Aspects of Muchnik's paradox in restricted betting
Abstract
Muchnik's paradox says that enumerable betting strategies are not always reducible to enumerable strategies whose bets are restricted to either even rounds or odd rounds. In other words, there are outcome sequences x where an effectively enumerable strategy succeeds, but no such parityrestricted effectively enumerable strategy does. We characterize the effective Hausdorff dimension of such $x$, showing that it can be as low as 1/2 but not less. We also show that such reals that are random with respect to parityrestricted effectively enumerable strategies with packing dimension as low as $\log\sqrt3$. Finally we exhibit Muchnik's paradox in the case of computable integervalued strategies.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 DOI:
 10.48550/arXiv.2201.07007
 arXiv:
 arXiv:2201.07007
 Bibcode:
 2022arXiv220107007B
 Keywords:

 Mathematics  Logic;
 Computer Science  Computer Science and Game Theory