  This paper considers the design of fully distributed Nash equilibrium seeking
strategies for multi-agent games. To develop fully distributed seeking
strategies, two adaptive control laws, including a node-based control law and
an edge-based control law, are proposed. In the node-based adaptive strategy,
each player adjusts their own weight on their procurable consensus error
dynamically. Moreover, in the edge-based algorithm, the fully distributed
strategy is designed by adjusting the weights on the edges of the communication
graph adaptively. By utilizing LaSalle's invariance principle, it is shown that
the Nash equilibrium is globally asymptotically stable by both strategies given
that the players' objective functions are twice-continuously differentiable,
the partial derivatives of the players' objective functions with respect to
their own actions are globally Lipschitz, the pseudo-gradient vector of the
game is strongly monotone and the communication network is undirected and
connected. In addition, we further show that the edge-based method can be
easily adapted to accommodate time-varying communication conditions, in which
the communication network is switching among a set of undirected and connected
graphs. In the last, a numerical example is given to illustrate the
effectiveness of the proposed methods.
