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We discuss a model of a $N$-person, non-cooperative stochastic game, inspiredby the discrete version of the red-and-black gambling problem introduced byDubins and Savage in 1965. Our main theorem generalizes a result of Pontiggiafrom 2007 which provides conditions upon which bold strategies for all playersform a Nash equilibrium. Our tool is a functional inequality introduced anddiscussed in the present paper. It allows us to avoid rather restrictiveassumptions of super-multiplicativity and super-additivity, which appear inPontiggia's and other authors' works. We terminate the paper with some exampleswhich in particular show that our approach leads to a larger class ofprobability functions than existed in the literature so far.

A new class of probabilities in the n-person red-and-black game