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

Abstract

We discuss a model of a two-person, non-cooperative stochastic game, inspired by the discrete version of the red-and-black gambling problem presented by Dubins and Savage. Assume that two players hold certain amounts of money. At each stage of the game they simultaneously bid some part of their current fortune and the probability of winning or loosing depends on their bids. In many models of the red-and-black game it is assumed that the win probability is a function of the quotient of the bid of the first player and the sum of both bids. In the literature some additional properties, like concavity or super-multiplicativity are assumed in order to ensure that the bold and timid strategies form a Nash equilibrium, which turns out to be unique in several cases. Our aim is to provide a generalization in which the probability of winning is a two-variable function which depends on both bids. We introduce two new functional inequalities whose solutions lead to win probability functions for which a Nash equilibrium is realized by the bold-timid strategy. Since both inequalities may appear as not quite intuitive, we discuss them in a separate section of the paper and we provide some illustrating examples.