  In a casino where arbitrarily small bets are admissible, any betting strategy
M can be modified into a savings strategy that, not only is successful on each
casino sequence where M is (thus accumulating unbounded wealth inside the
casino) but also saves an unbounded capital, by permanently and gradually
withdrawing it from the game. Teutsch showed that this is no longer the case
when a fixed minimum wager is imposed by the casino, thus exemplifying a
savings paradox where a player can win unbounded wealth inside the casino, but
upon withdrawing a sufficiently large amount out of the game, he is forced into
bankruptcy. We study the potential for saving under a shrinking minimum wager
rule (granularity) and its dependence on the rate of decrease (inflation) as
well as timid versus bold play.
