# Adaptive Learning in Large Populations

## TL;DR

This paper models adaptive learning in large populations using a PDE derived from Harley's rule, revealing that faster memory decay benefits evolution in certain 2x2 games.

## Contribution

It introduces a PDE framework for stochastic adaptive learning in large populations, connecting behavioral rules to population dynamics.

## Key findings

- Faster memory decay provides an evolutionary advantage in some games.
- The PDE model captures the conservation of stimuli in behavior selection.
- Analysis of 2x2 games shows conditions favoring different learning speeds.

## Abstract

We consider the adaptive learning rule of Harley (1981) for behavior selection in symmetric conflict games in large populations. The rule uses organisms' past, accumulated rewards as the predictor for the future behavior, and can be traced in many life forms from bacteria to humans. We derive a partial differential equation (PDE) that describes the stochastic learning in a population of agents. The equation has simple structure of the `conservation of mass'-type equation in the space of stimuli to engage in a particular type of behavior. We analyze the solutions of the PDE model for typical 2x2 games. It is found that in games with small residual stimuli, adaptive learning rules with faster memory decay have an evolutionary advantage.