If we direct our attention to the impact that agent actions have on others, we notice that if an agent's choice of best action is not impacted by the other agents' actions then its learning task reduces to that of learning to match the fixed function . That is, Mi will be fixed if agent i's choice of action depends solely on the world state w. There are many learning algorithms available that allow an agent to learn such a function. We assume the agent uses one such algorithm. From this reasoning it follows that: If agent i's actions are not impacted by other agents and it is capable of learning a fixed function, then it will eventually learn gi(w) = Mi(w) and will stay fixed after that.
If, on the other hand, the agent's choice of action is impacted by the other agents' actions and the other agents are changing their behavior, then we find that there is no constant Mi(w) function to learn. The MAS becomes a complex adaptive system. However, even in this case, it is still possible that all agents will all eventually settle on a fixed set of gi(w) = Mi(w). If this happens then eventually (and concurrently), the agents will all learn the set of best actions to take in each world state, and these will not change much, if at all. At this point, we say that the system has converged.
Convergence is a general phenomena that goes by many names. For example, if the system is an instance of the Pursuit Problem, we might say that the agents had agreed on a set of conventions for dealing with all situations, while in a market system, we would say that the system had reached a competitive equilibrium.
Unfortunately, we do not have any general rules for predicting which systems will converge, and which will not. These predictions can only be made by examining the particular system (e.g. under certain circumstances we can predict that a market system will reach a price equilibrium). Still, we can say that:
Proof If M is fixed then, eventually, the knowledge of type will become fixed, and so will the such that the agent will actually just have a (very complicated) function of w. Therefore, all the knowledge can be collapsed into knowledge of the form Ki(fi(w)) without losing any information.
If, on the other hand, the system has not converged then we find that deeper models are sometimes better than shallow ones, depending on exactly what knowledge the agents are trying to learn, how they are doing it and certain aspects of the structure of this knowledge, as we shall see in the next section.