The Song of the Martingale May be A Gambler's Siren Call

Every gambling aficionado has heard, read, or dreamed about betting strategies that somehow carve an advantage for the player out of games where the casino normally has an edge. A plethora of specific approaches has been proposed over the ages, although all tend to focus around a few familiar mathematical forms.

Options are especially rich on table games because bets can be varied over wide limits. One method that surfaces regularly is the Martingale geometric progression in which bets are doubled after every loss and returned to the original wager after every win. A less extreme example is the d'Alembert arithmetic progression in which bets are raised a fixed amount after every loss and lowered an equal increment after every win. The Labouchère cancellation procedure is similar, but has an aura of authenticity because it's more complicated. A fourth illustration is the Kelly proportional criterion in which bet size depends on the player's current bankroll and the probability of winning.

The slots afford less flexibility because betting spreads on machines are limited and players may hesitate to drop fewer than the maximum number of coins for fear of sacrificing opportunities for bonus jackpots. One strategy appropriate for slots as well as tables, however, involves making bets of constant size and terminating play after exceeding and dropping back to a target profit, experiencing a reasonable upswing - perhaps a dollar amount or a percentage recovery from a deep hole, or reaching a predetermined loss limit.

The bad news is that no wagering scheme can change the laws of mathematics; if the house has an inherent edge, players have negative expectations regardless of betting strategy. The good news is that betting strategies can affect the likelihood you'll be ahead or behind by any specified amounts after some number of plays; they therefore influence the probability you'll be in a position to quit as a winner within a designated time frame.

These two statements aren't mutually-exclusive. Negative expectation implies an average loss after extended play due to house advantage. But the statistical fluctuation and skewness characteristic of each game can be more significant than the edge for individual players during any particular session, pushing solid citizens way ahead or far behind the theoretical average. And these factors can be amplified or attenuated by betting strategy.

Casino players, deliberately or intuitively, often use positive betting strategies. These bias sessions toward high probabilities of small wins and low probabilities of large losses.

To understand what this means, say a player uses a Martingale progression on flat "don't pass" bets at craps, starting with $5. If a shooter makes eight straight passes, the sequence of bets is $5, $10, $20, $40, $80, $160, $320, $640, $1280. By the eighth bet, the player has lost the sum of the first seven wagers - $1275. The next wager is $1280, implying a bankroll at least $2555 and a casino willing to book bets over $1000 at a table with a $5 minimum. Winning on or before nine such bets has a high probability, over 997 out of 1000, and yields a $5 profit. Losing all nine and stopping because the well is dry or the house won't accept the next bet, $2560, has a correspondingly low probability, under three out of 1000. But it costs the player $2555.

Negative betting strategies are also possible, typically with wagers increased after wins and decreased after losses. These tend to yield occasional large payouts and numerous small losses. Gamblers eager to use their mad money for a shot at a big score often find these approaches attractive. They're especially common among the hot shots at the slots, but are also sometimes used by aggressive pressers at the tables.

Sumner A Ingmark, noted poet of the punting pits, had this to say about betting strategies:

Many small wins and a few big losses,
Or the other way around,
Both are the same to casino bosses,
Whose net revenues are sound.