[Paper Review] Learning Dynamics and the Co-Evolution of Competing Sexual Species

Georgios Piliouras, Leonard J. Schulman|arXiv (Cornell University)|Nov 18, 2017

Evolutionary Game Theory and Cooperation41 references4 citations

TL;DR

This paper proposes a game-theoretic model of co-evolution between two sexually reproducing species in a zero-sum competition, where fitness depends on Boolean functions of genetic traits. Using replicator dynamics, it demonstrates that the system exhibits robust, bounded periodic cycles—preventing genetic monoculture without requiring mutations or environmental changes—thereby explaining sustained genetic diversity through natural selection alone.

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 extit{periodic}. These cycles stay bounded away from the boundary and thus, extit{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.

Motivation & Objective

• To explain the persistence of genetic diversity in sexual species without invoking mutations, speciation, or environmental changes.
• To model co-evolution between two competing species (e.g., host-parasite) as a team game where genes act selfishly but coordinate via learning dynamics.
• To analyze the long-term behavior of replicator dynamics in a conservative, high-dimensional system arising from competing teams of genes.
• To establish that time-averaged play converges to optimal strategies in a zero-sum game, even though individual agents do not converge to equilibrium.

Proposed method

• Models two sexually reproducing species as teams of genes with aligned interests, competing via a 2×2 zero-sum game based on Boolean functions of their genotypes.
• Applies continuous-time replicator dynamics to model gene frequency evolution, where fitness depends on whether a Boolean function is satisfied.
• Introduces a (p, q, r, w)-planar dynamical system to describe the evolution of average gene frequencies, showing it is conservative and periodic.
• Uses Lyapunov functions and the Poincaré-Bendixson theorem to prove periodicity of trajectories for almost all initial conditions.
• Analyzes time-averaged behavior using integration over periodic orbits to show convergence to correlated equilibria and minmax strategies.
• Establishes that the time-averaged utility of agents matches the value of the team in the underlying zero-sum game, despite non-convergent, oscillatory dynamics.

Experimental results

Research questions

• RQ1 Can natural selection alone maintain genetic diversity in sexual species without mutations or environmental shifts?
• RQ2 What dynamical behavior emerges when two sexually reproducing species co-evolve in a zero-sum game based on Boolean fitness functions?
• RQ3 Do learning dynamics among selfish genes lead to coordinated, optimal population-level strategies despite non-equilibrium, oscillatory behavior?
• RQ4 Can time-averaged play in such systems implement minmax strategies of the team game, even when individual trajectories do not converge?

Key findings

• The system exhibits robust, bounded periodic cycles for almost all initial conditions, preventing allele frequencies from reaching fixation (i.e., 0 or 1).
• The time-averaged gene frequencies converge to the unique fully mixed Nash equilibrium of the 2×2 zero-sum game, ensuring optimal population-level coordination.
• The time-averaged expected utility of each agent converges to the value of the team in the zero-sum game, equivalent to playing the optimal minmax strategy.
• Despite non-convergent, oscillatory dynamics, the system implements a correlated equilibrium in the long run, with agents achieving the value of their team's optimal strategy.
• The dynamics are conservative, with a conserved quantity (Lyapunov function), and periodicity is proven via the Poincaré-Bendixson theorem applied to the planar system.
• The results hold under weak selection and in the infinite-population, continuous-time limit, with implications for finite populations being limited to moderate time scales.