Distributed Proportional-Integral Algorithms for Multiple Coalition Games Under Limited Communication Resources

This paper studies the algorithm design for multiple coalition games under limited communication resources, in which the players in the same coalition cooperatively optimize the summation of cost functions in this coalition and do not care about the costs of other coalitions. To address this game, we develop a distributed proportional-integral algorithm based on the coalition estimate strategy and the proportional-integral principle. Furthermore, when the communication resource is concretely quantified by bit rates in communication channels, we propose a coding-decoding-based distributed proportional-integral algorithm based on the distributed proportional-integral algorithm and coding-decoding rules for seeking the Nash equilibrium of multiple coalition games. It proves that both algorithms linearly and precisely converge to the Nash equilibrium in spite of limited communication resources. Then, the necessary and sufficient condition for the linear convergence of the proposed algorithm about the requirement of bit rates is presented. Moreover, the relationship between the bit rate and the convergence speed of the proposed algorithm is also theoretically explained. Lastly, the simulation in formation problems of unmanned vehicle swarms is presented to demonstrate the effectiveness of proposed algorithms.