Analysis Seminar: From Well-Mixed to Networked Populations: Stability Gaps in Replicator Dynamics

Abstract: Evolutionary game theory provides a mathematical framework to describe how the distribution of competing strategies evolves over time in a population under selection. In its classical formulation, the population is assumed to be well mixed, meaning that all individuals are intrinsically identical and interact uniformly with the population as a whole, without spatial or structural constraints. Under this assumption, the dynamics are governed by the replicator equation. For a game with $N$ strategies, this equation defines a nonlinear dynamical system on the $(N-1)$-simplex. While the general high-dimensional case remains challenging, the cases $N=2$ and $N=3$ admit a complete and well-developed theory: games can be fully classified, and fundamental concepts from game theory (such as dominance, Nash equilibria, and evolutionary stability) can be directly interpreted in terms of phase portraits, invariant sets, and stability properties of the associated dynamical systems.
In many realistic settings, interactions are not global but are constrained by an underlying structure, such as a network of interactions. This leads to evolutionary dynamics on graphs, where individuals or subpopulations interact only with their neighbors. The Evolutionary Game on Networks (EGN) model extends the classical replicator dynamics by coupling local population dynamics through the network structure. When interactions are homogeneous and the network is complete, the EGN model reduces to the classical replicator equation and shares the same equilibrium set.
In this talk, we derive the EGN model starting from basic population dynamics and progressively introducing additional layers of complexity, moving from well-mixed populations to structured interactions on networks. We first review the classical theory for evolutionary games with two and three strategies, emphasizing how the classification of games is achieved and how game-theoretic notions are encoded in the geometry and stability of the flow on the simplex. We then discuss the new challenges that arise when attempting to develop a comparable mathematical theory for the EGN model, even for games with a finite number of strategies. Despite the close relationship between the two frameworks, a fundamental qualitative difference emerges: interior equilibria that may be asymptotically stable in the classical replicator dynamics are never asymptotically stable in the networked model. While the existence of interior equilibria depends only on payoff parameters, their stability is entirely dictated by the network structure, revealing a structural stability gap between well-mixed and networked population models.

Host: Brett Wick