| We consider a kind of dynamical system that has a large number of interacting agents, who have a finite number of resources or strategies (called alternatives) from which they can choose in a given time period. The agents' collective choices lead to costs and benefits for each agent in the given period. Agents then adapt, possibly changing their choice of alternative for the next period, based on the information obtained in the current period. The information available to each agent is partial in the sense that an agent may observe the choices, or the effects of the choices, of only some of the other agents. Agents have different initial opinions about the state of the system, and must adapt relatively slowly since the system as a whole is non-stationary. The adaptive behavior of the agents can be described by stochastic approximation algorithms.; We are interested in studying the loading of such a system, namely what choices of alternative prevail and whether these are stable asymptotically. We consider a law of large numbers limit to capture the effect of large numbers of agents interacting. Because there are so many agents, the environment is noncooperative; agents do not negotiate among themselves before choosing the next alternative. Mathematically, the system is described by a measure-valued stochastic process, the measure-valued state space being necessary to describe the diversity of agents' opinions. This process is interpolated into continuous time, and we prove existence and uniqueness results for the limit process (after letting the number of users go to infinity and time become continuous). Given suitable conditions, we also establish stability results for the process.; Our framework is applicable to any environment that has a large number of interacting agents, who do not necessarily know or agree on the current state of the system. Possible areas of application include discrete choice theory in sociology, island models in economics, traffic assignment problems, and learning in game theory. |
