A class of distributed adaptive pricing mechanisms for societal systems with limited information

In this paper, we present a class of learning dynamics for distributed adaptive pricing in affine congestion games. We consider the setting where a large population of users is faced with the problem of choosing between a finite number of available resources, each resource having a particular cost function that depends only on the share of users using that particular resource. Since the mass of users is constant, their individual decisions affect the performance of all the available resources, thus generating a population game where each resource can be seen as a particular strategy in the game. Given the well-known fact that Nash equilibria in population games may not be socially optimal, a social planner is faced with the challenge of designing incentive mechanisms that induce a socially optimal Nash equilibrium. To achieve this, we present in this paper a class of model-free distributed pricing algorithms that guarantee convergence to the set of optimal tolls that induce a socially optimal Nash equilibrium. Our results allow us to consider populations of users that react instantaneously to tolls, as well as populations with social dynamics. Since the algorithms are distributed and data-driven, they can be implemented in settings where full information of the game is not available. By combining tools from game theory, robust set-valued dynamical systems, and adaptive control, a convergence result is established.