Distributed fictitious play in potential games of incomplete information

We consider a networked multi-agent system with common unknown state of the world. A potential payoff function, that depends on the actions of agents and the state of the world, captures the system´s global well-being. Agents with different information about the state of the world needs to reason about the actions of others to maximize the payoff. We introduce the distributed fictitious play algorithm as a decentralized decision-making model given only local network information in this setup. In the algorithm, agents observe past actions of their neighbors and keep an empirical distribution on the centroid population action. In addition, agents form beliefs on the state of the world through a parallel state learning process. At each stage, agents take an action that maximizes the expected global payoff assuming that others are going to play with respect to their estimated centroid empirical distribution given their belief on the state of the world. We show that this behavior converges to a consensus Nash equilibrium strategy when the potential payoff function is symmetric and agents reach a consensus in their beliefs on the state of the world fast enough. We exemplify the convergence behavior of the algorithm in a coordination game.