How to gamble against all odds |
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| Institution: | 1. Department of Agricultural Economics and Rural Development, Agricultural University of Athens, Iera Odos 75, 18855 Athens, Greece;2. INRA UMR210 Economie Publique, Avenue Lucien Brétignières, 78850 Thiverval, Grignon, France;3. Department of Natural Resources Development and Agricultural Engineering, Agricultural University of Athens, Iera Odos 75, 18855 Athens, Greece;1. School of Veterinary Science, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain;2. Wageningen UR Livestock Research, PO Box 338, 6700 AH Wageningen, The Netherlands;1. Faculty of Mathematics, Computer Sciences and Econometrics, University of Zielona Góra, Poland;2. Department of Economics, Arizona State University, USA;3. Department of Quantitative Economics, Warsaw School of Economics, al. Niepodległości 162, 02-554 Warszawa, Poland;1. INRA, UMR1213 Herbivores, Site de Theix, F-63122 Saint-Genès-Champanelle, France;2. Universidade Federal do Rio Grande do Sul, UFRGS, Av. Bento Gonçalves 7712, CEP 91501-970, Porto Alegre-RS, Brazil;3. Universidade Estadual de Maringá, UEM, Av. Colombo 5.790, CEP 87020-900, Maringá-Pr, Brazil |
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| Abstract: | 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). |
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| Keywords: | Repeated games Gambling Algorithmic randomness Pseudo-randomness Predictability |
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