Monotonous betting strategies in warped casinos

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.

Irreducibility of enumerable betting strategies

“…• the product of two computable fixed-outcome martingales [Barmpalias et al, 2020, §3] • the product of two computable martingales that bet on alternate stages [Muchnik, 2009, §2] so in a fully constructive environment, (a), (b) have positive answers. We refer to infinite binary sequences as reals.…”

“…We focus on supermartingales for the sake of generality; the above result remains true if we replace 'supermartingale' with 'martingale'. Barmpalias et al [2020] obtained a weak analogue of Theorem 1.2, which is restricted to effective countable mixtures of computable martingales, a subclass of left-c.e. martingales that corresponds to the condition that the wagers are left-c.e.…”

“…betting-strategies by a more refined examination of the complexity of the reals where they fail. In the case of the class M of effective mixtures of single-sided martingales, Barmpalias et al [2020] showed that there exists a real of effective Hausdorff dimension dim ( ) = 1/2 such that no betting-strategy in M succeeds on ; moreover, this is optimal in the sense that some betting-strategy in M succeeds on all reals with dim ( ) < 1/2.…”