Peretz, Ron and Bavly, Gilad (2014) How to gamble against all odds. Games and Economic Behavior. ISSN 08998256 (Submitted)

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Abstract
A decision maker observes the evolving state of the world while constantly trying to predict the next state given the history of past states. The ability to benefit from such predictions depends not only on the ability to recognize patters in history, but also on the range of actions available to the decision maker. We assume there are two possible states of the world. The decision maker is a gambler who has to bet a certain amount of money on the bits of an announced binary sequence of states. If he makes a correct prediction he wins his wager, otherwise he loses it. 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 Amartingale. A set of reals B anticipates a set A, if for every Amartingale there is a countable set of Bmartingales, such that on every binary sequence on which the A martingale gains an infinite amount at least one of the Bmartingales 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 r A, 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 (has no leftaccumulation points). Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012).
Item Type:  Article 

Official URL:  http://www.journals.elsevier.com/gamesandeconomi... 
Additional Information:  © 2014 Elsevier Inc. 
Divisions:  Mathematics 
Subjects:  Q Science > QA Mathematics 
JEL classification:  C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C73  Stochastic and Dynamic Games; Evolutionary Games; Repeated Games 
Date Deposited:  25 Sep 2014 13:54 
Last Modified:  31 Jul 2024 14:16 
Projects:  538/11 and 323/13, 2010253. 
Funders:  ISF, BSF 
URI:  http://eprints.lse.ac.uk/id/eprint/59542 
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