  In this paper, we establish the existence of optimal bounded memory strategy
profiles in multi-player discounted sum games. We introduce a non-deterministic
approach to compute optimal strategy profiles with bounded memory. Our approach
can be used to obtain optimal rewards in a setting where a powerful player
selects the strategies of all players for Nash and leader equilibria, where in
leader equilibria the Nash condition is waived for the strategy of this
powerful player. The resulting strategy profiles are optimal for this player
among all strategy profiles that respect the given memory bound, and the
related decision problem is NP-complete. We also provide simple examples, which
show that having more memory will improve the optimal strategy profile, and
that sufficient memory to obtain optimal strategy profiles cannot be inferred
from the structure of the game.
