Probabilistic vs deterministic gamblers

TL;DR

This paper explores the limits of probabilistic versus deterministic strategies in gambling within the framework of algorithmic randomness, introducing a new notion of almost everywhere computable randomness and demonstrating key separations.

Contribution

It introduces the concept of almost everywhere computable randomness and establishes its separation from partial computable randomness in certain Turing degrees.

Findings

Constructed a sequence that is partial computably random but not a.e. computably random.

Proved the separation between a.e. computable randomness and partial computable randomness.

Identified the exact Turing degrees where this separation occurs.

Abstract

Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion -- almost everywhere computable randomness. A binary sequence $X$ is a.e.\ computably random if there is no probabilistic computable strategy which is total and succeeds on $X$ for positive measure of oracles. Using the fireworks technique we construct a sequence which is partial computably random but not a.e.\ computably random. We also prove the separation between a.e.\ computable randomness and partial computable randomness, which happens exactly in the uniformly almost everywhere dominating Turing degrees.