Betting Systems and the House Edge

Copyright ©  Eliot Jacobson, Ph.D.

There is no mystery to the success of casinos.  People place wagers on games that have a built in house edge.  The players win or lose the individual bets, but that’s of no concern to the management.  The only concern is that players continue pumping wagers through the system.  In this case, the large variance of individual bets evens out and the house earns according to the following basic equation:

Earnings = (Total Wagers) x (House Edge)

A progression betting system is based on the belief that this equation is wrong.  It is an attempt to defy the laws of economics and mathematics by placing wagers according to a fixed pattern in an attempt to change the house edge. 

A progression betting pattern is one that bases the current wager on the previous amount wagered and the result of the previous hand, spin, toss of the dice, etc.  Many authors write books about these systems, claiming they will win if combined with stop-loss and money management methods.  Because of their simplicity, many people try them at the tables. 

Casino managers who see individuals win using betting systems may come to regard these systems as an advantage method (wrong!) and may even take action against players who use them (very wrong!).  Indeed, there are those who use betting systems and win big.  Moreover, the arguments in favor of betting systems appear logical.  But it is important to understand that it is the large variance of games that lures customers.  Of course many players will win.  There are always winners.  There have to be winners.  But others players will lose.  In the end, the losses will more than compensate for the wins so that the final result represents the house edge. 

On the other hand, the “authors” and “experts” who extol betting systems usually blame the losers for their losses.  They claim the losers are not disciplined or don’t fully understand the system (which essentially means that the loser is not able to foretell the future perfectly).  They point to the myriad of winners (there are always winners, that’s a given).   They claim that the computer simulations that show their systems are fraudulent don’t model the “real world.”  They invent fancy theories involving chaos and fractals and never fully explain it.  They hire publicists who send out press releases. 

Just because you win while playing a progression does not mean that you are a winner or that you are using a winning system.

Betting systems do not give the player an advantage over the casino.  If you are a casino manager, progression players should be encouraged to continue with their systems.  If you are a casino player, fallacious arguments, anecdotal accounts, and slick book covers cannot overcome the physical laws of the universe.   To believe otherwise is to believe the earth is flat and that gravity makes apples rise to the sky.

WAGERING LAW:   A betting system can not change the house edge; players using these systems as a whole lose at exactly the predicted rate.

However, progression betting systems do change the way in which losses occur.  To understand the appeal of these systems, we will look at two of them in detail.

POSITIVE PROGRESSIONS

The first progression we will consider is called the “Martingale system.”   This is the most common progression used by players and it is for even-money bets only.  In it, a player starts with a basic unit bet (say $10).  If a player loses a wager, then he doubles his wager for the next bet.  He continues doubling each wager until he wins.  After a win, his wager returns to its original value of $10. On a push, the wager stays the same.  By always leaving on a win, the player ensures himself a winning session.

This seems to be a sure thing. For example, if the sequence is lose, lose, win (LLW) then the player will bet $10, $20, $40. He lost the $10 and $20 bets for a net loss of $30, but he won $40 for an overall gain of $10. For a longer sequence, consider

LLWLWLLWWLLLLLLLLLLW

It is easy to figure out the profit for the player: it is his minimum bet times the total number of wins in the sequence, in this case $10 ´ 5 wins = $50.

How can there be anything wrong with this logic? Just leave on a win and the player walks away with profit in his pocket every time.

However, for many reasons, the player can’t always leave on a win.  For example, in the previous sequence, the player was actually down $10190 on the wager before the final win. Very few people can sustain this type of loss and keep on playing.  The player placed a wager of $10240 on the last bet in an effort to win $10.  The situation of losing 10 hands or spins in a row is not rare.  In blackjack it occurs about once every 1540 hands (or 15 hours of play).  In roulette, losing 10 consecutive red/black or even/odd wagers occurs about once ever 630 spins (or about 21 hours of play).  A series of 10 consecutive losses is almost a certainty on any prolonged trip to Las Vegas .  What if the sequence of losses was 15 hands or spins?  Then the player will need to wager $327680.  At blackjack, for example, a sequence of 15 losses in a row occurs on average about once in every 100 hours of play.  What if he needed to split and double down?  What if he lost that hand? 

The result of a Martingale system is that most players will come away a small winner most of the time because they did not endure a long string of losing wagers.  But, on those rare times the player cannot walk away a winner, his losses will be huge.

Other "positive progression" systems include:

• Super Martingale

• Fibonacci

• D'Alembert

• Labouchere

• "31" System

• "2-5" System

NEGATIVE PROGRESSIONS

A system where wagers are raised after a losing bet is called a “negative progression.”  The hook of this type of betting system is the frequency of winning sessions.  Almost every session, the player will be able to leave on a win and therefore bank a small win.  The sinker is that some rare losing sessions can be catastrophic for the player.  The player most fears a long series of losses.  In the long run, the wins and losses add up to the house edge.  The player cannot overcome the “Wagering Law.”

One final comment is appropriate about table limits.  Table limits are not in place to defeat the Martingale betting system.  It is a losing system.  Table limits are in place to protect the casino’s bank from the effects of a single large wager, where collusion might be involved.  These limits also help separate the premium players from the pack.  Nevertheless, most table limits allow at most seven or eight Martingale double-ups.

The other type of betting system that many players use is a “positive progression system,” also known as pyramiding your profit, a bet parlay, or letting it ride.  Players often think of their winnings as house money, and are willing to gamble more freely with this money in an effort to win even more.  In this type of betting system, the player is willing to accept many frequent small losses in an attempt to bank a single huge win.  The player needs less capital (since he is not chasing loses), so the downside risk is smaller for each session and the upside potential is huge.

For example, a common progression among blackjack and roulette players is to “almost” double up after a win.  The player will wager a base amount, say $10 on a blackjack hand (or on red/black, even/odd). After a single win, he will wager $15 (he doubles his $10 bet, and then removes $5 to add to his “winnings” pile).  After two wins, WW, he will wager $25 (he doubles the $15 bet, and then brings $5 back to his “winnings” pile).  After three wins in a row, WWW, his wager is $45 = $25 + $25 - $5. If he loses this bet, he then brings his bet back to $10 and starts again. 

A typical sequence of ten wagers might look something like WWLLWLLLWW.  Our player won bets of $10, $15, $10, $10, and $15.  He lost bets of $25, $10, $15, $10, $10.  His net for this sequence is a loss of $10.  He endured a small loss while looking for a big win.

What is the player hoping for?  How about WWWWWWWWWW, followed by leaving?  In this case the player has made and won wagers of $10, $15, $25, $45, $85, $165, $325, $645, $1285, and $2565.  And as far as he is concerned, it is all house money.  He has hit the jackpot, winning $5165 on his $10 initial wager.  He leaves the table with the applause of those around him.  He is humbly asked to share his wise and mysterious winning ways.  He writes a book about his system and the lore of this one magical evening helps him sell a million copies.  Don’t buy a copy!

The hook of a positive progression betting system is the lure of the big win.  The player is hoping for that rare but life changing event that will forever prove the power of the system: a long series of wins.  The sinker is that the big winning sessions are very rare.  The usual outcome of an evening’s play is a small loss.  The effect to the player is very much like a slot machine and the mathematics is the same as well.  In the long run, the wins and losses add up to the house edge.  The player cannot overcome the “Wagering Law.”

Other "negative progression" systems include:

• Dahl's Progression

• Up and Pull

• Hoyle's Press

• Oscar's Grind

• Reverse Martingale

Betting systems are not a tool of advantage players; they are used by naïve gamblers who have been sold a bill of goods about an easy “winning” system. 

The house edge is not just another number; it’s The Law.