Conceitos Básicos
Kolmogorov-Loveland betting strategies do not have the property that for any given bound, when betting on a binary sequence contained in an effective open set of small enough measure, at least one of the betting strategies in the set earns capital larger than the bound.
Resumo
The paper investigates the relationship between Kolmogorov-Loveland randomness (KLR) and Martin-Löf randomness (MLR). It has been shown that more general classes of betting strategies than Kolmogorov-Loveland ones, such as general betting strategies, exhaustive betting strategies, and balanced betting strategies, contain a finite set of strategies that can earn unbounded capital when betting on sequences in effective open sets of small enough measure.
The main result of the paper is that this property does not hold for the class of Kolmogorov-Loveland betting strategies. Specifically, the paper introduces a game between a Chooser and a Gambler, and shows that if the Chooser has a computable winning strategy in this game, then for every Kolmogorov-Loveland betting strategy, there is a bound on the capital such that for any given size parameter, the Chooser can construct an effective open set of measure less than the given size parameter that contains a sequence on which the maximal achieved capital of every Kolmogorov-Loveland betting strategy is below the strategy's bound.
The paper also introduces the concept of conservative betting strategies, where the difference between the maximal capital and the current capital is always less than 2. It is shown that if the Chooser has a computable winning strategy against conservative Gamblers, then the Chooser also has a computable winning strategy against any kind of Gambler.