  We discuss a model of a $N$-person, non-cooperative stochastic game, inspired
by the discrete version of the red-and-black gambling problem introduced by
Dubins and Savage in 1965. Our main theorem generalizes a result of Pontiggia
from 2007 which provides conditions upon which bold strategies for all players
form a Nash equilibrium. Our tool is a functional inequality introduced and
discussed in the present paper. It allows us to avoid rather restrictive
assumptions of super-multiplicativity and super-additivity, which appear in
Pontiggia's and other authors' works. We terminate the paper with some examples
which in particular show that our approach leads to a larger class of
probability functions than existed in the literature so far.
