  Purpose: In recent years Monte-Carlo sampling methods, such as Monte Carlo
tree search, have achieved tremendous success in model free reinforcement
learning. A combination of the so called upper confidence bounds policy to
preserve the "exploration vs. exploitation" balance to select actions for
sample evaluations together with massive computing power to store and to update
dynamically a rather large pre-evaluated game tree lead to the development of
software that has beaten the top human player in the game of Go on a 9 by 9
board. Much effort in the current research is devoted to widening the range of
applicability of the Monte-Carlo sampling methodology to partially observable
Markov decision processes with non-immediate payoffs. The main challenge
introduced by randomness and incomplete information is to deal with the action
evaluation at the chance nodes due to drastic differences in the possible
payoffs the same action could lead to. The aim of this article is to establish
a version of a theorem that originated from population genetics and has been
later adopted in evolutionary computation theory that will lead to novel
Monte-Carlo sampling algorithms that provably increase the AI potential. Due to
space limitations the actual algorithms themselves will be presented in the
sequel papers, however, the current paper provides a solid mathematical
foundation for the development of such algorithms and explains why they are so
promising.
