We study the computational complexity of deciding the existence of a Pure Nash Equilibrium or a subgame perfect Nash equilibrium with a given payoff and other related problems in finite multi-player extensive games with perfect information. We propose three ways of representing a game with different degrees of succinctness for the components of the game. We show that when the number of moves of each player is large and the player function and the utilities are represented succinctly the considered problems are
-complete. In contraposition, when the game is described extensively by means of its associated tree all the problems are decidable in polynomial time.