Stability and Diversity in Collective Adaptation

TL;DR

The paper addresses how collective adaptation emerges when many agents with memory loss reinforce actions in response to environmental feedback. It derives a general set of continuous-time differential equations from a discrete-time stochastic model, connecting dynamics to game-theoretic and information-theoretic ideas. The study reveals a rich spectrum of behaviors—from fixed points and neutral cycles to Hamiltonian and chaotic dynamics—across two-, three-, and multi-agent interactions, including nontransitive games and memory-driven dissipation. These results provide a unified, information-theoretic framework for understanding stability and diversity in adaptive collectives with potential applications to biology, social systems, and learning agents, while suggesting extensions to networks and stochastic settings.

Abstract

We derive a class of macroscopic differential equations that describe collective adaptation, starting from a discrete-time stochastic microscopic model. The behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior. We show that, although individual agents interact with their environment and other agents in a purely self-interested way, macroscopic behavior can be interpreted as game dynamics. Application to several familiar, explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors. The simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation. We also analyze the adaptation dynamics from an information-theoretic viewpoint and discuss self-organization induced by information flux between agents, giving a novel view of collective adaptation.

Figures (23)

• Figure 1: The time scale ($t$) of a single agent interacting with its environment and the time scale ($\tau$) of the agent's adaptation: $\tau \ll t$.
• Figure 2: A dynamic balance of adaptation and memory loss: Adaptation concentrates the probability distribution on the best action. Memory loss of past history leads to a distribution that is flatter and has higher entropy.
• Figure 3: The time scales of dynamic adaptation: Agent adaptation is slow compared to agent-environment interaction and environmental change is slower still compared to adaptation.
• Figure 4: Dynamics of single-agent adaptation: Here there are three actions, labeled $1$, $2$, and $3$, and the environment gives reinforcements according to ${\bf a}=(\frac{2}{3}\epsilon, -1-\frac{1}{3}\epsilon, 1-\frac{1}{3}\epsilon)$. The figure shows two trajectories from simulations with $\epsilon = 0.5$ and $\beta=0.1$ and with $\alpha = 0.0$ (right) and $\alpha = 0.3$ (left).
• Figure 5: Dynamics of zero-sum interaction without memory loss: Constant of motion $E = \beta_X^{-1} D({\bf x}^{\ast}\parallel{\bf x}) + \beta_Y^{-1} D({\bf y}^{\ast}\parallel{\bf y})$ keeps the linear sum of distance between the interior Nash equilibrium and each agent's state.