  The problem of multi-agent learning and adaptation has attracted a great deal
of attention in recent years. It has been suggested that the dynamics of multi
agent learning can be studied using replicator equations from population
biology. Most existing studies so far have been limited to discrete strategy
spaces with a small number of available actions. In many cases, however, the
choices available to agents are better characterized by continuous spectra.
This paper suggests a generalization of the replicator framework that allows to
study the adaptive dynamics of Q-learning agents with continuous strategy
spaces. Instead of probability vectors, agents strategies are now characterized
by probability measures over continuous variables. As a result, the ordinary
differential equations for the discrete case are replaced by a system of
coupled integral--differential replicator equations that describe the mutual
evolution of individual agent strategies. We derive a set of functional
equations describing the steady state of the replicator dynamics, examine their
solutions for several two-player games, and confirm our analytical results
using simulations.
