Bold Processes

Dr. Kyle Siegrist

Department of Mathematical Sciences
University of Alabama in Huntsville

September 4, 1998

Abstract

In the game of red and black, a player bets, at even stakes, on a sequence of Bernoulli trials with success probability p, until she either reaches a fixed goal or is ruined. A famous result holds that in the unfair case (p < 1/2), an optimal strategy is to play boldly, betting either the entire fortune or just what is needed to reach the target, whichever is smaller. The mathematical analysis of bold play leads to some exotic results concerning the win probability and the expected number of trials. Moreover, bold play is not uniquely optimal, but can be "re-scaled" in a certain way to produce infinitely many optimal strategies. In all of these results, the binary rationals play a critical role.

In this talk, we will explore a class of Markov processes that generalize and illuminate bold play in red and black. A bold process is one that follows a deterministic, discrete dynamical system until the (random) time that the process enters a fixed, terminal set of states. We will give results for the hitting distribution on the terminal set, the expected hitting time, and the construction of re-scaled processes. The states of finite rank relative to the deterministic map and the terminal set play the role that the binary rationals play in classical red and black. As a special case, we will explore bold play with k-players.