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.