Generalizations of bold play in red and black

The strategy of bold play in the game of red and black leads to a number of interesting mathematical properties: the playerʹs fortune follows a deterministic map, before the transition that ends the game; the bold strategy can be “re-scaled” to produce new strategies with the same win probability; the win probability is a continuous function of the initial fortune, and in the fair case, equals the initial fortune. We consider several Markov chains in more general settings and study the extent to which the properties are preserved. In particular, we study two “k-player” models.