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 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,
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.
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.
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.
The articles suggest flat betting mainly for the purpose of cover. i.e. the house won't suspect you of any advantage play techniques.
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.
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.
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.
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.
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.
), 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).
