In a casino where arbitrarily small bets are admissible, any betting strategy M can be modified into a saving 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.

在允许任意小额下注的娱乐场中，可以将任何投注策略M修改为一种储蓄策略，不仅可以在M所在的每个娱乐场序列上都成功（从而在娱乐场内积累无穷的财富），而且可以节省无数的资本，通过将其永久并逐渐从游戏中撤回。Teutsch证明，当赌场强加固定最低赌注时，情况已不再如此，从而体现了储蓄悖论玩家可以在赌场内赢得无限的财富，但是一旦从游戏中撤出足够多的钱，他就被迫破产。我们研究了在不断缩小的最低赌注规则（粒度）下的储蓄潜力，以及其对下降率（通胀）以及胆小与大胆的依赖。