Granularity of wagers in games and the possibility of savings

TL;DR

This paper investigates how the minimum wager size in casino games affects players' ability to save winnings, especially when the minimum wager decreases over time, revealing a paradoxical impact on wealth accumulation and withdrawal.

Contribution

It introduces a detailed analysis of savings potential under variable wager granularity and inflation, extending previous results to more general betting scenarios.

Findings

Savings potential diminishes with increasing wager granularity.

Decreasing minimum wagers can enable savings even with cautious play.

The rate of decrease in minimum wager influences the success of savings strategies.

Abstract

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.