Betting strategies with bounded splits

TL;DR

This paper investigates limitations of pairs of Kolmogorov-Loveland betting strategies in winning on non-Martin-Löf random sequences under certain bounded betting conditions.

Contribution

It establishes that such betting strategies cannot universally succeed if they are restricted by specific unbounded or sublinear bounds on the number of betting positions.

Findings

Pairs of betting strategies cannot win on all non-Martin-Löf random sequences under certain bounds.

The results apply to strategies with both unbounded and sublinear betting position constraints.

The work advances understanding of limitations in algorithmic randomness and betting strategies.

Abstract

We show that a pair of Kolmogorov-Loveland betting strategies cannot win on every non-Martin-L\"of random sequence if either of the two following conditions is true: (I) There is an unbounded computable function $g$ such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all $\ell$, bet on at most $\ell-g(\ell)$ positions among the first $\ell$ positions of the sequence. (II) There is a sublinear function $g$ such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all $\ell$, bet on at least $\ell-g(\ell)$ positions among the first $\ell$ positions of the sequence.