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

TL;DR

This paper introduces a generalized probabilistic model for an N-person red-and-black game, expanding the class of probability functions for Nash equilibria without restrictive assumptions, and broadening previous theoretical results.

Contribution

It generalizes existing results on Nash equilibria in multi-player red-and-black games by introducing a new functional inequality that relaxes previous assumptions.

Findings

Broader class of probability functions identified

Nash equilibrium conditions extended to more general settings

Examples demonstrate the wider applicability of the approach

Abstract

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.