True Nature of EV
So let me go ahead and give an example here. Let's say you have an EV of 100k after 1000 hours. Your ROR is 0.5%. You get your bankroll together and finally hit the tables to start your career. Your first session, you outperform and make 5k in a couple of hours. Is it not fair to say, your EV will now be much higher than the 50k? I mean if we're using the same argument regarding ROR and having to resize down or resize up, isnt it the same with EV? Our original 50k projection has now gone out the window, no? Shouldn't we now expect to make more than 50k?
Look forward to hearing some of the math guys on this topic.
If you project to play 1000 hours and project an EV of 100k (100/hour)....then if you play 5 hours and profit 5k, then you have 9995 hours to go. Your future EV is $99,500, but since you're already at +$5k, you could, at this point, say you expect to be at +$104,500 for the year.
Imagine a line graph: EV and actual. Your future EV is always going to be the same ($100/hour), assuming you don't alter your game plan. After 5 hours and being up $5k, your EV for the 1000 hours would still be $100k, but your projected profit would be $104,500.
For you to recognize that a shoe is in a state of higher or lower EV requires you to be counting cards (recording the past history). In craps and Roulette, that won't help you, no matter what the past history of rolls is.
So, I think this is a matter of terminology, confusing overall EV with states of variance that you can take advantage of from shoe to shoe by counting cards.
Quote: discflickerI'm not really a math guy, but this is question is simple basic statistics. No memory means no memory. Random means random. EV reflects the OVERALL expectation of value over an infinite number of iterations. When you say "cards", do you mean, for example, Black Jack? If all of the aces and tens are dealt from the top of the shoe, then yes, the EV changes for that shoe... but that is an expected fluctuation (a variance). EV is the average outcome of all of these variances over an infinite number of iterations, and so, by definition, never changes.
For you to recognize that a shoe is in a state of higher or lower EV requires you to be counting cards (recording the past history). In craps and Roulette, that won't help you, no matter what the past history of rolls is.
So, I think this is a matter of terminology, confusing overall EV with states of variance that you can take advantage of from shoe to shoe by counting cards.
He's counting cards.
No, your 5K win in a couple of hours is good variance. The next measure after hourly EV in Blackjack we usually look at the Standard Deviation or probable size of the swings in the game that you are playing. If you are playing the same game with the same rules betting the same fixed spread, doing everything the same, your Expected Value does not change in the sense that you were asking. James Grosjean wrote an article called "It's all in the Denominator." I believe Standford Wong's 'Professional Blackjack' book explains EV, standard deviation, etc verbally quite well and is a great book.
Being up $5k early just means you're more likely to be ABOVE the originally estimated EV... not that "you've got a lot of it early" etc.Quote: RSNot sure why you used 100k then later 50k EV figures, I assume this is an error and you meant the 100k for all of them.
If you project to play 1000 hours and project an EV of 100k (100/hour)....then if you play 5 hours and profit 5k, then you have 9995 hours to go. Your future EV is $99,500, but since you're already at +$5k, you could, at this point, say you expect to be at +$104,500 for the year.
Imagine a line graph: EV and actual. Your future EV is always going to be the same ($100/hour), assuming you don't alter your game plan. After 5 hours and being up $5k, your EV for the 1000 hours would still be $100k, but your projected profit would be $104,500.
I've come to realize that even if someone is experiencing positive variance and ends lets say +1SD for the year, it's just an illusion. I tend to have always played with some excitement in me because of the fact of shooting for being on the side of positive variance, but i just thought of something. Just like negative variance is an illusion and you will crawl your way back 'up' to EV, isnt positive variance an illusion as well and you will crawl back 'down' to the mean? Should I not even enjoy positive fluctuations anymore knowing i will just crawl back down to the mean over time just like i dont worry about negative fluctuations knowing i will crawl back up?
Now I have read Grosjean's article about the 'Denominator" and that you're never 'due' to lose BUT if EV is a fixed amount and never changes(assuming of course you dont change your bet sizes) wouldn't you always regress back to the mean if let's say you were on the far right side of the curve for the year? Wouldnt the next 500-1000 hours for example revert you back to the mean and under-perform for the next year?
Am i falling into the 'gamblers fallacy' argument?
Or does having an edge give you potential for being farther ahead than EV while also protecting you from ever being under EV given of course a long enough sample size?
Quote: ZenKinGWouldnt the next 500-1000 hours for example revert you back to the mean and under-perform for the next year?
Regressing to the mean after a positive swing will not always come from under-performing.
Quote: ZenKinGReviving an old thread as i started to think about EV again and positive fluctuations and hopefully someone can clear up my thoughts again.
I've come to realize that even if someone is experiencing positive variance and ends lets say +1SD for the year, it's just an illusion. I tend to have always played with some excitement in me because of the fact of shooting for being on the side of positive variance, but i just thought of something. Just like negative variance is an illusion and you will crawl your way back 'up' to EV, isnt positive variance an illusion as well and you will crawl back 'down' to the mean? Should I not even enjoy positive fluctuations anymore knowing i will just crawl back down to the mean over time just like i dont worry about negative fluctuations knowing i will crawl back up?
Now I have read Grosjean's article about the 'Denominator" and that you're never 'due' to lose BUT if EV is a fixed amount and never changes(assuming of course you dont change your bet sizes) wouldn't you always regress back to the mean if let's say you were on the far right side of the curve for the year? Wouldnt the next 500-1000 hours for example revert you back to the mean and under-perform for the next year?
Am i falling into the 'gamblers fallacy' argument?
Or does having an edge give you potential for being farther ahead than EV while also protecting you from ever being under EV given of course a long enough sample size?
Re-read my post that Romes quoted.
Quote: RSRe-read my post that Romes quoted.
You think you might be able to re-explain it? Im just trying to wrap my head around it, it would help if you can be a little more clear or someone else can chime in. I understand negative variance and having a big loss for the year you're eventually going to crawl back up to EV over time, but if you experience positive flux, i dont see why you would regress back to the mean since you have the edge and each time out you're technically 'expected' to make money. Using the 100k for 1000 hours example, playing 5 hours, you're expected to make $500 each time out and because the cards dont remember whether you won or lost, the EV is always the same, so im guessing for positive fluctuation and experiencing a big win, you actually WONT regress back to the mean.
Still confused. Just wondering whether or not i can still enjoy playing blackjack and shoot for getting a big win. If you always regress back to the mean, it almost takes the whole fun out of the game, which is still fine, EV is EV, but it would be a lot more enjoyable knowing you can always have a chance of outperforming and not regressing back to the mean.
Quote: ZenKinGYou think you might be able to re-explain it? Im just trying to wrap my head around it, it would help if you can be a little more clear or someone else can chime in. I understand negative variance and having a big loss for the year you're eventually going to crawl back up to EV over time, but if you experience positive flux, i dont see why you would regress back to the mean since you have the edge and each time out you're technically 'expected' to make money. Using the 100k for 1000 hours example, playing 5 hours, you're expected to make $500 each time out and because the cards dont remember whether you won or lost, the EV is always the same, so im guessing for positive fluctuation and experiencing a big win, you actually WONT regress back to the mean.
Still confused. Just wondering whether or not i can still enjoy playing blackjack and shoot for getting a big win. If you always regress back to the mean, it almost takes the whole fun out of the game, which is still fine, EV is EV, but it would be a lot more enjoyable knowing you can always have a chance of outperforming and not regressing back to the mean.
Bolded: Wrong. Hopefully this wonderfully illustrated MS Paint image will help you understand what I mean.
Red line is overall EV.
Blue line is actual.
The end of the blue line (where it turns into green) is "now" in time.
The green line is your future EV. Notice the EV (slope of green & red lines) are the same. But the green one, you're just starting at a different point (which would be "now" instead of the red one which was "6 months ago", for example).
Of course your overall EV is going to be that red line. But starting from "now", your EV going forward is that green line. You don't expect to get closer to that red line, you expect your results to parallel the red line (expect as in expectation, not expect as in "that's what's going to happen). If this makes any sense.
Quote: RSBolded: Wrong. Hopefully this wonderfully illustrated MS Paint image will help you understand what I mean.
Disagree. Given enough time, Expected Profits will converge with actual profits. Consider this scenario: someone who has an Expected Profit of $100 per hour. In the first hour of play they earn $100,000. Then every hour thereafter they earn $100. What is their hourly rate after x number of hours? Once x gets big enough it will eventually be $100.00.
Obviously none of us will be able to play cards for 100 million hours. But it does show a few things:
-we can enjoy our big wins for exactly what we won even with the understanding we will regress to the mean, because:
-we can regress to the mean without under-performing. And also:
-given the flimsy nature of these ideas it could just be ZenK searching for any excuse not to further his career
Quote: TomGDisagree. Given enough time, Expected Profits will converge with actual profits. Consider this scenario: someone who has an Expected Profit of $100 per hour. In the first hour of play they earn $100,000. Then every hour thereafter they earn $100. What is their hourly rate after x number of hours? Once x gets big enough it will eventually be $100.00.
Obviously none of us will be able to play cards for 100 million hours. But it does show a few things:
-we can enjoy our big wins for exactly what we won even with the understanding we will regress to the mean, because:
-we can regress to the mean without under-performing. And also:
-given the flimsy nature of these ideas it could just be ZenK searching for any excuse not to further his career
As a %, yes, it gets closer or you expect to get closer. But in a strictly finite or dollars sense, no.
But if you're up $100k in 100 hours on a $100/hour play ($10k in value), if you play for 1 million total hours (999,900 more hours), you should be up $100k + 999,900*$100, NOT $100 million.
https://en.wikipedia.org/wiki/Random_walk
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Compared to most people's intuitions, we tend to hang above EV or below EV for VERY long times.
Positive side:
1) If we get a good hit or run very early, we can raise our bets and make a higher hourly rate than anticipated.
When I first started vulturing UX in 2009, I hit a $4,000 Aces w Kicker after six months. This put me way over EV for maybe over a year...
Negative side:
2) One reason I've heard that professional teams play 1/4 or 1/2 Kelly...instead of full Kelly...
If a round of bad luck hits very early, then the bankroll may drop by 50%, so the hourly rate drops by 50%, and people may have less incentive to play.
There is "regression to the mean", but you will tend to swing right back out to "above EV" to "below EV".
The Expected divergence from the mean for a simple 1-dimenional random walk is Order(Sort(n)), where n is the number of steps.
So after 1,000,000 hands of flat betting 1 unit on a coin-flip, we would expected to be Order(1,000) units away from our EV.
For a Blackjack press, we probably want to multiply by 2-4 (2,000-4,000 units away from EV).
With a 1% advantage for basic counting, our EV on 1,000,000 hands is 10,000 units ... and we expect to be Order(3,000) units away from +10,000 units.
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This isn't the same as an SD calculation, which is more common.
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What most people have been taught about "bell curves" is misleading, as we might expected most of the time gambling to be in the middle of the bell curve close to the mean.
What most long-term gamblers will tell you is that we seem to stay in very long "hot streaks" or "cold streaks" where we are "way above" or "very below" EV.
...which matches what the mathematics says.
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To answer the OP question: In some sense both sides are right about EV changing/not-changing after some wins/losses. The original EV, and the corrected EV after a bankroll move are both correct. It gets complicated to explain what's happening....because it gets into the precise definitions of "expectation", "predictor", etc... In some sense it gets philisophical. (For example, look up "frequentist" vs. "Bayesian". A "confidence interval" is not what most people think it is...which is more like a Bayesian "credible interval").
Soon after they will die in some freak long-shot accident.Quote: TomGConsider this scenario: someone who has an Expected Profit of $100 per hour. In the first hour of play they earn $100,000.
If instead the opposite happens and they lose 100k.... soon after they will meet and marry a rich supermodel.
This is your better/best example.Quote: RS...But if you're up $100k in 100 hours on a $100/hour play ($10k in value), if you play for 1 million total hours (999,900 more hours), you should be up $100k + 999,900*$100, NOT $100 million.
I too agree that after 10 million hands you will hit EV, regardless of the variance experienced early. It's the same reason when we run SIM's for 100,000,000 shoes they will always converge to the SAME number (EV). It doesn't matter all that much if in the short run (earlier when variance effects us) if we dip up or down. Speaking from mathematics, after a certain amount of time (hands) you will converge to your EV +/- SD, and that's a mathematical fact. But what RS and we're discussing are two different lines. He's basically "restarting" his expectations after a big win or loss and projecting EV from there. Thus, apples and oranges.
I believe what RS is saying, is taking this data REAL TIME your starting points (and thus final EV) differ. If you start with a $10k loss, then your EV trajectory is exactly the same, but you're starting from $10k lower now so your EV at the same point in time will be $10k lower (even if it's parallel and rising all the same).
https://ibb.co/cgwedv
Again though, I think we're comparing apples to oranges. If you start at $X bankroll, after millions of hands your EV is $Y... regardless if you start with a win or loss after millions and millions of hands you WILL hit your original EV.
https://ibb.co/f8i35a
*Not sure why my images wouldn't load inside an img tag.
Quote: TomGDisagree. Given enough time, Expected Profits will converge with actual profits. Consider this scenario: someone who has an Expected Profit of $100 per hour. In the first hour of play they earn $100,000. Then every hour thereafter they earn $100. What is their hourly rate after x number of hours? Once x gets big enough it will eventually be $100.00.
Obviously none of us will be able to play cards for 100 million hours. But it does show a few things:
-we can enjoy our big wins for exactly what we won even with the understanding we will regress to the mean, because:
-we can regress to the mean without under-performing. And also:
-given the flimsy nature of these ideas it could just be ZenK searching for any excuse not to further his career
TomG has it right and it's basically what Grosjean said as well. You're never expected to be 'due' for a loss if you have an edge regardless if you had positive flux or negative flux. The EV is always the same per session. If you have a $100 an hour play, your EV every session is $100 an hour and you're 'expected' to make that $100. Of course that almost never happens but for theoretical purposes that's how it works and you're 'expected' to make that $100.
I think the problem people are missing here is exactly what Grosjean said. The numbers 'fix' themselves in the denominator and NOT the numerator. What happens is just because you won 5k quickly in 5 hours where you should have won only 500, it doesnt mean you're all of a sudden now going to hit a losing streak to make up for it. What happens is that your wins will continue to hit the $500 EV mark consistently enough to 'dilute' the sample size overtime and thus ending you with the original 100k EV.
Take a 50/50 quarter for example. You might flip 100 heads and 20 tails. The gamblers fallacy argument is that tails will eventually hit a 'streak' to make up for those 80 lost tails and that is purely incorrect. What happens is you start hitting the 50/50 EV and it 'dilutes' the sample size enough to where now the overall results will = 50/50.
Hope this makes sense. I believe I have it right now.
You're trying to reconcile the apples and oranges being discussed.Quote: ZenKinG...I think the problem people are missing here is exactly what Grosjean said. The numbers 'fix' themselves in the denominator and NOT the numerator. What happens is just because you won 5k quickly in 5 hours where you should have won only 500, it doesnt mean you're all of a sudden now going to hit a losing streak to make up for it. What happens is that your wins will continue to hit the $500 EV mark consistently enough to 'dilute' the sample size overtime and thus ending you with the original 100k EV.
Take a 50/50 quarter for example. You might flip 100 heads and 20 tails. The gamblers fallacy argument is that tails will eventually hit a 'streak' to make up for those 80 lost tails and that is purely incorrect. What happens is you start hitting the 50/50 EV and it 'dilutes' the sample size enough to where now the overall results will = 50/50...
With the coin flip... If you flip 90 heads and 10 tails in the beginning, that's just variance. AS you continue to get more flips and move towards the long run your ups and downs will be "random variance" from that one particular point in time. However, on the grand scale of "The Long Run" (let's say 100,000,000 flips) your numbers WILL balance out to 50/50. The TWO effects this has is 1, as you said, it dampens the previous variance spike (that 90-10 spike now doesn't look like a spike at all), and 2 you will SPREAD OUT OVER THE COURSE OF 100,000,000 FLIPS see more up and down spikes that will "correct" (hate using that word) the data as you described. You will NEVER be "expected" to win or lose more/less, but over the long run these tiny infractions will balance out. Basically, the math will work itself out so long as it's a true random coin flip.
Read my post above, with the images... I'm saying the same thing yet another way.
It is what it is. Less talking, more playing.Quote: ZenKinGQuote: TomGDisagree. Given enough time, Expected Profits will converge with actual profits. Consider this scenario: someone who has an Expected Profit of $100 per hour. In the first hour of play they earn $100,000. Then every hour thereafter they earn $100. What is their hourly rate after x number of hours? Once x gets big enough it will eventually be $100.00.
Obviously none of us will be able to play cards for 100 million hours. But it does show a few things:
-we can enjoy our big wins for exactly what we won even with the understanding we will regress to the mean, because:
-we can regress to the mean without under-performing. And also:
-given the flimsy nature of these ideas it could just be ZenK searching for any excuse not to further his career
TomG has it right and it's basically what Grosjean said as well. You're never expected to be 'due' for a loss if you have an edge regardless if you had positive flux or negative flux. The EV is always the same per session. If you have a $100 an hour play, your EV every session is $100 an hour and you're 'expected' to make that $100. Of course that almost never happens but for theoretical purposes that's how it works and you're 'expected' to make that $100.
I think the problem people are missing here is exactly what Grosjean said. The numbers 'fix' themselves in the denominator and NOT the numerator. What happens is just because you won 5k quickly in 5 hours where you should have won only 500, it doesnt mean you're all of a sudden now going to hit a losing streak to make up for it. What happens is that your wins will continue to hit the $500 EV mark consistently enough to 'dilute' the sample size overtime and thus ending you with the original 100k EV.
Take a 50/50 quarter for example. You might flip 100 heads and 20 tails. The gamblers fallacy argument is that tails will eventually hit a 'streak' to make up for those 80 lost tails and that is purely incorrect. What happens is you start hitting the 50/50 EV and it 'dilutes' the sample size enough to where now the overall results will = 50/50.
Hope this makes sense. I believe I have it right now.
(1) In 100,000,000 hands, you are only expected to see 4,000 zero-crossings (for a Bernoulli coin flip, sqrt(n/6.28)).
Chances are you are in a 25,000 hand hot-streak (or cold-streak) and won't be at EV....either order (10,000) units up or o(10,000) units down.
EV compared to wins vs. total money bet will tend to coverage,
but most of us live in a world with real dollars. o(10,000) units up/down for $25 units is +/- $250,000 above or below EV,
which is sizable for most people.
We mostly don't care that in 100 million hands, we have bet $2.5 billion...and that $250,000 is 0.01%.
(2) If you record your daily bankroll and project your future bankroll from the "peaks" or the "lows"...using EV from those points, chances are those projections will not be as good as ones from the center.
It's how you draw the "observation point" for calculating future bankroll.
