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Why Any Progression Must Fail For Negative Expectancy Games In the Long Run

I was chatting with my friend Garrett last week about using progressions in a negative expectancy game such as baccarat, and I thought our discussion would make an interesting post.

In this post, I will explain why in the long run, any progression, no matter how mild or aggressive, must fail when used to bet in a negative expectancy game such as baccarat. I’ve mentioned this fact in many of my previous posts, but I haven’t yet devoted an entire post to explaining why.

First, here is a reminder about the three basic elements of any method of play:

1. Bet Placement: where to place your bet, for example, in baccarat, whether to bet Banker or Player.

2. Bet Selection: how much to bet for each of your bet placements. Flat betting (always betting 1 unit) and progressions (increasing your bet beyond your base bet in some manner as you win or lose the last bet) are examples of bet selection methods.

3. Money Management: when to start, pause, or stop betting. Some call this “discipline.â€

Boiled down to their essences, all baccarat approaches (methods, systems, etc.,) consist of at least one of these three elements, usually, all three. Developers, players, and enthusiasts spend countless hours researching and discussing each or any of their combinations in their search for the holy grail, a system that can consistently win in the long run. However, each of the three elements can be rigorously shown to fail in the long run for negative expectancy games such as baccarat.

Here, I will explain why any type of progression (an example of 2. Bet Selection) must fail for negative expectancy games in the long run. No matter how mild or aggressive, simple or complex the progression is, it will not be able to make a long term negative expectancy method positive. Not only do my computational results clearly demonstrate this, but the rationale for why this must be true can be easily and intuitively grasped.

The key is simply this:

In the long run, any progression is simply a set of many different levels of flat bets.

So, if flat betting is long term negative, then progressions must be, too.

An example may help.

Let’s skip right to the top to the king of kings of progressions, the most aggressive progression possible, the revered and feared standard Martingale, which doubles the size of the bet upon each loss.

The Martingale goes like this:

Start betting at 1 unit (1u).

If you win, keeping betting 1u.

If you lose, double your bet to 2u.

If you win, you win back your -1u lost in the previous bet, plus you gain 1u, so you’re ahead by +1u.

If you lose again, double your bet to 4u.

If you win, you win back the -3u lost in the last two bets (1u and 2u) and gain an additional unit (+1u).

If you lose, double again.

And keep doubling until you finally win.

So, the Martingale progression is 1u, 2u, 4u, 8u, 16u, 32u, etc … as long as your bankroll lasts and you don’t hit the table limits.

In the Martingale, every time you win, you gain +1u, and every time you lose, you double the amount of your last bet. With bottomless pockets and no table limits, the Martingale is the most effective way to stay ahead of your losses in the long run. All other progressions can be regarded as some dilution of the Martingale.

Now, let’s pretend we’ve played a million decisions of a perfectly fair, 50/50 coin-flip game, and we’ve thus placed a million bets using the Martingale.

The following table summarizes the number of wins and losses per bet amount level:

Bet Amount

Number of Wins

Number of Losses






















Of course, in reality, the actual results won’t be so nice, neat, and exact, but I use these rounded off, whole numbers to facilitate discussion and to emphasize the main conceptual point. The important feature about the results is that for each level of bet amount, the number of times you won will be roughly the same as the number of times you lost. In the limit of an infinite number of bets, the number of wins and losses will approach exact equality for a perfectly fair, 50/50 game (a zero expectancy game), while for a negative expectancy game, the losses (net after any commissions, as in baccarat) will always outweigh the wins.

Presenting the total results in the above table should help you see the key point: Even though you were using a progression of many different bet amounts, you were effectively simply flat betting at various amounts. That is, there’s no mathematical difference between using a Martingale over 1,000,000 decisions or flat betting 1u over 500,000 decisions, flat betting 2u over 250,000 decisions, flat betting 4u over 125,000 decisions, etc. For each of those bet amounts, the wins and losses will cancel each other out at best, just as would have happened had you simply flat bet the same amount throughout.

Moreover, it’s easy to generalize the above to any kind of progression, no matter how mild or aggressive, simple or complex (convoluted?), because in the final analysis, tabulating all of the bets of any progression in the same manner as above will show the same thing in the long run: the number of wins essentially cancel the losses at each and every bet amount level.

So, if you can’t stay ahead flat betting at any single amount, how can flat betting at various amounts (which is what a progression really is) be any better? It can’t!

Comparing 3 progressions:

Blue: Flat betting

Red: U1D2M2

Green: Martingale

From Series 14 Results

My computational results consistently show that in the long run, progressions don’t change or worsen the flat bet expectancies. For example, in baccarat, the flat bet expectancies are roughly -1%ish. That is, for every $100 you bet, you expect to lose about $1 in the long run. Applying a progression, the long term expectancies can still remain -1%ish, or drop to -2% to -4%ish, (lose on average $2-$4 per $100 bet), or in some cases, depending on when the simulation is stopped, be much worse, exceeding -10%.

To understand this more intuitively, consider a capped Martingale, where a maximum sized bet is set, for example 1u, 2u, 4u, 8u, 16u, 32u, and no more. If you lose the 32u, start the progression over at 1u for your next bet. The capping makes the Martingale more practical, because it lengthens your bankroll’s lifetime, as well as respects the casino’s table limits. Just as we had discussed previously, in the long run, the number of wins and losses of each bet amount will be equal. But realize this: Every time the peak bet of 32u wins, your bankroll gains only +1u, but every time the peak bet of 32u loses, your bankroll suffers a huge -63u drop! And then you have to go back to betting only 1u to start the progression over. The figure below illustrates this quite dramatically.

Another way to see the same result: Statistically, winning 6 in-a-row will occur just as often as losing 6 in-a-row. When you win 6 in-a-row, your bankroll gains only +6u, but when you lose 6 in-a-row, your bankroll crashes -63u! So clearly, because the expected losses are so much greater than the expected wins, the long term expectancies will be worse than simply flat betting. This is likewise true for any other progression capped at a peak bet limit.

Characteristic “stair step†pattern of bankroll balance when peak bet of capped Martingale is lost.

From Series 16 Results

You might try to be clever and think, if my long term expectancies are significantly worse than expected for a nearly break-even game, I’ll just reverse the bet placements which had yielded -2% to -4% to become +1% to +3% (pivoting around the expected -1%). Why that won’t work is due to the progression. If a certain bet placement method yielded -4% flat betting, then, yes, reversing the bets would indeed yield +3% (and it will be a Holy-of-Holy-Grails!). But using a progression, reversing the bets won’t change the final answer. Why? Simply because everything reverses symmetrically. That is, while reversing your bets will change a 6 in-a-row loss of -63u into a 6 in-a-row win of +6u, it will symmetrically change your previous 6 in-a-row win of +6u into a 6 in-a-row loss of -63u!

(Incidentally, traders often make the same error in reasoning, thinking they can make a winning trading method out of a losing one by simply reversing the entry logic!)

Gambling gurus often teach progressions as essential to stay ahead in a game such as baccarat. By doing so, they are acknowledging the underlying negative expectancy of the game. But their advice would be valid only in the short term. This is because in the short term, luck may favor you and winning that larger bet can instantly erase a whole host of smaller losses which you’ve suffered up to that point.

So, if your objective is to leave the casino once you’re profitable and never return to the tables again, then you would be wise to use a progression, if you can afford to, a Martingale, and pray to your lucky stars you win before your bankroll dries up. In the short term, progressions are double-edged swords: When you win, you dig yourself out of the hole faster, but when you lose, the hole becomes your grave.

If your intention is to keep playing (and thus dwell within the realm of the long term), understand that using any kind of progression can be much worse than flat betting, and it will never flip the polarity of the underlying expectancies.

E. Clifton Davis Says:

December 16, 2012 at 6:34 am

Virtuoid, I agree with you on every facet of your explanation. Every bet of a progression is in fact part of a series of flat bets. Therefore progressions do you no good in the long run in random cards and commission (negative expectancy) will eventually rule, But, therein lies an assumption. You are assuming that all cards are always random – that bet placement can’t be engineered to an advantage. You are negating any skill on the part of the gambler. This is probably still a correct assumption for somewhere around 999 out of 1000 gamblers. But that one out of a thousand knows how to achieve a hands won rate better than 50/50. Moreover, he knows how to select the games wherein he can achieve the highest hands won rate or, perhaps more importantly, he knows how to avoid games where he knows he can’t achieve a positive hands won rate. He knows when to play and when not to play. He knows how the casino operates. He knows his enemy.

Take Blackjack, for instance. Basic Strategy alone beats random cards. The casino knows this full well. Therefore they see to it that the cards are NOT random. Perfect Basic Strategy loses perfectly in clumped cards. Therefore the casino sees to it that the cards are clumped. They do this by picking up the mostly low break cards first and then the mostly high non break cards each round producing large mostly low and mostly high clumps in the discard shoe. The standard two or three hand shuffles won’t even begin to neutralize these clumps. The unwitting gambler plays perfect basic strategy on Saturday night against maximum clumping and gets killed against the dealer who seldom breaks – nowhere near the normal 28% break rate in random cards. Had he simply waited and played new (random) cards on Monday morning his basic strategy would have performed much better. But alas, he is oblivious to these basic facts of gambling.

Baccarat is similar. Nearly every shoe dealt with normally shuffled cards (not factory preshuffled) ended up favoring either chop, streak or neutral. If the gambler correctly matches his highly practiced chop, streak, and neutral bet placement systems to his shoes along with good cash mgt, he wins. If he attempts to play only one bet placement system against all shoes, as most do, he loses to the worldwide average of 26% of his buyin money as impossible as that may seem given that commission is only 1.25% of his money bet.

As proof I offer this little nugget: I was able to win nearly every Bac shoe at Gold Strike, Tunica to the tune of 20 units per shoe for 3 years straight by sticking to new cards in the morning because I knew that their standard morning card prep favored OTB4L. (Opposite time before last) (neutral).

Playing with you, I was able to achieve a 26% Player Advantage for 6 shoes, probably a world record, because I knew that the first casino favored OTB4L (neutral) and the 2nd casino favored S40 (chop).

Through skilled table selection my 30 year avg hands won rate in BJ is 55% against an overall average basic strategy hands won rate of 43%.

Any devoted player with average intelligence can learn these skills. But, for the most part, they won’t. Too much work. They would rather use the strange excuse that losing is fun. Well, one thing we agree on – they are in for a lot of fun!

Last edited by E. Clifton Davis; 12-16-2012 at 06:38 AM

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