A theory for adaptation and competition applied to logistic map dynamics

We present a theory for adaptive, predictive, and competitive agents in an evolving chaotic environment. The agents are simple algorithmic agents designed to model, predict, and exert open loop control on their environment. The environment is the time iteration of the logistic map with external noise added. We find that passive agents can make accurate single step predictions even if the environmental dynamics is chaotic, while accurate multiple step predictions are possible only if the Liapunov exponent of the environmental dynamics is negative. Multiple step predictions with high precision can be made over a broad range of conditions when one agent exerts control on the environment. When two agents are simultaneously attempting control of the environment an agent will achieve the smallest prediction error when the second agent's goal dynamics has a stable fixed point which coincides with a stable or unstable fixed point of the goal dynamics of the first agent. When the fixed points of the goal dynamics of the two agents do not match, we find that the prediction errors of both agents approach a constant value while the amplitudes of the driving forces grow at a constant rate. Further, our studies suggest that generally the agent with the more complicated goal dynamics may achieve an extremely small prediction error by a perfect entrainment of the environmental dynamics.