Distributed Nash Equilibrium Seeking for Multiple Coalition Games by Coalition Estimate Strategies

This article studies the Nash equilibrium (NE) seeking problem for multiple coalition games over unbalanced directed graphs, where the players in the same coalition aim to achieve the optimal consensus cooperatively, but different coalitions competitively seek the NE. A distributed algorithm is proposed based on the coalition estimate strategy (CES) and the gradient tracking method. The CES directly estimates the consensus decision in a coalition rather than all decisions of other players, and the decision consensus inside a coalition can also be accomplished by the CES without the extra dynamic. The gradient tracking method is used for estimating the gradient summation inside a coalition. Based on the weighted Frobenius norm and the established linear system of inequalities, it is shown that the proposed algorithm linearly converges to the NE if the maximum value of uncoordinated step sizes is smaller than derived upper bounds for strongly convex cost functions. Furthermore, it is illustrated that the proposed algorithm also applies to distributed optimization problems and networked noncooperative games. Lastly, simulations in formation problems of unmanned vehicle swarms are performed to verify the effectiveness of proposed algorithms.