A Functional Inequality and a New Class of Probabilities in the N-Person Red-and-Black Game

In this paper, we explore a model of an N-player, non-cooperative stochastic game, drawing inspiration from the discrete formulation of the red-and-black gambling problem, as initially introduced by Dubins and Savage in 1965. We extend upon the work of Pontiggia from 2007, presenting a main theorem that broadens the conditions under which bold strategies by all players can achieve a Nash equilibrium. This is obtained through the introduction of a novel functional inequality, which serves as a key analytical tool in our study. This inequality enables us to circumvent the restrictive conditions of super-multiplicativity and super-additivity prevalent in the works of Pontiggia and others. We conclude this paper with a series of illustrative examples that demonstrate the efficacy of our approach, notably highlighting its ability to accommodate a broader spectrum of probability functions than previously recognized in the existing literature.