A THEORY FOR ADAPTATION AND COMPETITION APPLIED TO LOGISTIC MAP DYNAMICS

We present a theory for adaptive, predictive, and competitive agents i n an evolving chaotic environment. The agents are simple algorithmic a gents designed to model, predict, and exert open loop control on their environment. The environment is the time iteration of the logistic ma p with external noise added. We find that passive agents can make accu rate single step predictions even if the environmental dynamics is cha otic, while accurate multiple step predictions are possible only if th e Liapunov exponent of the environmental dynamics is negative. Multipl e step predictions with high precision can be made over a broad range of conditions when one agent exerts control on the environment. When t wo 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 stabl e or unstable fixed point of the goal dynamics of the first agent. Whe n 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 constan t value while the amplitudes of the driving forces grow at a constant rate. Further, our studies suggest that generally the agent with the m ore complicated goal dynamics may achieve an extremely small predictio n error by a perfect entrainment of the environmental dynamics.