Aspects of Muchnik's paradox in restricted betting

TL;DR

This paper investigates Muchnik's paradox in restricted betting strategies, characterizing the effective Hausdorff dimension of sequences where unrestricted strategies succeed but parity-restricted ones do not, revealing dimension bounds and randomness properties.

Contribution

It provides a detailed characterization of the dimensions and randomness of sequences exhibiting Muchnik's paradox under various restrictions, including parity and computable integer-valued strategies.

Findings

Effective Hausdorff dimension can be as low as 1/2 for such sequences.

Sequences can be random with respect to parity-restricted strategies with packing dimension as low as log√3.

Muchnik's paradox also occurs with computable integer-valued strategies.

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 parity-restricted 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 parity-restricted effectively enumerable strategies with packing dimension as low as $\log\sqrt3$. Finally we exhibit Muchnik's paradox in the case of computable integer-valued strategies.