Monotonous betting strategies in warped casinos

TL;DR

This paper investigates the potential for successful, separable betting strategies in biased roulette scenarios, demonstrating the existence and limitations of such strategies based on the casino sequence’s complexity.

Contribution

It introduces the concept of separable strategies, shows their effectiveness on certain sequences, and establishes their limitations based on Hausdorff dimension.

Findings

Effective mixtures of separable strategies succeed on sequences with dimension < 1/2.

No effective mixture of separable strategies succeeds on sequences with dimension = 1/2.

Results extend to broader classes of strategies.

Abstract

Suppose that the outcomes of a roulette table are not entirely random, in the sense that there exists a successful betting strategy. Is there a successful `separable' strategy, in the sense that it does not use the winnings from betting on red in order to bet on black, and vice-versa? We study this question from an algorithmic point of view and observe that every strategy $M$ can be replaced by a separable strategy which is computable from $M$ and successful on any outcome-sequence where $M$ is successful. We then consider the case of mixtures and show: (a) there exists an effective mixture of separable strategies which succeeds on every casino sequence with effective Hausdorff dimension less than 1/2; (b) there exists a casino sequence of effective Hausdorff dimension 1/2 on which no effective mixture of separable strategies succeeds. Finally we extend (b) to a more general class of strategies.