Things can go bad for a gambler and he can lose all he can afford. There are various ways to determine the possibilities of ruin happening and a legitimate way of figuring it when dealing with progressive betting systems is to calculate the probable number of plays which will most likely bring the player to betting all the money he has to bet on the next bet.And a gauge to determine these numbers is to look at an even game with the knowledge the real game against the casino will be slightly worse.
Negative and positive progressions can be mathematically emulated by the Martingale and the Antimartingale. Other progressions may be more subtle and less quickly resolved but they all amount to either raising the bet based on the last outcome or lowering the bet based on the outcome. While the mixed systems do both, depending on the circumstances, mathematically in the long run, one or the other of such mixed decisions will override the other.
The simplist Antimartingale is simply bet two. If lose bet two. If win bet 1.
The outcomes, WW +3 , WL, +1, LW 0, LL-4 point out the resolutions. The average bet is 1.75 unit and probable ruin can be calculated simply. It is the number of plays which multiplied by the average bet and multiplied by the unfavorability of the game equals the initial Bankroll. That is:
Bankroll=B, n=number of plays, Probability of win subtracted from Probability of loss=q-p, and average bet =b.
B=(n) * (q-p) * (b) or n=B/((q-p) * (b)).
Positive progressions are harder to calculate because such progressions should win much more often than they lose and even in a negative expecctation game the bankroll should grow before wipeout occurs from a long string of losses.
A simple mathematical proof based on Random Walk theory shows why the Martingale cannot work. Kim Lee furnished this proof after it was found a proof I had included in this article was not accurate.
The expected change in a random walk at any finite bounded stopping time is zero. We can define the "T"th stopping time as the first time the bankroll hits T or is ruined starting with a bankroll of N. Since the expected change is zero, the probability of ruin must be at least (T-N)/T and the probability of success must be at most N/T. As T grows large the probability of success shrinks to zero.
This reasoning works for any random walk and any betting strategy that has a sequence of bankroll levels that occur at bounded stopping times. If your expectation is zero and you try to make an unbounded amount of money then the probability of success must go to zero while the probability of ruin goes to 1.
And what is true of the Martingale is true of all positive progressions. At some time the crunch will come and the bankroll will be gone even in an even game.
After trashing progressions for much of this article, and trashing positive progressions here, it is probably time to use the word positive in relation to gambling in another sense. There can be some merit in certain progressions and there is certainly some merit in the gambler knowing something about them. This will be my next (and final) topic.
Saying something positive.
A summary of the positions of people who like progression and advocate playing them along with some comment follows:
Progressions are more fun. They probably are. If you are betting only a small portion of your available money, progressions would be fun. Their nature often allows the player to win fairly well or lose money slowly and allows big bets now and again.
Progressions work well for comps. They are bound to. Again betting only a small portion of your available money and making reasonably high average bets is bound to get decent comps. Your bets, if large enough would justify comps and the casinos would want you back. And, with available money and limiting the duration of the progressions, the doubles and splits at high bets could be played normally, putting the player at the reasonably low risk basic strategy presents.
Also, in other games with fixed, reasonably small house advantages, some progressions (while not cutting the overall disadvantage) would probably give more winning trips. Oscar may have been telling the truth.
The blackjack player should know and understand progressions for a number of reasons. Casinos recognize progressions and like progression players while disliking counters. The more bet size changes can be made to resemble a progression, the better the cover.
Also, at some points, a mild progression can be used which does not hurt the player at all. Take the area between +1 and +2, for example. The deck is favorable but not enough to warrent big bets. Further, this range's frequency is large as opposed to most ranges. Suppose your system justified a two bet at +1. The player could do a 1, 2, 3 progression or a 1,2 at the lower ends and a 1,2,4 at the higher ends. This could be extended even higher in higher counts such as a 2,4 progression. It is something to think about.
A personal note.
As I told Richard Reid when I volunteered to write this article, I enjoy research and writing for publication. I've enjoyed researching and writing this series of articles and have learned a lot doing it.
I hope these articles will be helpful to gamblers and, especially, blackjack players because I would like to repay the many people on the internet, especially on bj21, for what I have been taught about what Griffin calls this intriguing game.
Regards,
Mike