A gambling system and a Markov chain

Abstract

"Oscar's system" is a gambling system in which the aim is to win one betting unit, at least with high probability, and then start over again. The system can be modeled by an irreducible Markov chain in a subset of the two-dimensional integer lattice. We show that the Markov chain, which depends on a parameter p representing the single-trial win probability, is transient if $p < 1/2$ and positive recurrent if $p \geq 1/2$.