Evolution as a Mathematical “Game” / Comment

A basic conflict exists: it’s a problem of hierarchy in systems: can evolution be “accounted for” by the extreme abstract reductionism of mathematics? That is, what we encounter as living organisms; the complex details, patterns, and consequences of evolutionary processes over billions of years, is explained by computer models based on a (human selection) of “assumptions” that are not proven nor provable.

Although mathematics are often assumed to offer a “pure” representation of universal “isness” these “languages” may be our most significant inventions; an extremely useful set of tools that attempt to “remove” faulty human cognition from technological endeavors.

Can computer models “explain” the specific consequences of evolutionary processes? No – computer models function on the elimination of “the tough stuff” – that surprisingly, our faulty brains can handle; the messy nitty-gritty of growing and operating a “good-enuff for now” version of life; matter-energy transformations that are insignificant in scale, and yet, of ultimate meaning to us.

As the underlying engine of technical dominance and exploitation of nature, mathematics may also be our worst invention. The seductive and real power of mathematics has exponentially extended the consequences of human behavior, but technology has not improved prediction of the far-reaching results of that behavior. We may say that mathematical systems are devoid of content, therefore, the denial  that nature (the universe of consequences) has content.

 

Algorithms, Games, and Evolution

Erick Chastain, Adi Livnat, Christos Papadimitrious, Umesh Vazirani

Crossmark, April12, 2014 http://www.pnas.org/content/111/29/10620.full

Significance

Theoretical biology was founded on the mathematical tools of statistics and physics. We believe there are productive connections to be made with the younger field of theoretical computer science, which shares with it an interest in complexity and functionality. In this paper, we find that the mathematical description of evolution in the presence of sexual recombination and weak selection is equivalent to a repeated game between genes played according to the multiplicative weight updates algorithm, an algorithm that has surprised computer scientists time and again in its usefulness. This equivalence is informative for the pursuit of two key problems in evolution: the role of sex and the maintenance of variation.

Sample: “Precisely how does selection change the composition of the gene pool from generation to generation? The field of population genetics has developed a comprehensive mathematical framework for answering this and related questions (1). Our analysis in this paper focuses particularly on the regime of weak selection, now a widely used assumption (2, 3). Weak selection assumes that the differences in fitness between genotypes are small relative to the recombination rate, and consequently, through a result due to Nagylaki et al. (4) (see also ref. 1, section II.6.2), evolution proceeds near linkage equilibrium, a regime where the probability of occurrence of a certain genotype involving various alleles is simply the product of the probabilities of each of its alleles. Based on this result, we show that evolution in the regime of weak selection can be formulated as a repeated game, where the recombining loci are the players, the alleles in those loci are the possible actions or strategies available to each player, and the expected payoff at each generation is the expected fitness of an organism across the genotypes that are present in the population. Moreover, and perhaps most importantly, we show that the equations of population genetic dynamics are mathematically equivalent to positing that each locus selects a probability distribution on alleles according to a particular rule which, in the context of the theory of algorithms, game theory, and machine learning, is known as the multiplicative weight updates algorithm (MWUA). MWUA is known in computer science as a simple but surprisingly powerful algorithm (see ref. 5 for a survey). Moreover, there is a dual view of this algorithm: each locus may be seen as selecting its new allele distribution at each generation so as to maximize a certain convex combination of (i) cumulative expected fitness and (ii) the entropy of its distribution on alleles. This connection between evolution, game theory, and algorithms seems to us rife with productive insights; for example, the dual view just mentioned sheds new light on the maintenance of diversity in evolution.”

Game theory has been applied to evolutionary theory before, to study the evolution of strategic individual behavior (see, e.g., refs. 6, 7). The connection between game theory and evolution that we point out here is at a different level, and arises not in the analysis of strategic individual behavior, but rather in the analysis of the basic population genetic dynamics in the presence of sexual reproduction. The main ingredients of evolutionary game theory, namely strategic individual behavior and conflict between individuals, are extraneous to our analysis.

We now state our assumptions and results. We consider an infinite panmictic population of haplotypes involving several unlinked (i.e., fully recombining) loci, where each locus has several alleles. These assumptions are rather standard in the literature. They are made here to simplify exposition and algebra, and there is no a priori reason to believe that they are essential for the results, beyond making them easily accessible. For example, Nagylaki’s theorem (4), which is the main analytical ingredient of our results, holds even in the presence of diploidy and partial recombination.

Much, much more…

 

Note: Wikipedia: Evolutionary Algorithm is a generic term used to describe computer-based problem solving methods based on the concept of biological evolution. They are search algorithms that maintain a population of structures and evolve according to rules of selection as well as other genetic operators like crossover and mutation [8]. Feb 12, 2007

 

 

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