Title: Learning Dynamics and the Co-Evolution of Competing Sexual Species,
Abstract: We analyze a stylized model of co-evolution between any two purely competing
species (e.g., host and parasite), both sexually reproducing. Similarly to a
recent model of Livnat \etal~\cite{evolfocs14} the fitness of an individual
depends on whether the truth assignments on $n$ variables that reproduce
through recombination satisfy a particular Boolean function. Whereas in the
original model a satisfying assignment always confers a small evolutionary
advantage, in our model the two species are in an evolutionary race with the
parasite enjoying the advantage if the value of its Boolean function matches
its host, and the host wishing to mismatch its parasite. Surprisingly, this
model makes a simple and robust behavioral prediction. The typical system
behavior is \textit{periodic}. These cycles stay bounded away from the boundary
and thus, \textit{learning-dynamics competition between sexual species can
provide an explanation for genetic diversity.} This explanation is due solely
to the natural selection process. No mutations, environmental changes, etc.,
need be invoked.
The game played at the gene level may have many Nash equilibria with widely
diverse fitness levels. Nevertheless, sexual evolution leads to gene
coordination that implements an optimal strategy, i.e., an optimal population
mixture, at the species level. Namely, the play of the many "selfish genes"
implements a time-averaged correlated equilibrium where the average fitness of
each species is exactly equal to its value in the two species zero-sum
competition.
Our analysis combines tools from game theory, dynamical systems and Boolean
functions to establish a novel class of conservative dynamical systems.