Kolmogorov-Loveland Betting Strategies Fail to Achieve Unbounded Capital on Sequences in Effective Open Sets

核心概念

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.

摘要

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.

統計資料

None.

引述

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從以下內容提煉的關鍵洞見

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

by Tomi... 於 arxiv.org 04-01-2024

https://arxiv.org/pdf/2403.19817.pdf

深入探究

What are the implications of the main result for the relationship between Kolmogorov-Loveland randomness and Martin-Löf randomness

The main result of the paper indicates that Kolmogorov-Loveland betting strategies do not have the property of achieving unbounded capital on certain sequences within effective open sets. This result suggests that there are limitations to the effectiveness of Kolmogorov-Loveland randomness in predicting and capitalizing on certain sequences. In terms of the relationship between Kolmogorov-Loveland randomness and Martin-Löf randomness, this result implies that there are differences in the behavior and capabilities of betting strategies based on these randomness concepts. It raises questions about the equivalence or differences between Kolmogorov-Loveland randomness and Martin-Löf randomness in the context of algorithmic randomness.

Can the techniques used in the paper be extended to other classes of betting strategies beyond Kolmogorov-Loveland strategies

The techniques used in the paper could potentially be extended to other classes of betting strategies beyond Kolmogorov-Loveland strategies. By analyzing the properties and behaviors of different betting strategies in the context of algorithmic randomness, similar methodologies could be applied to explore the effectiveness and limitations of other classes of strategies. This extension could provide insights into the capabilities and constraints of various betting strategies in predicting and capitalizing on sequences within different sets.

How might the concept of conservative betting strategies be useful in other areas of algorithmic randomness or computability theory

The concept of conservative betting strategies, as discussed in the paper, could be useful in other areas of algorithmic randomness or computability theory. Conservative strategies, which ensure that the difference between maximum capital and capital is limited, could be valuable in scenarios where risk management and controlled growth are important. In algorithmic randomness, conservative strategies could be applied to optimize betting behaviors and minimize potential losses. Additionally, in computability theory, conservative strategies could play a role in designing efficient algorithms that prioritize stability and consistent performance. The concept of conservative betting strategies offers a structured approach to managing risk and maximizing returns in various computational contexts.