Progression Betting

1. Introduction
Progressive betting or betting patterns based on prior bet out comes or the present level of wins or losses on which the better stands occur to nearly everyone who becomes interested in gambling. After all, you would not be in the game if there was not some reasonable chance of winning and all you have to do is make sure you have a big enough bet out when the inevitable win comes to cover prior losses. After all, the world is fair isn't it?

This article suggests that, so far as gambling goes, the world isn't fair and progressive betting systems, whatever their complexity or allure cannot win in the long term. This article will not be a diatribe but will, instead; discuss the mathematics and statistics involved; describe different progressive betting systems, the historical ones, the ones described in popular "gambling" books, and some of the ones used as part of commercial blackjack and other gambling "systems;" discuss how progressive betting systems apply to blackjack; offer a detailed explanation of why one system, the Martingale, is doomed to failure no matter the size of bankroll; give some reasons why and how some progressions may not be the worst thing in the world: and, finally, make more general observations on Progressive Betting.

Mathematics and Statistics of Progressive Betting.
There are mathematical Theorems which all mathematicians accept and have proven. Richard A. Epstein in the Theory of Gambling and Statistical Logic (Academic Press) expresses these theorems.

Theorem I:
If a gambler risks a finite capital over a large number of plays in a game with constant single-trial probability of winning, losing, and tying, then any and all betting systems ultimately lead to the same value of mathematical expectation of gain per unit amount wagered.

In nonmathematical terms, what is being said is a bettor can vary bets any way he desires but, in the long run, no matter what method of varying bets is used, he is going to win or lose the percentage of all the money he bets that the probability of the game predicts he will win or lose.

Without going into the formal proof of the theorem, some other statistical concepts and some examples can show why this theorem has to be true. The statistical Law of Large Numbers and a field of statistics known as combinatorial analysis must be used to demonstrate the examples.

The Law of Large Numbers has been known for a long time. It was first stated by Jacob Bernouli in the early 1700's. Algebraically what it says is the number of specific outcomes (n) equals the total number of trials (N) multiplied by the probability (p) of the specific outcome occurring as N gets larger and larger or n=pN and p=n/N. The equation is rearranged to point out how uncertainty is involved. In a game such as dice (or blackjack) long term probabilities are known and favorable outcomes divided by total outcomes will tend to equal the known probability as the number of outcomes become larger and larger. But will it be exactly equal? We are talking about statistics and gambling here and most likely it will not. n/N will only get closer to p. Does that tell us anything? Yes, because the likelihood of how close n/M will be to p can also be calculated. So we change the formula to reflect this uncertainty and come up with a formula p=n/N +u (uncertanty) where u can be either a plus or minus number. As N gets bigger it can be proven u gets smaller so it is fair to say n/N approaches p as N gets larger. (In statistical notation, my u for uncertainty is called variance and is generally expressed as sigma squared. This is a different topic which is also discussed at this site and I am not going to get into it.)

An aside:
The statistical fact that u gets smaller as N gets larger does not have to mean the total number of outcomes is closer in number to the predicted number of outcomes as N gets larger. That is, the gambler can be winning or losing more money as time goes on and the Law can will still prevail and be valid. Example: Imagine a game where the player has a 2% disadvantage (craps more or less if no crazy bets are made). After 100 plays it is not unreasonable for the player to be down six bets. p=.49, n=47, N=100. .49=47/100+u. u=-.02. After 1000 plays, assume the player is down ten bets. He has lost four more. .49=495/1000+u. u=-.005, a number a fourth of the size of .02 .

The Law of Large Numbers is directly related to Theorem I. No matter how many bets are made and no matter what the size of the bets, after enough time (And, when dealing with statistics things much more often than not end up in the short term very close to to where the end up in the long term. If they did not, polling would not work), the gambler is going to come close to winning or losing the percentage of his bets which probability dictates.

But, it is said, the long term is not important. If you have a system which will win in the short term, the long term is just a sum of your short term wins. I will return to that in the discussion of Theorem II but now I will discuss it in terms of combinatorial analysis.

Combinatorial analysis is also legitimate mathematics and statistics. It deals with with the probabilities of final outcomes of a defined series of events. Since it does that, it is perfectly suited to analyze gambling.

All series of events branch. That is, after one thing happens, something else unrelated to what had happened before will happen. In gambling there is a win, a lose, or a tie. A progressive betting system says the win, loss, and maybe the tie should lead to different next bets.

Combinatorial analysis shows betting with any system is fruitless.

Take an even game which does not have ties and a series of three plays.

The outcomes are WWW, WWL, WLW, LWW, WLL, LWL, LLW, LLL.

There are eight different combinations, each of equal liklihood, or a probability of 1/8.

Clearly, if one bet were made at a time, this equal game would have an equal number of wins or losses. One time in eight it would win 3 units and three times in eight it would win one unit. One time in eight it would lose 3 units and 3 times in eight it would lose one unit.

But what happens if you bet progressively? Let's first look at the Martingale where the goal of a series is to win one unit. If you won the last time you bet one unit. If you lose the last time you bet one unit more than your total losses which means you double your bet after each loss
 

WWW	+3		LLL	-7
WWL	+1		LLW	+1
WLW	+2		LWL	0
LWW	+2		WLL	-2

What is shown above explains the allure and seductiveness of Progressive Betting Systems. There are more paths to wins so wins happen more often but the two paths to losses create as many losses as the five paths to wins create. And no betting pattern can be divised which does not have equal amounts of total wins and losses for the same size series in an even game.

But most games are not even. Going back to the 2% unfavorable game means out of every hundred plays 51 will be losses and 49 will be wins. Think of the series as pathways. Each time you have a choice of 100 pathways, 51 leading to a loss and 49 leading to a win

And to come out with even numbers, lets suppose 12,000,000 series of three.

Flat Betting
 

W	5,880,000 pathways		L	6,120,000 pathways
WW	2,881,200 pathways		LL	3,121,200 pathways
WWW	1,411,878 pathways		LLL	1,591,812 pathways

And since the probabilities of only one loss or only one win is the same whatever the order,
 

WWL	1,469,412 pathways		LLW	1,529,388 pathways
WLW	1,469,412 pathways		LWL	1,529,388 pathways
LWW	1,469,412 pathways		WLL	1,529,388 pathways

These figures are provided for anyone who wishes to do the arithmetic, but if you do the arithmetic for flat betting you will find wins exceed losses exactly 2% of total bets and if you do the arithmetic for the Martingale you will find wins exceed losses by 2% of total bets.

The groans from the systems players can be heard, "What's this damn fool doing talking about 12,000,000 hands. Gambling is short term. I'm going to use my system and my stop loss limits and stop win limits and quit at the right time."

Such a thing might be true except for the mathematical validity of Theorem II and its Corollary.

Theorem II:
No advantage accrues to the process of betting only some subsequence of a number of independent repeated trials forming a complete sequence.

Corollary:
No advantage in terms of mathematical expectation accrues to the gambler who possesses the option of discontinuing the game after each play.

Again proofs are detailed and depend on the mathematical fact that the probability relations for a subsequence is equivalent to the probability relations for the total sequence.

Also, the Theorem and the Corollary can be understood with common sense.

If a player stays at a craps table for an hour or goes to the buffet to eat, he would be facing identical probabilities at the end of the hour and these probabilities would be the same as they had remained during the hour. His absence or presence made no difference.

Neither does it make a difference whether the player can "stop when he is ahead". If he sets a modest goal, 10 units for example, at which to quit, he will often win his goal. But at times he will lose and, in the even game, the losses will equal the wins. If his loss limit is 10, half the time he will quit winners and half the time he will quit loser. If his loss limit is 20, 2/3 of the time he will quit winner and one third of the time he will quit loser. And if the game is not even, he will quit loser in a proportion to his disadvantage. Statistics allows no other expectation.

One more point needs to be addressed. Some progressive betting systems raise bets only when the player is "ahead" in the progression. An example would be to flat bet until one is three bets ahead, start a new count if the player is three bets down (or two bets or four bets or whatever) and then raise the bet a little when the three bets are won. Example. Basic bet $15.00. If lose $45.00 start over. If win $45 bet $20. If win bet $25. If win bet $30, etc. or quit at some point. Since in a nearly even game, the player will win $45 about as often as he will lose $45, the extra $20 bet appears to be "free". If he loses he has $25 winnings and if he wins the next bet he will be $40 up and can bet $20. He is going to win some of these $20 bets so he is going to be $65 or $60 up as often as he is going to be down $45 or so the reasoning goes.

The paths of this type of progression are harder to follow, but if you follow them you will come out at the same place. In gambling there is such a thing as a free lunch and free drinks but there is never a free lunch at the tables. If mathematics and statistics could be violated, casinos would not be in business.

Despite the mathematical and statistical certainties that progressive betting systems lose (or win) at the rate the game dictates, they do have the allure and the seductiveness talked about earlier. Nearly every gambler thinks about them and many use them. The variations are countless. Some variations have been described and used for hundreds of years. Books are published and sold which have hundreds of pages and the only original thing in them is a different progressive betting system. Gambling marketeers sell systems for hundreds and sometimes thousands of dollars which are simply progressive betting systems and people buy them.