Does EV ever Trump Game Quality?
- Thread starter blackjack avenger
- Start date
Because you are NOT putting out $400. Two hundred dollars is already on the table and you cannot take it back. If it's correct to bet $200 with a 2.5% edge, and if your double down will yield an increase of >2.5% over NOT doubling down; then it's correct to put down $200 MORE. Your FIRST $200 is already gone and in the hands of the blackjack gods. All you're doing NOW is making another $200 bet with a 2.5% advantage.
BTW, this is the reason why no one with a lick of sense will play FULL Kelly when counting cards; because blackjack is not always an even money game. If you make a $200 bet you're actually risking MORE than $200.
BTW, this is the reason why no one with a lick of sense will play FULL Kelly when counting cards; because blackjack is not always an even money game. If you make a $200 bet you're actually risking MORE than $200.
This logic is just incorrect. I addressed this in my first post. I will try to do so again in a more illustrative way.
This is what I said in my first post about this matter:
Think of it this way. It is well-known that the advantage gained for making a departure when the index is close is absolutely minimal. So, assume we have just placed our max-bet and have been dealt a hand that we may want to double-down. We calculate the TC and find it to be just above the EV-maximizing index. Granted the EV gained for doubling here is probably extremely small (as opposed to not doubling), why would you be willing to slide out another max bet here to do so? After all, you would not slide out a max-bet for your max-bet when you only had an advantage a very slight advantage; if you are like most of us, you'd wait for an edge of 2-3%.
So, you're right about not being able to consider what we already have on the table. We have to think on the margin, as do economists. For simplicity consider the following choices with made-up EV's:
(1) Put up another max-bet to complete the double-down so that our the return on our initial bet is slightly higher than what it would be by not following the EV-maximizing indext.
(2) Take a hit (and play the hand from there), which results in a slightly lower return with respect to our initial bet than if we were to follow the EV-maximizing departure.
As you can see, we are putting up another max-bet for only a tiny gain. You don't put up a max-bet as your initial bet when the TC is only marginally good, so why would you do it here? Now, before you start claiming that one loses a significant amount of EV by waiting until the RA index, please go check out out how small the additional EV is for completing a play like this when the index is greater than the EV-maximizing index, but less than the RA index. You'll find that that the additional EV is very small indeed and that it does not call for another max-bet to be made.
Best,
SP
This is what I said in my first post about this matter:
Think of it this way. It is well-known that the advantage gained for making a departure when the index is close is absolutely minimal. So, assume we have just placed our max-bet and have been dealt a hand that we may want to double-down. We calculate the TC and find it to be just above the EV-maximizing index. Granted the EV gained for doubling here is probably extremely small (as opposed to not doubling), why would you be willing to slide out another max bet here to do so? After all, you would not slide out a max-bet for your max-bet when you only had an advantage a very slight advantage; if you are like most of us, you'd wait for an edge of 2-3%.
So, you're right about not being able to consider what we already have on the table. We have to think on the margin, as do economists. For simplicity consider the following choices with made-up EV's:
(1) Put up another max-bet to complete the double-down so that our the return on our initial bet is slightly higher than what it would be by not following the EV-maximizing indext.
(2) Take a hit (and play the hand from there), which results in a slightly lower return with respect to our initial bet than if we were to follow the EV-maximizing departure.
As you can see, we are putting up another max-bet for only a tiny gain. You don't put up a max-bet as your initial bet when the TC is only marginally good, so why would you do it here? Now, before you start claiming that one loses a significant amount of EV by waiting until the RA index, please go check out out how small the additional EV is for completing a play like this when the index is greater than the EV-maximizing index, but less than the RA index. You'll find that that the additional EV is very small indeed and that it does not call for another max-bet to be made.
Best,
SP
I don't know all exact terms using in the gambling world, but I know a little of statistics and i think to look out for is the simple logarithmic return estimate
Say you play a game (or a session) with bet unit b, which have a different return per bet r_i with known probabilities p_i. Further, you do have a bankroll
.
Then the thing to look out for is
LBG = sum_i p_i * log(bankroll + b * r_i)
(I'm not sure of the name in literature, I just called it Logarithmic Bankroll Growth)
The best game or session you can play would be that with the highest value of LBG - which in turn also dictates your bet size b.
Now lets take a few tests on this formula.
If you have a sure money printing machine, that is a positive EV game without any fluctuations. Then there is only one single return r, and hence p = 1.
Maximizing LBG yields maximizing b for any bankroll if r is positive, and comes with b=0 if r is negative.
The (trivial) interpretation is to put all money you can grap (even beyond your bankroll - so take loans) to put in your money printing machine if it actually prints money (r > 0), or don't use it if it burns money (r < 0).
If you now take a real game with two-way outcome (win or loss) with probabilites p_1 and p_2 and returns r_1 > 0 (a win), r_2 = -1 (a loss), LBG is maximized exactly at b = Kelly betsize when you take (r_1+1) as the "odds" given for that game.
So if you only have just that single game available, LBG gives you the Kelly betsize. So it advises you the correct amount to bet. That at least looks promising.
Now the thing with Blackjack is slightly different - for two reasons.
With pushes and doubles/splits, you actually have 5 or more possible results (win/push/loss and double win / double loss ...) with corresponding probabilities.
So the betsize you get for Blackjack is NOT a Kelly bet - since it's not a simple two-way game.
You have to include the possibility of doubling/splitting beforehand.
Furher while card counting (a) those probabilities will change with the count. And even further the count drift has specific probabilities. To complicate things, those are not entirely independent.
It's never said to be an easy decision of which game to play - and it neglects time (although just substract log(time) from the LBG). But mathematicaly there is a definite choice of best game. The art is however, how to estimate rather than calculate best game.
Say you play a game (or a session) with bet unit b, which have a different return per bet r_i with known probabilities p_i. Further, you do have a bankroll
.Then the thing to look out for is
LBG = sum_i p_i * log(bankroll + b * r_i)
(I'm not sure of the name in literature, I just called it Logarithmic Bankroll Growth)
The best game or session you can play would be that with the highest value of LBG - which in turn also dictates your bet size b.
Now lets take a few tests on this formula.
If you have a sure money printing machine, that is a positive EV game without any fluctuations. Then there is only one single return r, and hence p = 1.
Maximizing LBG yields maximizing b for any bankroll if r is positive, and comes with b=0 if r is negative.
The (trivial) interpretation is to put all money you can grap (even beyond your bankroll - so take loans) to put in your money printing machine if it actually prints money (r > 0), or don't use it if it burns money (r < 0).
If you now take a real game with two-way outcome (win or loss) with probabilites p_1 and p_2 and returns r_1 > 0 (a win), r_2 = -1 (a loss), LBG is maximized exactly at b = Kelly betsize when you take (r_1+1) as the "odds" given for that game.
So if you only have just that single game available, LBG gives you the Kelly betsize. So it advises you the correct amount to bet. That at least looks promising.
Now the thing with Blackjack is slightly different - for two reasons.
With pushes and doubles/splits, you actually have 5 or more possible results (win/push/loss and double win / double loss ...) with corresponding probabilities.
So the betsize you get for Blackjack is NOT a Kelly bet - since it's not a simple two-way game.
You have to include the possibility of doubling/splitting beforehand.
Furher while card counting (a) those probabilities will change with the count. And even further the count drift has specific probabilities. To complicate things, those are not entirely independent.
It's never said to be an easy decision of which game to play - and it neglects time (although just substract log(time) from the LBG). But mathematicaly there is a definite choice of best game. The art is however, how to estimate rather than calculate best game.
Ok; accepting your figure of a 2.1% gain on the initial bet; here is what is REALLY happening:
If you put up another max-bet to complete the double-down; yes - the return on the INITIAL bet is 2.1%. HOWEVER, you are ALSO earning 2.5% on that DOUBLE DOWN max bet. For a $200 bet you are earning $4.00 by not doubling, but you will be earning a total of $9.20 by doubling. This works out to a gain of 2.6%, not .1%.
I do not mean to come across as "ridiculing" you, and if it appears that way; I apologize. This is nothing more than a heated, and IMO, very productive discussion. If think I see an error in your work, I won't hesitate to tell you; and I expect the same out of you if you find an error in MY work.
If you put up another max-bet to complete the double-down; yes - the return on the INITIAL bet is 2.1%. HOWEVER, you are ALSO earning 2.5% on that DOUBLE DOWN max bet. For a $200 bet you are earning $4.00 by not doubling, but you will be earning a total of $9.20 by doubling. This works out to a gain of 2.6%, not .1%.
I do not mean to come across as "ridiculing" you, and if it appears that way; I apologize. This is nothing more than a heated, and IMO, very productive discussion. If think I see an error in your work, I won't hesitate to tell you; and I expect the same out of you if you find an error in MY work.
Sorry for a double post, but merging would destroy context.
Your reasoning is not correct for two reasons.
First, if you have a max bet out, in terms for the second bet (as the money from first bet is not in your bankroll anymore) your bankroll is less than before the first bet.
So if $250 is your Kelly bet - say on a $10k bankroll), your doubling bet is on a $9750 bankroll. Your Kelly bet for this reduced bankroll would be ~ $245.
Luckily this is not much of a problem, as a casino usually allows you to also double for LESS than the original bet.
Now the second problem is much more severe. When it comes to Kelly betting, the essence of a Kelly bet is to proper size fluctuations. Doubling down is NOT equivalent to another bet. It is for EV, but not for variance.
The simple reason is that both bets are not independent. You either win the original bet and the doubled bet, or you loose both. This quadruples the variance compared to two independent bets (which only doubles them).
If you have Kelly bet out as initial bet, you have exactly no room for increased fluctuations for doubling. The only additional money you can put on the table when your initial bet is already out is a insurance bet (only when favourable). Because insurance lowers fluctuations of the combined bets.
Your reasoning is not correct for two reasons.
First, if you have a max bet out, in terms for the second bet (as the money from first bet is not in your bankroll anymore) your bankroll is less than before the first bet.
So if $250 is your Kelly bet - say on a $10k bankroll), your doubling bet is on a $9750 bankroll. Your Kelly bet for this reduced bankroll would be ~ $245.
Luckily this is not much of a problem, as a casino usually allows you to also double for LESS than the original bet.
Now the second problem is much more severe. When it comes to Kelly betting, the essence of a Kelly bet is to proper size fluctuations. Doubling down is NOT equivalent to another bet. It is for EV, but not for variance.
The simple reason is that both bets are not independent. You either win the original bet and the doubled bet, or you loose both. This quadruples the variance compared to two independent bets (which only doubles them).
If you have Kelly bet out as initial bet, you have exactly no room for increased fluctuations for doubling. The only additional money you can put on the table when your initial bet is already out is a insurance bet (only when favourable). Because insurance lowers fluctuations of the combined bets.
This is EXACTLY what I mean when I say that no one with a lick of sense plays FULL Kelly when counting cards. I seem to recall reading somewhere that the OPTIMAL bet size for BJ is something like .61 Kelly. This will compensate for the times when you have to make the borderline doubles & splits.
Here is a mathematical piece of trivia: If a card counter insists on playing full Kelly, and if for some reason there happens to be NO SUCH THING as a table limit; he has a 100% MATHEMATICAL probability of tapping out at SOME time in the future, even if it's billions of years from now.
Here is a mathematical piece of trivia: If a card counter insists on playing full Kelly, and if for some reason there happens to be NO SUCH THING as a table limit; he has a 100% MATHEMATICAL probability of tapping out at SOME time in the future, even if it's billions of years from now.
Overbet-busting is a typical singularity in the infinite play limit. If someone doesn't believe in that fact, he also doesn't believe in boiling water he makes his coffee with every morning. Both are phase transitions not only physicists are quite interested in (an in fact is already well studied, as they merely turn out to be most the same).
As a "gambler" one wouldn't spent so much time in figuring out the best betting stake if it would only be a matter of efficiency (then underbetting would be same ineffective as overbetting, yet underbetting is not much of a problem).
I don't know if 0.61 is the correct size (let me check with LBG from above tonight) - just to make sure your bankroll does not get evaporated away in fluctuations (unlike boiling water) stick to 0.5.
As a "gambler" one wouldn't spent so much time in figuring out the best betting stake if it would only be a matter of efficiency (then underbetting would be same ineffective as overbetting, yet underbetting is not much of a problem).
I don't know if 0.61 is the correct size (let me check with LBG from above tonight) - just to make sure your bankroll does not get evaporated away in fluctuations (unlike boiling water) stick to 0.5.
A Sim Shall Lead Them
Sucker if you have the ability
run a sim with ev indices
run a sim with ra indices
The ev will earn more money
The ra will show a higher SCORE, higher DI and lower NO. So ra indices are superior in regard to risk.
Whatever fraction of kelly bettor one is, they should use RA indices.
:joker::whip:
Sucker if you have the ability
run a sim with ev indices
run a sim with ra indices
The ev will earn more money
The ra will show a higher SCORE, higher DI and lower NO. So ra indices are superior in regard to risk.
Whatever fraction of kelly bettor one is, they should use RA indices.
:joker::whip:
Sucker,
I am glad to hear that we are on good terms in the academic arena.
Now permit me to explain why your reasoning is incorrect here, as relates to to this example.
You are confusing TBA (Total Bet Advantage) with IBA (Initial Bet Advantage), and therein lies the problem. You are not considering the fact that the advantage on your hand changes when you decide to split or double-down. In fact, you are making it less likely that you'll win the hand, at the exchange of being able to put more money on the table. In the example I gave, the advantage relates to your initial bet.
So, back to the example. You'd have a 2% advantage with respect to your initial wager on your hand if you decided to take a hit (and play your hand from there. However, if you were willing to put up another max-bet by doubling-down, then you'd have a 2.1% advantage, with respect to your initial wager.
However, by doubling-down, you do not have an advantage of 2.1%, with respect to your total wager (IOW, your initial bet plus double-down bet). Indeed, by doubling down, you are decreasing your chance of actually winning the hand, and thus, you are decreasing the advantage with respect to your total wager. However, so long as you are not decreasing your advantage by more than a factor of 2, then the EV is still positive because you are doubling your wager. Therefore, the EV-maximizing index is determined by the point where you are giving up less than half of your edge to be able to double your wager. With this enlightenment, let's revisit the example.
By taking a hit and playing our hand from there we have an advantage of 2% with respect to our initial wager, but if we are willing to double the money on the table then we'd have an advantage of 2.1%, with respect to our initial wager. This then means that our advantage on the hand has only become 1.05%, but by doubling our wager, we get a 2.1% return with respect to our initial wager.
So, if we assume our max-bet is $200, we have an expectation of acquiring $4.00 by not doubling (since we have a 2.0% advantage). However, if we put out another $200 to double-down our total wager becomes $400, but doubling decreases our chance of winning the hand to 1.05%. Therefore, by doubling we expect to earn $4.20. Hence, we have jeopardized another $200, just for an additional expectation of $0.20.
As you can see, your advantage decreases by doubling, but so long as you are able to double your bet and your advantage decreases by less than a factor of 2, then the EV will be positive. However, looking at things from only an EV perspective leads to some non-SCORE maximizing outcomes that greatly increase RoR for a disproportionate return.
Furthermore, TBA (Total Bet Advantage) is not to be confused with IBA (Initial Bet Advantage).
Best,
SP
I am glad to hear that we are on good terms in the academic arena.
Now permit me to explain why your reasoning is incorrect here, as relates to to this example.
You are confusing TBA (Total Bet Advantage) with IBA (Initial Bet Advantage), and therein lies the problem. You are not considering the fact that the advantage on your hand changes when you decide to split or double-down. In fact, you are making it less likely that you'll win the hand, at the exchange of being able to put more money on the table. In the example I gave, the advantage relates to your initial bet.
So, back to the example. You'd have a 2% advantage with respect to your initial wager on your hand if you decided to take a hit (and play your hand from there. However, if you were willing to put up another max-bet by doubling-down, then you'd have a 2.1% advantage, with respect to your initial wager.
However, by doubling-down, you do not have an advantage of 2.1%, with respect to your total wager (IOW, your initial bet plus double-down bet). Indeed, by doubling down, you are decreasing your chance of actually winning the hand, and thus, you are decreasing the advantage with respect to your total wager. However, so long as you are not decreasing your advantage by more than a factor of 2, then the EV is still positive because you are doubling your wager. Therefore, the EV-maximizing index is determined by the point where you are giving up less than half of your edge to be able to double your wager. With this enlightenment, let's revisit the example.
By taking a hit and playing our hand from there we have an advantage of 2% with respect to our initial wager, but if we are willing to double the money on the table then we'd have an advantage of 2.1%, with respect to our initial wager. This then means that our advantage on the hand has only become 1.05%, but by doubling our wager, we get a 2.1% return with respect to our initial wager.
So, if we assume our max-bet is $200, we have an expectation of acquiring $4.00 by not doubling (since we have a 2.0% advantage). However, if we put out another $200 to double-down our total wager becomes $400, but doubling decreases our chance of winning the hand to 1.05%. Therefore, by doubling we expect to earn $4.20. Hence, we have jeopardized another $200, just for an additional expectation of $0.20.
As you can see, your advantage decreases by doubling, but so long as you are able to double your bet and your advantage decreases by less than a factor of 2, then the EV will be positive. However, looking at things from only an EV perspective leads to some non-SCORE maximizing outcomes that greatly increase RoR for a disproportionate return.
Furthermore, TBA (Total Bet Advantage) is not to be confused with IBA (Initial Bet Advantage).
Best,
SP
Are you saying that IBA does NOT take anything into consideration but the possibility of making a single good hand? IOW, it does not contemplate the possibilities of either splitting or doubling down? Where can a definition of IBA be found? Theory of Blackjack? It just seems to fly in the face of our entire concept of dollar advantage being due to getting more dollars on the table (splits, dd's) when we have an advantage. Would we even be playing that kind of money with a scant fractional percentage advantage?
I'm not sure what you're trying to say here, but I'll try my best to answer it.
There is a determined IBA for the hand is dealt based on the TC. This assumes that we will follow our strategy, whether it be EV-maximizing or SCORE-maximizing. However, once we get dealt our hand, then we ignore what the TC told us because we now have a better indicator of what our advantage will be on the hand. We can simply refer to the advantage related to our initial bet, depending on how we play our hand.
IBA and TBA are both statistics that CVData provides in simulations. I may be wrong, but I do not believe CVCX provides both of these. When CVCX calculates your spread, it uses IBA to do so. My definition of these two terms come from the literature that comes from the manual for CVData.
See my next post for a discussion of putting up more money for a split / double down only for a marginally better return.
SP
There is a determined IBA for the hand is dealt based on the TC. This assumes that we will follow our strategy, whether it be EV-maximizing or SCORE-maximizing. However, once we get dealt our hand, then we ignore what the TC told us because we now have a better indicator of what our advantage will be on the hand. We can simply refer to the advantage related to our initial bet, depending on how we play our hand.
IBA and TBA are both statistics that CVData provides in simulations. I may be wrong, but I do not believe CVCX provides both of these. When CVCX calculates your spread, it uses IBA to do so. My definition of these two terms come from the literature that comes from the manual for CVData.
See my next post for a discussion of putting up more money for a split / double down only for a marginally better return.
SP
Johann Cruyff once said " playing football is simple, playing simple football is the hardest"
Sometimes i get a headache from reading some replies not because they contain misinformation but because they over complicate the subject.
First I need to correct a few misconceptions:
Using RA indexes WILL NOT reduce your RETURN (will be further discussed), using RA indexes will enable to bet slightly more while maintaining the same risk thus will slightly increase your RETURN.
Playing a game with fewer decks and smaller spread WILL NOT necessarily reduce your variance, variance is complex and it depends on your playing strategy, betting strategy , the true count frequencies (penetration, number of decks) or the frequency at which you identify advantageous situations.
When we talk about expectation value (ev or EV), we have to clearly define which expectation value we are talking about, expectation value in units or RETURN, or expectation value as a percentage (also called ADVANTAGE). Unfortunately expectation value as percentage is the most commonly used however it could be misleading at times because of "information smearing" due to averaging.
What a player really cares about is the RETURN or expected amount of money won OVER A TIME PERIOD (win rate) not the expectation value expressed as a percent. The return is given by the following equation.
Return or EV in units = Total Action * EV (percentage)
if one includes time we will get a win rate (using one hour for convenience), (win rate is still an expectation value )
win rate= hourly action * EV (percentage)
hourly action is as important as the EV because it has information about the number of hands played, how frequently advantageous situations occur, how frequently the player identifies them.
Bottom line for a player what is important are both his hourly action and the advantage he has, which are both accounted for in the win rate.
SCORE is just a standardized win rate ($10,000 bankroll, full Kelly fixed betting and 100 hands an hour) so one can make a good comparison based on the intrinsic attractiveness of two games.
Sometimes i get a headache from reading some replies not because they contain misinformation but because they over complicate the subject.
First I need to correct a few misconceptions:
Using RA indexes WILL NOT reduce your RETURN (will be further discussed), using RA indexes will enable to bet slightly more while maintaining the same risk thus will slightly increase your RETURN.
Playing a game with fewer decks and smaller spread WILL NOT necessarily reduce your variance, variance is complex and it depends on your playing strategy, betting strategy , the true count frequencies (penetration, number of decks) or the frequency at which you identify advantageous situations.
When we talk about expectation value (ev or EV), we have to clearly define which expectation value we are talking about, expectation value in units or RETURN, or expectation value as a percentage (also called ADVANTAGE). Unfortunately expectation value as percentage is the most commonly used however it could be misleading at times because of "information smearing" due to averaging.
What a player really cares about is the RETURN or expected amount of money won OVER A TIME PERIOD (win rate) not the expectation value expressed as a percent. The return is given by the following equation.
Return or EV in units = Total Action * EV (percentage)
if one includes time we will get a win rate (using one hour for convenience), (win rate is still an expectation value )
win rate= hourly action * EV (percentage)
hourly action is as important as the EV because it has information about the number of hands played, how frequently advantageous situations occur, how frequently the player identifies them.
Bottom line for a player what is important are both his hourly action and the advantage he has, which are both accounted for in the win rate.
SCORE is just a standardized win rate ($10,000 bankroll, full Kelly fixed betting and 100 hands an hour) so one can make a good comparison based on the intrinsic attractiveness of two games.
Doubling-down is only going to decrease your advantage if you are forgoing opportunities by doing so. It will never increase your advantage on the hand, although it may increase your EV (because you are putting more money on the table). What does this mean? Permit me to explain.
Doubling down 10 v. 6 will not decrease your chance of winning the hand (if the TC is positive, we had a max-bet out in the example) because there are no circumstances where we would want to take an additional card. Unless you are a tensplitter type, you do not hit 12 v. 6 in the case of drawing a deuce. However, when doubling 10 v. 10, you are giving up the opportunity to draw the third card (one that you may want if your first card is low), thus you are decreasing your disadvantage on that hand. Consequently the RA index is equal to the EV-index for 10 v. 6, but not for 10 v. 10.
SP
Doubling down 10 v. 6 will not decrease your chance of winning the hand (if the TC is positive, we had a max-bet out in the example) because there are no circumstances where we would want to take an additional card. Unless you are a tensplitter type, you do not hit 12 v. 6 in the case of drawing a deuce. However, when doubling 10 v. 10, you are giving up the opportunity to draw the third card (one that you may want if your first card is low), thus you are decreasing your disadvantage on that hand. Consequently the RA index is equal to the EV-index for 10 v. 6, but not for 10 v. 10.
SP
iCnT,
Thank you for joining the conversation. Your views are always enlightening.
I agree that variance is not dependent on the number of decks. However, for all intents and purposes, considering the typical DD game and 8-deck game, if you are to play separate strategies on each of these games, each of which results in an equivalent percent-return (TBA), then it is almost certainly going to be the case that the variance associated with the DD game will be lower than that of the 8-deck game. This will hold true in all, but the most bizarre cases, that do not reflect the typical DD game and typical 8-deck game.
Best,
SP
Thank you for joining the conversation. Your views are always enlightening.
I agree that variance is not dependent on the number of decks. However, for all intents and purposes, considering the typical DD game and 8-deck game, if you are to play separate strategies on each of these games, each of which results in an equivalent percent-return (TBA), then it is almost certainly going to be the case that the variance associated with the DD game will be lower than that of the 8-deck game. This will hold true in all, but the most bizarre cases, that do not reflect the typical DD game and typical 8-deck game.
Best,
SP