  This paper studies an $N$-coalition non-cooperative game problem, where the
players in the same coalition cooperatively minimize the sum of their local
cost functions under a directed communication graph, while collectively acting
as a virtual player to play a non-cooperative game with other coalitions.
Moreover, it is assumed that the players have no access to the explicit
functional form but only the function value of their local costs. To solve the
problem, a discrete-time gradient-free Nash equilibrium seeking strategy, based
on the gradient tracking method, is proposed. Specifically, a gradient
estimator is developed locally based on Gaussian smoothing to estimate the
partial gradients, and a gradient tracker is constructed locally to trace the
average sum of the partial gradients among the players within the coalition.
With a sufficiently small constant step-size, we show that all players' actions
approximately converge to the Nash equilibrium at a geometric rate under a
strongly monotone game mapping condition. Numerical simulations are conducted
to verify the effectiveness of the proposed algorithm.
