2014/08/26 by Ruta Mehta, Ioannis Panageas, Mehta, Ruta +3
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Computational Engineering #Dynamical Systems (math.DS) #FOS: Biological sciences #FOS: Computer and information sciences #FOS: Mathematics #Finance #Quantitative Methods (q-bio.QM) #and Science (cs.CE) #cs.CE #math.DS #q-bio.QM
paper · pdf · doi:10.48550/arxiv.1408.6270
18 pages, 1 figure
arxiv created 2014/10/08 · arxiv updated 2014/10/09
In a recent series of papers a surprisingly strong connection was discovered between standard models of evolution in mathematical biology and Multiplicative Weights Updates Algorithm, a ubiquitous model of online learning and optimization. These papers establish that mathematical models of biological evolution are tantamount to applying discrete Multiplicative Weights Updates Algorithm, a close variant of MWUA, on coordination games. This connection allows for introducing insights from the study of 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 MWUA converges 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.