  In this paper, Nash equilibrium seeking among a network of players is
considered. Different from many existing works on Nash equilibrium seeking in
non-cooperative games, the players considered in this paper cannot directly
observe the actions of the players who are not their neighbors. Instead, the
players are supposed to be capable of communicating with each other via an
undirected and connected communication graph. By a synthesis of a
leader-following consensus protocol and the gradient play, a distributed Nash
equilibrium seeking strategy is proposed for the non-cooperative games.
Analytical analysis on the convergence of the players' actions to the Nash
equilibrium is conducted via Lyapunov stability analysis. For games with
non-quadratic payoffs, where multiple isolated Nash equilibria may coexist in
the game, a local convergence result is derived under certain conditions. Then,
a stronger condition is provided to derive a non-local convergence result for
the non-quadratic games. For quadratic games, it is shown that the proposed
seeking strategy enables the players' actions to converge to the Nash
equilibrium globally under the given conditions. Numerical examples are
provided to verify the effectiveness of the proposed seeking strategy.
