learning and optimization. These papers establish that, under specific assumptions, mathematical models of biological evolution can be reduced to studying discrete replicator dynamics, a close variant of MWUA, in coordination games. This connection allows for introducing insights from game theoretic dynamics into the field of mathematical biology. Using these results as a stepping stone, we show that mathematical models of haploid evolution imply the extinction of genetic diversity in the long term limit, a widely believed conjecture in genetics. In game theoretic terms we show that in the case of coordination games, under minimal genericity assumptions, discrete replicator dynamics converge to pure Nash equilibria for all but a zero measure of initial conditions. This result holds despite the fact that mixed Nash equilibria can be exponentially (or even uncountably) many, completely dominating in number the set of pure Nash equilibria. Thus, in haploid organisms the long term preservation of genetic diversity needs to be safeguarded by other evolutionary mechanisms such as mutations and speciation