Hello you all. Please forgive me Ken and my family here. However, you have to check this article out. Please let me know what you all think. I have to hear your comments because this one will turn heads. http://www.huffingtonpost.com/2011/05/24/don-johnson-gambling-winnings_n_866194.html?ncid=webmail
Yeah, we've been discussing this quite a bit over at BlackjackInfo. It's nice to see a sharp player pull down those kinds of numbers. Very impressive!
Wish I could get a rebate. If Johnson gets a 20% rebate after losing a million dollars then big betting helps him. With a casino edge at BJ of 0.5% and if he were limited to betting a constant $100 (10,000 betting units) a hand then the chance of him winning $1,000,000 before losing $1,000,000 would be 1 chance in 2.69x10^43 and any casino should be happy to refund him $200,000 after a $1,000,000 loss. However if Johnson is limited to betting a constant $100,000 (10 betting units) then he will lose the million 52.50% of the time and win a million 47.50% of the time. However with the rebate he only loses $800,000. Therefore for each visit to a casino where he can bet $100,000 until he doubles up busts out he gains on the average: .4750x1,000,000 - .5250x800,000 = +55,000 If he were allowed to bet 1,000,000 (one betting unit) a hand then the average win rate per session is : .4975x1,000,000 - .5025x800,000 = 95,500 However I think that at that level of betting the casinos catch on that a rebate is a bad thing for them. The formula comes from the probability of ruin formula that can be found in the book "CHANCE, LUCK AND STATISTICS" p.85. ...............................BlueLight
The mathematics of loss rebates This seems like an appropriate thread to revive with some questions that have been nagging away at me for a while - I first encountered the idea of loss rebates and how they may be exploited in an article by Peter Griffin (printed in the book 'Gambling Ramblings'). And then after the Don Johnson affair, some articles appeared at Blackjack Insider: http://www.bjinsider.com/newsletter_138_rebate.shtml http://www.bjinsider.com/newsletter_138_discount.shtml http://www.bjinsider.com/newsletter_139_discount.shtml My questions are a little hard to articulate clearly, but here goes... 1. Choosing how long to play, and how much to bet. Everything I've read seems to imply an assumption that there is nothing extra to be gained by varying your bet size or deviating from the number of bets you initially set out to make, based on your results so far. So for instance, Griffin calculates that with a 50% loss rebate, betting on a single number in roulette, the optimal number of spins to play is 234, and if you continue beyond this then after 930 spins the house has the edge again. But intuitively I would have thought that an early success ought to make you change your plans. E.g., If you get as lucky as it is possible to get and win your very first bet, then your next 35 bets (in this example) have no rebate associated with them, since you are now playing entirely with your own money. So I'd be tempted to either quit and lock in my profit for this session, or (if my bankroll and the table limits allowed it, increase my bet size so that the impact of the profit made so far on the EV of the remaining play is lessened.) And in the general case, whenever I find myself with some level of profit this would seem to call for a dynamic reassesment of how (or if) to proceed from there. Is my intuition leading me astray here? 2. Choosing which game to play. In Griffin's article he explains how variance is highly desirable. It increases the number of bets you should make and the overall EV of the rebate. The implication is that the best bet available is the number bet in roulette, with its 35:1 payoff. For the 50% rebate example, the even-money roulette bets give their maximal profit (c. 0.28 units) after only 7 spins, compared to c. 8.80 units for the number bet after 234 spins. (This assuming a flat bet of 1 unit throughout.) What's confusing me here is that the BJ Insider articles seem to be talking up the suitability of blackjack for exploiting a rebate. I know blackjack's variance is raised by the fact that you can double and split and by the 3:2 payoff for a natural, but I was under the impression that it was still broadly similar to a simple even-money game like betting on a colour in roulette. But the figures quoted in BJ Insider seem to imply that blakckjack is a much better choice than betting on a number in roulette. Am I misunderstanding something? (or indeed many things? )
The key here may be the definition of "success". If your goal is simply to have a winning session, then reducing your bet or limiting the number of hands played after an initial win (or when you're ahead) will certainly increase the chances of this happening (stopping all together will guarantee it). Changing your bet or limiting the number of hands played does not change the EV of the game here, but it does improve your odds of reaching this particular goal. This is one reason why a progression like the Martingale is of no use for general table play, but is an extremely powerful tool in a tournament situation. However, if your goal is squeeze out the maximum advantage over a period of time, then the referenced articles suggest that one good strategy might be to flat bet for as few hands as possible before applying the loss rebate. 1 lost hand/spin/etc is optimal here, but clearly not practical. The articles suggest flat betting mainly for the purpose of cover. i.e. the house won't suspect you of any advantage play techniques. As far as varying your bet while playing with a loss rebate, I think it should be approached in the same way as any other advantage situation. You are playing with an advantage over the house for the initial number of hands before the advantage returns to the house. You should therefore bet the maximum you can afford given your desired risk of ruin. In this case, your advantage is determined by the number of hands you must play before you can apply your loss rebate. Increasing your bet when ahead would make sense if the amount you have won has increased your bankroll to the point that your risk of ruin can support a higher bet. There should be no other consideration when sizing your bets. Think of it this way. Say you're playing blackjack and counting the cards. The count goes up and you raise your bet according to your advantage. If you win the first hand do you consider lowering your bet, even though the count is still high, in order to "lock in" your profit for that high-count sequence? If you do, you will increase the chances of that particular sequence being a winning one, but you will lower your overall EV over time. The misleading term here is "even-money". The so called even-money bets in roulette are far from it. They have the same 5.26% house edge as all of the other roulette bets. Most blackjack games have a house edge of under 0.5%. The optimal number of hands/spins is indeed related to variance, but it is also related to the ratio of the loss rebate compared to the house edge which is massively different for blackjack and roulette.
Gronbog, Thanks for the reply; I appreciate you taking the time. However, I think I disagree with much of what you've said. I found it pretty difficult to concisely sum up what my concerns are in my original post, and I'm not sure how good a job I did. Also, it's a shame I could not reference the Griffin article with a link, as I did with the BJ Insider ones, as I think it is that which most helps explain where I am coming from. I'll try and elaborate on things and answer one or two of your points as I go... [P.S. Having now done so, this has turned into a rather mammoth post. Hope it doesn't all sound like the ramblings of a madman.] The goal is to maximize the expected return from each session, where the player gets to call a halt to a session whenever he chooses. While there may be a number of practical, real-world complications (such as a minimum session length required by the casino, or risk-reducing and camouflage measures deemed appropriate by the player), none of that should come into play in this purely abstract analysis. I think the only constraint that we need to keep in mind is that you cannot simply halt one session and immediately dive into the next. If you could do that then the whole concept of a 'session' would disappear and you would simply be playing each hand (or spin or whatever) with the same fixed level of rebate. You could (and should) treat each hand as a separate session. (This would be logically identical to having an infinite stack of matchplay coupons, and no casino has yet been known to be that generous.) And, as I said, it seems to me that changing the bet size can indeed change the EV. Consider the example I gave of winning the very first spin of roulette. Your EV on that spin was (1/38)*35 - (37/38) + (37/38)/10 = +4.47% [where (37/38)/10 is the 10% loss rebate applied when you lose.] But, having won, the EV on the next spin (and in fact on at least the next 35 spins) is just the conventional house edge for roulette: (1/38)*35 - (37/38). This is because you can lose all those bets and still be in profit for the session. If, on the other hand, you were to be an extraordinarily flamboyant gambler and not only let your winnings ride, but top up your bet to some degree, then losing that bet would mean your bankroll going into negative territory, and thus the EV calculation for this spin is once again boosted by the loss rebate. A less flamboyant approach might be to switch to betting 2 'units' so that although the next few bets will still have no rebate associated with them, it will take half the number of losses to get to the point where the rebate reasserts itself. Or of course you could simply call a halt to the session and walk away with your winnings. Although I've focused on the expectations of two individual spins in the above, that is just to demonstrate the dynamics of what is going on. The overall goal is still to maximize the expectation for the entire session. In places the BJ Insider articles calculate the per-hand EV. I'm not sure why; this seems like a pointless and confusing step to me. Unless we get into discussions of hourly win-rate, then we don't care whether we play 10 hands or 100 hands; we just want to come away with more profit at the end. 1 lost coup (I'll say coup from now on to mean a generic hand/spin/etc) is not optimal (unless you can start a new session immediately, as I discussed above). 11 spins (each betting one unit) maximizes the return at +0.24 units for the roulette number bet (with a 10% rebate rate). This is with no regard to which of the 11 will be winners or losers, or which of them have been winners or losers if you are mid-way through the session. You can imagine placing your series of 11 bets blindfolded, and only being told at the end how much you have won/lost. The whole crux of my questioning has been to speculate whether you should in fact take note of your current profit/loss when deciding whether or not to continue the session and whether or not to stick to betting 1 unit. If you were to stop after one spin (if this were to be optimal after a loss, then it must surely also be done after a win, since there is no further rebate available until you lose all your winnings), then you only expect to win about +0.04 units, from my above calculation of the 4.47% EV. Are you just inferring that or is it stated somehwere? I don't recall seeing it stated in those artilcles, and it certainly isn't part of what Griffin is talking about. Griffin gives a couple of formulas. One is a clever way of getting an approximation for the optimal number of coups, involving statistical techniques somewhat beyond me, including use of the 'UNLLI'. The other is a more brute-force technique of calculating the overall expectation for n coups, which you can then use to discover what value of n maximizes the expectation. Bear in mind that this is all apparently under the assumption that you should always then play exactly n coups, each of 1 unit. There is no doubt in my mind that this will yield the calculated expectation, but I'm querying wether it in fact maximizes the expectation. Unfortunately, the formula cannot be applied directly to blackjack because it relies on a simple, two-outcome bet with a fixed payout (e.g., win 35 units or lose 1 unit at roulette.) I don't know how to reproduce mathematical symbols in here, so this could get a little tricky, but I'll try to show you the formula. n = number of coups k = number of wins p = probability of a win q = (1-p) = probability of a loss C(n,k) means n choose k The formula for roulette number bets with a 10% rebate is - SUM(k=0 to n/36) C(n,k) (1/38)^k (37/38)^(n-k) (n-36k)/10 - n/19 It took me a while to figure it out, but what the above is doing is this - Sum for every possible losing session (if you win more than n/36 spins then you make a profit for the session, so limit k to n/36) The number of ways you can have k wins from n spins [i.e. C(n,k)] times The probability of k wins and n-k losses [i.e. (1/38)^k (37/38)^(n-k)] times The rebate for a session of k wins and n-k losses [i.e. (n-36k)/10] The above gives you the rebate that is due after n spins. So add the house edge (-n/19) to this to give the bottom-line expectation of the session. Phew; hope that makes sense. If it does, then consider this. Suppose we only find out about the rebate offer after we have already been playing in the casino for a little while. We thus have a current session profit/loss balance which it ought to be fairly straightforward to incorporate into the above formula. If we are in profit then that that will surely affect the answer that the formula gives, since all our losing C(n,k) sessions will now either have a smaller rebate or no longer be losing at all and hence have no rebate. However, since our profit to-date is a dollar amount, we could get answers that are yet again different by experimenting with different 'unit' sizes to be used in the playing of the rebate. The bigger the unit, the more it suppresses the impact of the initial profit. But if all the above is true in the case where we enter into the promotion already with an initial running balance, so to speak, then surely it is just as true during our actual play of the promotion? That's the basis for my (quite possibly faulty) reasoning that the expectation for the session isn't maximized unless you reassess your plans after every coup. Hmm. I suppose that could be it, although something doesn't seem quite right. The difference in variance is more stark than that in EV. The average squared result for one coup in roulette is 33.2, whereas in blackjack I recall it is 1.26. It would be an interesting exercise to run the above roulette formula for a 'fair' table (with just 36 numbers on it). If that doesn't massively outperform the stated returns for blackjack then there is surely something amiss. I'm wondering if it could also be applied to blackack using some variant of the approximation techniques hinted at in the articles.
And just to complete this thought, the EV of the next bet following a loss is also impacted and can also be changed by varying the bet size. When you think about it, any individual bet, the losing or winning of which will not cause your balance to move from positive to negative or from negative to positive, will be subject to the normal house edge for whatever game you are playing - If you stay in positive territory then your receive no loss rebate on that bet. If you stay in negative territoty then a loss will receive the rebate, but a win will undo previous rebates, and the net result is still the house edge. Intuitively (once again), this seems like another factor in favour of roulette. You get to start out with 35 losing bets before you reach the house edge on the next bet. Whereas for blackjack you are almost there after one loss (it's only the possibility of a 3:2 payoff or doubling/splitting that means you may win back more than the one unit you have already lost). It seems quite paradoxical that the majority of individual bets must be subject to the normal house edge (particularly in the case of blackjack) and yet the overall net effect is a substantial player edge.
It's all good. Hearing differing opinions and ideas and having our errors corrected are how we learn from one another. This is always one of my fears when posting here. It took a while for me to work up the confidence to post at all and, when I do now, I still read and re-read each post for evidence of my ignorance before finally pushing the button. This post has been brewing for a few days now, which turns out to be a good thing, because my position has gone back and forth a few times as I thought about it more. I've re-read your original post and your latest restating of the questions and I'm starting to think that there might be something to this after all. The light started to go on when I read and understood the formula for the expected value of a session of n hands of roulette with a given loss rebate percentage. We see that the EV of n-hand sessions varies with n and so it would make sense that the EV of the remaining n hands of a session would also vary with n. Also, as you have pointed out, the EV of the remaining n hands of a session would depend on the starting bank roll and the size of the bet, since the application of the loss rebate and therefore the EV of playing the n hands depends on the probabilities of ending those n hands with losses of all possible sizes. Since the EV of playing out the required session hands varies as the session progresses, it now makes sense (to me) that some variations in bet sizes are in order. The question is how to determine how much to bet and for how many additional hands? Here's one way to potentially quantify it. It seems to me that it should be possible to modify the original formula in order to obtain the expected value of continuing a session for n additional hands starting from a given bankroll with a given bet amount. For a given starting bank roll, some combination of n (>= 0), which meets the casino's session constraints, and a bet amount will yield the maximum EV. This is the optimal bet and number of hands to play for that bank roll. This process would be repeated for each hand until n becomes 0 at which point the session should be halted. The original formula would end up being a special case of the general formula for which the starting bank roll is zero and the actual amount bet is therefore irrelevant, so we can arbitrarily use the value 1 as is done in the original formula. If the EV for the selected n is positive then the bet should be the maximum, otherwise it should be the minimum. In order to develop a similar formula for blackjack, one would need to consider every possible way to end up with every possible result of playing n hands. However a series of simulations could easily compute the optimal combination of n and bet amount for each starting bank roll. Simulation could also yield the same tables for other games/bets. This process could therefore also be used to answer your second question which was which bet to make in which game. For a given starting bankroll, some combination of n, a betting amount and a particular game/bet will yield the maximum EV. This would not only tell you which game/bet to start with, but could conceivably result in recommending switching games/bets as the session progresses (and if the casino allows it). Is this algorithm optimal? It's hard to say. However, via simulation we can say whether it's an improvement on flat betting, which would answer your original question 1. We can also compare it to other strategies the you might come up with, provided you can describe them to me in precise terms. I would be happy to run the required simulations. I think that, based on the formula your presented, it's more general than that. I think that any bet which makes it possible to cross the line between profit and loss over the next n hands is affected by the application of the loss rebate. Only when you are ahead by more than n max bets or behind by more than n max winning payouts is the consideration of the loss rebate removed from the equation. I'm guessing that it's because of all the factors affecting the strength of the loss rebate, variance is the most important.
I'm in a similar position, trying to come up with a response to your response. The more I think about this topic, the more I confuse myself. In the meantime it's worth saying that one reason for my initial post was the hope that people might be able to point me towards other published material. I think there may be some other articles lurking on BJ Insider, and I expect there probably is a chapter devoted to this kind of thing in the book 'Beyond Counting' (though from what I have heard, it seems unlikely I would be able to get hold of a copy). As it happens, I've tracked down another source, the last article in the list at http://wizardofodds.com/gambling/. I wonder if the author, 'G.M.', is The GameMaster? I've only briefly scanned through it. (Any document in which the mathematical symbols outnumber the English words always takes me a long time to decypher!) If I'm reading it right, it looks like he's taken the approach of experimenting with different levels of stop-loss and stop-win, running simulations to find optimum values. I'll pluck one of the conclusions out from the text, (though there may be some important context to this which I am missing, such as a bet-sizing scheme) -
I've read and reread the 'G.M.' articles but can't follow much of the content, although some of what's in there has given me some new ideas. I've also found some more material here - http://www.beatingbonuses.com/cashback.htm, including a simulator java applet. It now seems to me that any comparison of different betting schemes does in fact have to account for different levels of risk from the very beginning, rather than just focus on maximising the EV of the session. That is, in order for the comparisons to be meaningful they need to be standardised in some way. The factors making me think this are - Since simply flat betting with a larger unit size will increase the $EV of a session, then measuring the effect of any more complex variation of bet sizes is going to be clouded by this fact unless you somehow account for the change in average bet size and/or total amount bet. After your first bet, for all subsequent bets for which your current bankroll is different to your starting bankroll, no matter what bet size you choose the EV can always be improved by the choice of an even larger bet. So if there was no table maximum, the optimal bet, based purely on EV, would apparently be your entire fortune. But this is clearly a nonsense. Since the long-term advantage comes from repeated sessions, ideally with results polarised towards big wins and big losses, it seems that a necessary first step is to come up with a session bankroll, the fraction of your overall bankroll which you can cope with losing (after allowing for the rebate). It then becomes a question of how best to employ that session bankroll (sbr), and from what I've been reading the optimal approach is essentially to set a target, some multiple of your sbr, appropriate to the game you are playing, and keep going all in (or betting the table max) until you either reach your goal or bust your sbr. Apparently the above can be improved upon somewhat if your current sbr is not an exact multiple of the max bet, choosing a bet size that may take you to an exact multiple. The casino's stipulations and/or the need to disguise what you are doing may make the repeated all-in approach too extreme, though. Did you look at the G.M articles? Maybe you would be able to make more sense of the algorithm being described there than I have been able to. Regarding your proposed algorithm. One idea might be to start with a simplification - limit the analysis to the case of a high roller such as Mr. Johnson, who is continually betting the table max. The issue then becomes when he should stop. Should it be only after the prescribed n hands, or after reaching a particular goal? Varying the bet size runs into all the issues I outlined above. I'm struggling to think what an appropriate standardisation method would be. In fact even the choice of the session bankroll doesn't seem very straightforward. Its size ought presumably to depend on the risk of ruin associated with how you intend to play it, as well as the EV which you derive from playing in that way. A low roller, playing well within the table maximum, will have a much higher risk of ruin, following the repeated all-in approach, compared to the high roller flat betting the table max. But this also means a higher EV (in percentage terms), thanks to more polarised results. Thanks for the offer of running sims. I'm still much too confused by the whole topic to come up with any meaningful requests, and I could probably fashion some ad hoc sims of my own. But there may well come a time when I come to you seeking a second oppinion on some wild result or other I may have obtained.
