TWO-PERSON RED-AND-BLACK GAME WITH LOWER LIMIT

Abstract

In this article we consider a two-person red-and-black game with lower limit. More precisely, assume each player holds an integral amount of chips. At each stage, each player can bet an integral amount between a fixed positive integer ℓ and his possession x if x ≥ ℓ; otherwise, he bets all of his own fortune. He might win his opponent's stakes with a probability that is a function of the ratio of his bet to the sum of both players' bets and is called a win probability function. The goal of each player is to maximize the probability of winning the entire fortune of his opponent by gambling repeatedly with suitably chosen stakes. We will give some suitable conditions on the win probability function such that it is a Nash equilibrium for the subfair player to play boldly and for the superfair player to play timidly.