[Paper Review] Multiplicative Updates in Coordination Games and the Theory of Evolution

Erick Chastain, Adi Livnat+2 more|arXiv (Cornell University)|Aug 15, 2012

Evolution and Genetic Dynamics13 references4 citations

TL;DR

This paper establishes that under weak selection, natural selection in sexual reproduction operates as multiplicative updates in a coordination game among genes, where allele frequencies evolve to maximize mixability—the average fitness across genetic partners. The key result is that equilibria can have large supports, implying high genetic diversity is sustained, with a lower bound on the number of equilibria shown to be exponential in the number of alleles.

ABSTRACT

We study the population genetics of Evolution in the important special case of weak selection, in which all fitness values are assumed to be close to one another. We show that in this regime natural selection is tantamount to the multiplicative updates game dynamics in a coordination game between genes. Importantly, the utility maximized in this game, as well as the amount by which each allele is boosted, is precisely the allele's mixability, or average fitness, a quantity recently proposed in [1] as a novel concept that is crucial in understanding natural selection under sex, thus providing a rigorous demonstration of that insight. We also prove that the equilibria in two-person coordination games can have large supports, and thus genetic diversity does not suffer much at equilibrium. Establishing large supports involves answering through a novel technique the following question: what is the probability that for a random square matrix A both systems Ax = 1 and A^T y = 1 have positive solutions? Both the question and the technique may be of broader interest. [1] A. Livnat, C. Papadimitriou, J. Dushoff, and M.W. Feldman. A mixability theory for the role of sex in evolution. Proceedings of the National Academy of Sciences, 105(50):19803-19808, 2008.

Motivation & Objective

• To formalize the connection between population genetics under weak selection and game-theoretic multiplicative update dynamics.
• To rigorously demonstrate that mixability—average fitness across genetic partners—is the key driver of evolutionary dynamics under sex.
• To prove that equilibria in such systems can have large supports, implying sustained genetic diversity.
• To develop a novel mathematical technique for analyzing the existence of positive solutions in linear systems derived from random matrices.
• To extend insights from coordination games and multiplicative updates to evolutionary genetics, particularly in the context of recombination and sex.

Proposed method

• Modeling evolution under weak selection as a coordination game where genes are players, alleles are strategies, and population frequencies are mixed strategies.
• Showing that allele frequency updates follow multiplicative update rules, with the utility being the allele's mixability.
• Using the Wright manifold approximation to reduce genotype dynamics to allele frequency tracking under weak selection.
• Applying a transformation technique to analyze the probability that both Ax=1 and A^T y=1 have positive solutions for random matrices.
• Proving that the set of matrices with positive inverse row and column sums has measure at least 2^{-(2n-1)} in the unit cube.
• Deriving a lower bound on the number of equilibria using combinatorial approximations and matrix analysis.

Experimental results

Research questions

• RQ1 How does the multiplicative update rule in coordination games relate to allele frequency dynamics in population genetics under weak selection?
• RQ2 What is the role of mixability in evolutionary dynamics under sexual reproduction, and how does it differ from traditional fitness maximization?
• RQ3 Under what conditions do equilibria in evolutionary systems have large supports, implying high genetic diversity?
• RQ4 What is the probability that both Ax=1 and A^T y=1 have positive solutions for a random square matrix A?
• RQ5 Can the diversity and equilibrium structure in multi-gene systems be characterized using similar techniques?

Key findings

• The dynamics of allele frequency change under weak selection are mathematically equivalent to multiplicative updates in a coordination game between genes.
• The utility maximized in this game is precisely the mixability of the allele, defined as its average fitness across genetic partners.
• Equilibria in two-player coordination games can have large supports, implying that genetic diversity is not significantly reduced at equilibrium.
• The probability that a random matrix A in [-1,1]^{n×n} satisfies both Ax=1 and A^T y=1 having positive solutions is at least 2^{-(2n-1)}.
• The expected number of k×k equilibria in an m×n weak selection fitness matrix is at least 2(mn/(4k²))^k, which is exponential in k for k < (1/2)√(mn).
• The model suggests that sex is not a bug but a feature of evolution, as recombination promotes high-mixability alleles rather than fixed optimal combinations.