  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.
