Assume an N-persons game where each player i has a pay-off function f_i(X). X is the joint strategy of all players. So pay-off function f_i depends on the i-th player's strategy and on the strategies of the other players. (Strategy sets are continuous).

Now assume that there exists a Nash equilibrium X* so that EACH player is in a global maximum of its pay-off function.

(this means there exists an X* for which f_i(X*)=max f_i(X) for all i)

Does this case exist somewhere in literature? I would like to find some sufficient conditions for convergence for these kind of games.