If You Must Gamble: Minimizing Expected Loss

We consider a typical gambling situation such as red/black roulette bets of a fixed size, with a limited bankroll, a fixed goal (such as doubling one’s money), and a maximum time of play. Through the use of Markov chains, we analyze the expected value of this game for various bet sizes. Unlike in the classical Gambler’s Ruin problem with no time constraint, we will see there really are “good” and (very!) “bad” bet sizes, which depend on the values of the various constraints.