A new look at old betting systems.

This is a reply to the question of how to handle 100 events from which it is known that one will win precisely 30 times.

Reposted from http://www.rouletteforum.cc/index.php?topic=13422.60 .

[PART I]

Okay, thank you, Skakus.

What makes this solution so simple is that it is part of broader, generalized gambling theory.

I will give only the relevant part at hand here for now, and allow others to try to build onto that. This quickly becomes more difficult, so I'll likely show up later to help out. (I was posting for fun at the BetSelection site last month, when I had some time off, and was on my way out for another year, but Kav's question caught my eye.)

To do this, observe that the cancelation or "Labby" betting system is the most versatile, and hence susceptible to generalization. And, that it's that very versatility which is to blame for it's being commonly misunderstood with regard to its simplest, logical form. We must rework it's arbitrary betting to begin, and then get a handle on its broader application to the different L:W ratios.

Let's bet one unit under the pretend assumption that we lost our last bet, however. Upon a win, we stroke that bet out, and start over. There is not a matter of how many bets to cross out, because we had only one. So, let's consider a series of losses to have the "modified", nay, actually regular, cancelation betting system grow. We'll do it for a mid-range L:W ratio of two to one, in which case it's logical that we stroke off (up to) two thus previously recorded losses for each successive win. Now, we'll have to do one other thing as each new loss is recorded to the end of the previous sequence of lost bets. To not merely have the wins account for or balance out the losses (by strokes), let's add an additional one onto each new amount of the bet. When the wins even out the losses, we'll be left with the "additional" unit ones which we added on at each new bet stage. Of course, and there are a lot of other specifics which come into play, the L-W ratio may be reassessed and adjusted for for continued accuracy as go along. But degrees of those specifics aren't important to the system, itself, for short-run considerations to do with its illustration, e.g..

Start at 1 unit, and upon a loss, go to 1 + 1 unit. There were no losses before the 1 unit, so we add only the "additional" one unit which is done in general. The sequence of losses, or in other words new bets, has been recorded as 1, 2. Upon another loss, this time of 2 units, the next bet becomes 1 + 2 + 1 "additional" = 4. Upon another loss, this time of 4 units, the sequence of bets has become 1, 2, 4, and 6. Here, 6 = 1 + 4 + "1". We took the last lost bet, and added that to the number of bets from the front end of the sequence which make a total of two bets. The two was from the L of the L:W ratio, in this case, two to one. Upon yet another loss, the bet becomes 6 + 1 + "1" = 8. Lose the 8, and the next bet will become 10. And so on, until a win.

Suppose that we win that bet of 10. Well, we don't add that to the sequence of lost bets, but instead stroke out two bets from the previous sequence of 1, 2, 4, 6, 8. Namely, stroke out the last bet, 8, and the first bet, 1. Then, we are left with lost bets of 2, 4, 6. The next bet becomes 6 + 2 + "1" = 9. Upon a loss, we record the 9, and so have 2, 4, 6, 9. Next bet, 12. Win, so stroke out the 9, and the 2. Next bet, 4 + 6 + "1" = 11. Win, so stroke out the 4, and 6. Notice that we are set to begin again with another pretend loss of one unit, but with a profit of three units from the above three wins.

It's note worthy at this time to point out a couple of items, at least in principle. This regular "Labby" reduces to the D'Alembert at the one to one L:W ratio. But it expands to the full-blown Martingale as the L:W becomes much worse and/or the number of trials becomes much smaller.

The more intuitive reader may be left to wonder, how is this regular "Labby" generalized to the favorable L:W ratios (beyond one to one from the worse ratios)? Say, one loss to two wins? A trivial way may be to show, with particular other restrictions, how it is possible to lose with the better ratios.

We must express this ratio in terms of integers because it's not possible stroke out, or add, a fractional number of lost bets. For simplicity, we take one side of the ratio as one. However, there are a few techniques around this integer limitation as an issue, or the betting inaccuracies which might accrue from it. Incidentally, the bet amount is subject to also other sorts of criteria. (Interesting stuff for the new people who will take the time and make the effort to purse this.)

Hope that I haven't made any errors so far. Bear with me. It's one thing to know this stuff on the fly, but tedious to have to go through the small steps across the linear written page. And, to keep of which I write here consistent with the stuff from further down the road. (A lot of cross referencing.)

Another ex, while I leave this to others for now. Set the L:W ratio at three to one.

Lose the first four bets, to arrive at a the lost sequence of bets, 1, 2, 4, and 8. The 8 comes from adding the two front-end bets with the last bet. The next bet is 8 + 1 + 2 +"1" = 12. Win that, then stroke out the 8, and the 1, 2, (but keep the "1"). This leaves this ratio's regular "Labby" sequence at 4, and with the next bet at 4 + 0 up front + "1" = 5. Lose that, then the sequence is 4, 5. Next bet, 10 units. Lose that, then we have 4, 5, 10. Win the next bet of 20, and we are left to begin again, but with a profit of two units, one for each possible win within the allowed for number of trials at the given L:W ratio.

P. S. Better use of my time can be made by allowing some of my proofreading to the other members. Thanks.

Reposted from http://www.rouletteforum.cc/index.php?topic=13422.75 .



[PART II]

Kav,

Your hastily contrived Fantasy formula "B=(LU/(33-WS)) +1 (we always round up)" changes little until after most of the W's have occurred, at which point it fast degrades to the Martingale. So, why bet anything until most of the W's have past, at which point you can apply the Martingale from scratch to limit the bets in the worst cases. You still have to be able to cover those sets of outcomes at some point, so work with the worst possible set every time.

Forget about all the other cases in which your formula might put you in a worse spot then before by betting a string of two units, three units, etc, upon the usual case of the W's slowly falling behind the L's. You wrote, "... after 40 spins we are down 100 units and we have won 10 spins." What when that's after 14 or more W's, and so you're in a worse spot then before with the 100... with 41 L's, and 19 W's left. (No longer 1/3 rd of the 60 remaining events can be W's.) Bet from the outset, you might get lucky, and win one unit, and then "head for the hills". Big waste of time, and all the money you'll have to keep on hand to keep that setup rolling. It's funny that you mention this single problem as some sort of bug in YOUR (one and only winning) system, one which happens to pop up on the end of it in all places. But that's the only real gist of it. You went on to write, "The simple solution would be to increase the divider by 1 or 2, like the "safety break on the divisor system." Classic system seller/web guy talk. Sounds like, back to the (fourth grader's) drawing board? No, nobody discovered perpetual motion yet either.

My solution is based on first principles, a method which not only directly works to a maximum guaranteed return over each and every set of outcomes, or indirectly to a guaranteed maximum return averaged over all outcomes for any outcome, but also tends to lead to broader implications in the gambling theory, both in general and specific. Specifically, a known gain of at least so much for each and every W surely beats out a tentative gain of only one unit for all, and W's left on the table. Can't properly develop that which wasn't properly defined and governed to begin with.

"Test for yourself and share your comments." - People like you are the main reason that I don't post even my significant work on the internet. Takes too much time to find out that ALL problem gamblers - all gamblers have a growing problem with the beast - want only to continue to delude themselves until they are so "far gone" that they take up some other type of "religion" on one of the anti-gambling psychology boards, e.g.. "No skin off my back", I've taken as many "funky" ideas from others here and there as possible, and made the most of a select few of those (along with a couple of my own). If they're not interested... let's me off easy.

People wonder about the reasons that I criticize a lot of the internet stuff. NONE of it is completely correct; but far and away, MOST of it is garbage, baby talk. What else is there to do then? I don't work for anybody, let alone a bunch of crazy-dumb internet people. However, I do like to post for myself for my own motivation about something which I have just noticed, and probably criticized. And, occasionally after seeing something so dumb that it just begs for some sort of correction, knowing full well in advance that it will be either completely ignored, misunderstood or trashed. Almost perfect record there, I have to delight. So each gets what each deserves. That my friends is gambling as it was meant to be. (A beauteous thing to behold even close up!)

__________________________________________________

That dispensed with one final time, please allow me to finish off an other of my own posts/replies for completeness, and of course the record. This time, I think that I'll cut to the chase w/o a lot of explanation. Well, that's the natural luxury of having took the time and made the effort to properly set the thing up right.

I believe that there was a question of how to proceed from the L>1:1 ratios to those of 1:W>1. I corrected the "Labby" progression, and then, at least in concept, extended that business to the then also consequently properly redefined D'Alembert (for the ratios of 1:W>1, specifically for 1<W<2,) and then in the other direction for L to the Martingale for the worst ratios of L:W. The incorrect "Labby" is discussed here, http://en.wikipedia.org/wiki/Labouch%C3%A8re_system . I point this out to only not have to explain also its (for all legitimate purposes) incorrect inverse.

The answer is that the L of the L:W ratio stands for the number of losses to be stroked out after a win; and that the W stands for the number of prebets before, during, or after a loss, as recorded in the sequence of losses up to a given time in play. The pre-bets are given by the sequence (1), 1, 2, 4, 8, and so on. Those fit in with the manner in which the unfavorable ratios' sequences of losses are recorded so that the Parlay betting system may be logically approached with the same corrected "Labby". (Imagine those sequences of losses in a stepwise progression to the right of where those were in the vertical list of corresponding unfavorable ratios.)

Example for an L:W at 2:1. Two losses are stroked off after each win; but there will be only one pre-bet, the "pretend assumption that we lost our last bet" one. No matter which L>1:W=1 ratio, we begin with the pre-bet as in the two examples from my first reply.

Example for an L:W at 1:3. One loss is stroked off after each win; but there will be three pre-bets, the "pretend assumption that we lost our last bet" one (1) , 1, and 2. The next, in this case the first actual, bet would be 4, corresponding to the L:W sequence of recorded losses of 3:1. Lose, and go to 8. Lose again, and go to 12. Stroke of one of the losses for each win. In this case, each W will amount to four units gained upon the "Labby's" resolution. Restart things the same way. Note that this is the correct manner by which to not-so "arbitrarily resolve" also the L-terms before the front end of a sequence of losses for a given unfavorable ratio. Those pre-bets, however many, are considered resolved after the last actual recorded loss has been stroked off.

What else to say?

I should re-emphasize that if the "Labby" isn't properly constructed from the get go, then not only it is unworkable in practice, there will be catastrophe in theory. As we can now conclude, the only time to flat bet is when the "pretend assumption that we lost our last bet", it was zero. Ie, flat betting never had a future in higher level research/context. Same for the "in betweeners", the ones which call for an increase after a win but no change in bet after a loss, e.g.. Even the stepped positive progressive betting is left behind in its native sense. Sure, that progression may have ad hoc usages, but those are somewhat esoteric and rare applications. More standard is its mitigated application by Kelly to positive expectation. (The positive nature is retained by the mid-range criterion compromise of whole steps to very small changes in bet with regard to the available perceived and/or actual edge.) So what of the ridiculous fantasy betting systems which magically spring from everywhere but nowhere (of substance) on the internet? Ever stupidly transforming until being reinvented again, or involuntarily "put out to pasture" as those posters literally die off.

Guaranteed maximum math doesn't really apply here. Can't really have both that, and the maximum guaranteed math. And, can't recover from having gone broke, at least not in any meaningful sense (, not able to go broke on a salary, e.g.. That rules out the former. Game theory talk. Nice place to visit, but I wouldn't want to live there. Some of the hardest math on the horizon, what with quantum mechanics drawing a big breath, hence there being fewer places to hide, specifically with regard to to where to cheat.)

I suppose that no one was applying the Martingale correctly either when it comes to the unit's size. Have to find a manner of play, whichever, by which to limit your L:W ratios in either direction. By that, the unit's size follows easily given a BR, though not quite so clearly from within the favorable practical ratio's boundary. We needed to work our new negative progression up into the favorable ratios anyway, to make the most of those instead of the one unit for each W which we would have had by it unimproved before. There are a lot of outright advantages with the knowledge of exactly where you are at each time, if only for an indication to start/stop at those boundary ratio's. Specifically, with variance, e.g., later on, if you don't get greedy.

In closing, maybe there is a practical way to beat also plain, old randomness. I think that it was the Wizard of Vegas/Odds who first publically realized after a number of years that the loser math has no value at all, and then went on to sell out every one he could by that very math instead of try to work in some corrections - at least, eh - and, more importantly, a couple of new ideas here and there. (A couple is all anyone can hope for in this life.)