TomG
Joined: Sep 26, 2010
• Posts: 835
March 14th, 2017 at 5:09:58 AM permalink
Quote: ZenKinG

Wouldnt 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.
RS
Joined: Feb 11, 2014
• Posts: 4671
March 14th, 2017 at 8:30:34 AM permalink
Quote: ZenKinG

Reviving 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.
"What would Brian Boitano do?"
ZenKinG
Joined: May 3, 2016
• Posts: 167
March 15th, 2017 at 8:03:26 PM permalink
Quote: RS

Re-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.
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RS
Joined: Feb 11, 2014
• Posts: 4671
March 15th, 2017 at 8:29:29 PM permalink
Quote: ZenKinG

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.

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.
"What would Brian Boitano do?"
TomG
Joined: Sep 26, 2010
• Posts: 835
March 15th, 2017 at 8:46:03 PM permalink
Quote: RS

Bolded: 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
RS
Joined: Feb 11, 2014
• Posts: 4671
March 15th, 2017 at 10:08:42 PM permalink
Quote: TomG

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

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.
"What would Brian Boitano do?"
mamat
Joined: Jul 13, 2015
• Posts: 245
March 15th, 2017 at 10:31:09 PM permalink
Some math keywords are "zero-crossings' and "random walks".
https://en.wikipedia.org/wiki/Random_walk

------
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.

------
This isn't the same as an SD calculation, which is more common.

-----
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.

-----
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").
AxelWolf
Joined: Oct 10, 2012
• Posts: 11847
March 16th, 2017 at 12:50:52 AM permalink
Quote: TomG

Consider this scenario: someone who has an Expected Profit of \$100 per hour. In the first hour of play they earn \$100,000.

Soon after they will die in some freak long-shot accident.

If instead the opposite happens and they lose 100k.... soon after they will meet and marry a rich supermodel.
♪♪Now you swear and kick and beg us That you're not a gamblin' man Then you find you're back in Vegas With a handle in your hand♪♪ Your black cards can make you money So you hide them when you're able In the land of casinos and money You must put them on the table♪♪ You go back Jack do it again roulette wheels turinin' 'round and 'round♪♪ You go back Jack do it again♪♪
Romes

Joined: Jul 22, 2014
• Posts: 3605
March 16th, 2017 at 7:39:48 AM permalink
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.
Playing it correctly means you've already won.
ZenKinG
Joined: May 3, 2016
• Posts: 167
March 16th, 2017 at 8:42:52 AM permalink
Quote: TomG

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

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.
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