Irreducibility of enumerable betting strategies

TL;DR

This paper demonstrates that certain classes of enumerable betting strategies are fundamentally irreducible, showing that complex strategies cannot be simplified into more restricted forms, with implications for understanding randomness and supermartingales.

Contribution

It proves the existence of irreducible members within effectively enumerable betting strategies and extends results on the limitations of simplified supermartingales in detecting randomness.

Findings

Existence of irreducible enumerable betting strategies.

Certain non-1-random reals cannot be succeeded on by restricted supermartingales.

Supports conjecture that natural classes of supermartingales defining 1-randomness cannot be simplified.

Abstract

We study the problem of whether a betting-strategy can be decomposed into an equivalent set of simpler betting-strategies, such as betting-strategies that bet on a restricted set of stages or bet on a restricted of favorable outcomes. We show that the class of effectively enumerable betting-strategies has irreducible members which cannot be decomposed into an equivalent set of simpler betting-strategies. We answer questions of Kastermans and Hitchcock by constructing a real on which no kastergale (which is a left-c.e. supermartingale whose favorable outcomes are effectively determined) succeeds, but some unrestricted left-c.e. supermartingale succeeds on it. We generalize a result of Muchnick by showing that there is a non-1-random real such that no muchgale (which is a left-c.e. supermartingale who does not bet on certain stages) succeeds on it. Our methodology is then used to obtain further irreducibility results, strongly supporting a conjecture that if a natural class of left-c.e. supermartingales defines 1-randomness, then a single member of that class can do so. In another word, the class of left-c.e. supermartingales cannot be reduced to a simpler, natural subclass of it. For example, we show that there is a non-1-random real such that no kastergale or muchgale succeeds on it.