Infinite winnings at a fair roulette table

Casino games are devised to make casinos money. Elements of game play including suspense and speed of payout are used to manipulate adrenaline levels and to make people feel like they have a chance of becoming rich quick. Even though we all know this, people still play and try to devise systems to “beat the house”.

One of the simplest games in a casino is roulette where the addition of 0 (and often 00) are used to tilt the odds in the casinos favour. However, there exist gambling strategies that guarantee massive payouts if the casino plays fair. For instance, the martingale strategy of doubling the stake after each loss leads to the mathematically pleasing (but practically implausible) \[ \begin{align*} \text{E(Winnings)} &= \text{Stake}\times \Pr(\text{Winning 1st time}) + \text{Stake}\times \Pr(\text{Winning 2nd time}) + \cdots\\ &= \text{Stake}\sum_{n=1}^\infty \Pr(\text{Winning }n\text{th time})\\ &= \text{Stake}\sum_{n=1}^\infty \frac{1}{2^n} ~=~ \text{Stake},\\ \text{Var(Winnings)} &= 0. \end{align*} \] Another simple strategy is attributed to Jean-Baptiste le Rond d’Alembert, and there are examples of it being employed to good effect. It is again a system for manipulating subsequent bets based on performance, but, instead of doubling the stake, one unit is added to the stake on a loss and a unit taken away on a win. The following two simulations show the danger of employing it on a roulette wheel offering fair odds on red:

Research angles

• Calculating expected winnings and variability in winnings for various strategies,
• Linking gambling strategies to Markov theory,
• Understanding the decision making in play through utility theory and psychology.

Prerequisites

• An understanding of the calculations needed for expectations and variances.

The following could be useful but are not necessary:

• Statistical computing using R,
• Familiarity with Markov chains.

Some references

• Ayton, P. and Fischer, I. (2004). The hot-hand fallacy and the gambler’s fallacy: Two faces of subjective randomness? Memory and Cognition, 32.
• Bethell, V. (1910). Monte Carlo anecdotes and systems of play. London: William Heinemann.
• Grimmett, G. and Stirzaker, D. (2001). Probability and Random Processes (3rd ed.). Oxford University Press.