New inequalities for probability functions in the two-person red-and-black game

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

“…This, joined with inequality (10) applied for functions f and g for both players, after an easy calculation leads to the same functional inequality that appeared in [4] in a bit different situation, where bold strategy was shown to be best response to the timid strategy ([4, Theorem 1 and inequality ( 7) therein]).…”

“…In this paper we will explore further the last discussed game, namely N -person game, where the winning sum in non-constant. Based also on [4], where a generalization for results of [1] has been given, we will show a less restrictive model. Then, we present examples of the functions that meet assumptions of our main theorem and thus we show that we extend the main result of [8].…”

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

“…A special case of the above theorem is when N = 2 and with no possibility of winning by the casino. In this situation, condition (5) simplifies, which together with inequality (12) applied to functions f j and g j for both players after an easy calculation, leads to the same functional inequality that appeared in [8] in a bit of a different situation, where the bold strategy was shown to be the best response to the timid strategy ([8] (Theorem 1 and inequality (7) therein)).…”

“…This paper delves deeper into the previously discussed N-person red-and-black game with a non-constant winning sum, expanding upon the work presented in [7] by L. Pontiggia. We draw inspiration from [8], which generalizes the results of Chen and Hsiau [3]. By leveraging these insights, we aim to construct a more general model that surpasses the limitations of previously explored scenarios.…”

“…We will establish a more general model that surpasses the generality of existing models by incorporating a non-constant winning sum and building upon the advancements outlined in [8].…”