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Irreducibility of enumerable betting strategies
- Publication Year :
- 2021
-
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.
- Subjects :
- Mathematics - Probability
Mathematics - Logic
03D80, 68Q30, 03D32
F.4.1
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2112.14416
- Document Type :
- Working Paper