How to gamble against all odds

• 成果类型: Article
• 署名作者: Bavly, Gilad; Peretz, Ron
• 署名单位: Bar Ilan University
• 刊物名称: GAMES AND ECONOMIC BEHAVIOR
• ISSN/ISSBN: 0899-8256
• DOI: 10.1016/j.geb.2015.10.006
• 发表日期: 2015
• 页码: 157-168
• 关键词: Repeated games; gambling; Algorithmic randomness; Pseudo-randomness; predictability

摘要

We compare the power of betting strategies (aka martingales) whose wagers take values in different sets of reals. A martingale whose wagers take values in a set A is called an A-martingale. A set of reals B anticipates a set A, if for every A-martingale there is a countable set of B-martingales, such that on every binary sequence on which the A-martingale gains an infinite amount at least one of the B-martingales gains an infinite amount, too. We show that for two important classes of pairs of sets A and B, B anticipates A if and only if the closure of B contains rA, for some positive r. One class is when A is bounded and B is bounded away from zero; the other class is when B is well ordered. Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012). (C) 2015 Elsevier Inc. All rights reserved.