Dynamic control of agents playing aggregative games with coupling constraints (1609.08962v2)

Abstract: We address the problem to control a population of noncooperative heterogeneous agents, each with convex cost function depending on the average population state, and all sharing a convex constraint, towards an aggregative equilibrium. We assume an information structure through which a central coordinator has access to the average population state and can broadcast control signals for steering the decentralized optimal responses of the agents. We design a dynamic control law that, based on operator theoretic arguments, ensures global convergence to an equilibrium independently on the problem data, that are the cost functions and the constraints, local and global, of the agents. We illustrate the proposed method in two application domains: network congestion control and demand side management.

Dynamic Control in Aggregative Games with Coupling Constraints

The paper "Dynamic Control of Agents playing Aggregative Games with Coupling Constraints" by Sergio Grammatico focuses on controlling a population of competitive, heterogeneous agents interacting in an aggregative game setting, featuring coupling constraints. In this context, each agent seeks to minimize its convex cost function influenced by the average state of the population and constrained by shared resources. The paper presents a dynamic control law devised to guide the collective behavior of these agents towards an aggregative equilibrium.

Key Contributions and Results

1. Framework Description: The paper describes the setup of competitive aggregative games where individual cost functions are influenced by other agents' strategies. A central coordinator broadcasts control signals based on the aggregate population state, facilitating optimal decentralized responses.
2. Dynamic Control Law: Grammatico introduces a dynamic control law using operator theoretic approaches. The law guarantees global convergence to an equilibrium irrespective of the problem data, namely the cost functions and the constraints involved.
3. Mathematical Insights: The paper reveals a multivariable mapping whose zero corresponds to the equilibrium sought. This mapping is a sum of monotone operators, allowing the application of splitting methods from monotone operator theory to ensure convergence.
4. Numerical Simulations: Practical applicability is demonstrated through simulations in domains such as network congestion control and demand side management. The results confirm the efficacy of the proposed dynamic control under varying parameters and population sizes.

Implications

The research implications are twofold:

• Theoretical Developments: The findings contribute to the theory of aggregative games by providing a dynamic mechanism to compute equilibria under coupling constraints, which is traditionally challenging due to non-cooperative agent interactions.
• Practical Applications: The convergence guarantee of the dynamic control law implies potential applications in fields requiring coordination of decentralized agents, like smart grids and telecommunication networks. The agents remain autonomous, needing minimal information exchange, which is crucial for scalability and privacy.

Future Directions

Future research might explore extensions to asynchronous updates, where agents react to control signals at differing times, and stochastic game settings, wherein the agents operate under uncertainty. Moreover, the optimization of control parameters for enhanced convergence rates remains an open area for inquiry. Such advancements could broaden the applicability of this framework in more complex, real-world systemic interactions.

In conclusion, the paper delivers a robust analytical and algorithmic framework for controlling agents in aggregative games, presenting significant theoretical insights and practical methodologies for handling coupling constraints within decentralized systems.