Generalizations of bold play in red and black.
In the game of red and black the player starts with an initial fortune \(x\) (normalized to lie in \([0,1]\)), and then bets on a sequence of Bernoulli trials, until he either reaches the target (1), or is ruined (0). A famous result is that in the subfair case, an optimal strategy is ``bold play'' whereby at each trial the player bets \(\min (x,1-x)\), where \(x\) is the fortune. In addition to optimality, the bold strategy has a number of interesting properties. In particular before the game ends, the fortune process is deterministic, following the map \(x\mapsto 2x\) mod 1. The purpose of this paper is to study relevant Markov chains in a more general setting that preserves some of these properties. A general class of partially deterministic chains is introduced. These chains follow a deterministic map before entry into a given set of states \(D\). Results for the distribution of the hitting time and place are obtained. Further, chains on the sequence space \(\{0,1,\dots ,k-1\}^\infty \) are studied. When \(k=2\) and the sequences are interpreted as the binary coordinates of the player's fortune, the chain corresponds to standard bold play in red and black. For \(k>2\) the models obey similar properties but, unfortunately, the chains on the sequence space do not seem to have a natural gambling interpretation. Finaly, Markov chains that naturally generalize bold play with \(k\) players are studied.
scientific article; zbMATH DE number 2102380