This paper investigates the relation between Nash equilibria and non-dominated solutions in a special class of games, namely
path player games
. Nash equilibria are situations in a game where none of the players is able to obtain a better outcome by himself. On the other hand, a situation is
if there does not exist a situation which is really better for one of the players, and at least the same for all others. We provide two classes of path player games in which each non-dominated situation is a Nash equilibrium, and one class in which also the reverse is true.