Kolmogorov-Loveland betting strategies lose the Betting game on open sets

Abstract

Whether Kolmogorov-Loveland randomness (KLR) is the same as Martin-L\"of randomness (MLR) is a major open problem in the study of algorithmic randomness. More general classes of betting strategies than Kolmogorov-Loveland ones have been studied in \cite{MMS, Rute, TP}. In each case it was proven that the class induces a notion of randomness equivalent to MLR. In all of those proofs, it was shown that the class contains a finite set of betting strategies such 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. We show that the class of Kolmogorov-Loveland betting strategies does not have this property.