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Author Topic: Just-Dice.com : Invest in 1% House Edge Dice Game  (Read 433269 times)
infested999
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September 22, 2013, 03:09:12 AM
 #1901

I think an even bigger issue is it is a Saturday night, the biggest normal time for players to gamble and the site has been down for hours with no word from Doog.  My guess is he is out and unaware.  This DDoS issue is quite out of control.

I'm logged onto JD with no issues right now.

What IP?

Code:
ping just-dice.com

Not sure how to get the IP address, I'm on an iPad.  Probably doesn't help you, but the URL is:
https://just-dice.com/

Well I know what the URL is, I just told you to ping it!

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shawshankinmate37927
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September 22, 2013, 03:21:15 AM
 #1902

I think an even bigger issue is it is a Saturday night, the biggest normal time for players to gamble and the site has been down for hours with no word from Doog.  My guess is he is out and unaware.  This DDoS issue is quite out of control.

I'm logged onto JD with no issues right now.

What IP?

Code:
ping just-dice.com

Not sure how to get the IP address, I'm on an iPad.  Probably doesn't help you, but the URL is:
https://just-dice.com/

Well I know what the URL is, I just told you to ping it!
Just wasn't sure if the "https://" made a difference.  Are you not able to access JD?  Seems like it might be a little slow, but bets are being placed and players are chatting.

"It is well enough that people of the nation do not understand our banking and monetary system, for if they did, I believe there would be a revolution before tomorrow morning."   - Henry Ford
infested999
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September 22, 2013, 03:34:44 AM
 #1903

I think an even bigger issue is it is a Saturday night, the biggest normal time for players to gamble and the site has been down for hours with no word from Doog.  My guess is he is out and unaware.  This DDoS issue is quite out of control.

I'm logged onto JD with no issues right now.

What IP?

Code:
ping just-dice.com

Not sure how to get the IP address, I'm on an iPad.  Probably doesn't help you, but the URL is:
https://just-dice.com/

Well I know what the URL is, I just told you to ping it!
Just wasn't sure if the "https://" made a difference.  Are you not able to access JD?  Seems like it might be a little slow, but bets are being placed and players are chatting.

The site is back up now

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elm
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September 22, 2013, 04:13:30 AM
 #1904

The Kelly Criterion derives from a model that differs from JD's real situation in a couple of important ways.

First, in the Kelly model, the player with the edge controls the betting.  Since he has an edge, he keeps on betting.  In contrast, with JD the house has the edge but must wait passively for the whales to bet.  Since the whales don't have an edge, they can and should stop when they're ahead.

Second, the Kelly model runs on "bet time", where the unit of time is one bet.  The Kelly Criterion maximizes the return over the number of bets.  In contrast, JD runs on "calendar time".  Investors count their return in percent per day or month or year, and count their opportunity cost the same way.

Because of these differences, it does not follow that setting the maximum bet based on the Kelly Criterion will maximize JD's return in calendar time. 

The maximum bet policy has been questioned before, and it's always been answered by an appeal to the Kelly Criterion, or to simulations based on the Kelly model.  I'm suggesting that the model doesn't match the reality, so it's time for a fresh look.


thank You for that perfect explanation, imho. I couldnt explain it this way, I tried it with simple words Smiley
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September 22, 2013, 06:04:43 AM
 #1905

Kelly wouldnt say risk 1% of capital right? More like 25% if my built in guesstimator is correct.
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September 22, 2013, 10:33:43 AM
 #1906

I deposited some btc with inputs.io and got nothing in my balance.

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September 22, 2013, 03:02:05 PM
 #1907

I deposited some btc with inputs.io and got nothing in my balance.

did you remember to type in your account# for just-dice.com into the inputs.io transaction?
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September 22, 2013, 03:17:55 PM
 #1908

Charlotte is doing some serious betting right now.  Cheesy
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September 22, 2013, 03:43:15 PM
 #1909

The Kelly Criterion derives from a model that differs from JD's real situation in a couple of important ways.

First, in the Kelly model, the player with the edge controls the betting.  Since he has an edge, he keeps on betting.  In contrast, with JD the house has the edge but must wait passively for the whales to bet.  Since the whales don't have an edge, they can and should stop when they're ahead.

Second, the Kelly model runs on "bet time", where the unit of time is one bet.  The Kelly Criterion maximizes the return over the number of bets.  In contrast, JD runs on "calendar time".  Investors count their return in percent per day or month or year, and count their opportunity cost the same way.

Because of these differences, it does not follow that setting the maximum bet based on the Kelly Criterion will maximize JD's return in calendar time.  

The maximum bet policy has been questioned before, and it's always been answered by an appeal to the Kelly Criterion, or to simulations based on the Kelly model.  I'm suggesting that the model doesn't match the reality, so it's time for a fresh look.


You are correct. But the problem is much much deeper than this. Let me begin by asking the simple question; since the KC maximizes profit over number of bets, and since variance decreases with bet size, how many max bets would we need to make in order to decrease variance to get, say, 0.9% < profit < 1.1% assuming all bets were max bets? Going with a set max bet size, say 500 BTC, guarantees we will find a sample size more than sufficient to limit profit in this way. Let's further simplify by going after the RNG and not the house edge.

So now we have simplified the problem into determining how many coin flips we need to make to show whether or not a coin is fair. Which is actually a well known problem. If we calculate this number, and determine that actually, just-dice has "flipped the coin" more times, we have then proven that just-dice is not a fair coin. It does not matter that we are not using the actual formulas for just-dice's statistics; our results are a superset of theirs. In short, if just-dice's numbers are within the sample size we require it may or may not be fair (we won't know) but if their numbers lie outside of ours we have proven that they are unfair. This proves using actual just-dice statistical formulas will merely create numbers x and y such that our figures bracket them as such; 0.9% < x < profit < y < 1.1%.

I'll even draw a picture. We will end up with a number (sample size) which will appear in one of the following places: A, B or C:
Code:
0 ...======================================================================... infinity
        (A)       JUST-DICE-STATS    (B)   OUR-SIMPLE-STATS         (C)
   

If our number shows up as A or B, we will not know which one it is (since we are calculating a simplified version of the statistics). In the case of A and B all we know is that just dice has not yet achieved the sample size we require to limit profit to 0.9% < profit < 1.1%. If, however, just dice has a sample size which falls at (C) -- which is greater than what we require -- we have guaranteed that profit should be limited to 0.9% < profit < 1.1%.

The formula for required sample size is (Z*Z)/(4*E*E), where E is the desired error (ex. 0.01 for 1%) and Z is how many standard deviations you want. 3 standard deviations gives a 99.7% level of confidence, which is less frequently broken than 1/300. A quick glance at a chart which shows how likely you are to die from various causes shows that it is far more probable that you will die by falling down (1:246) if you don't first die from committing suicide (1:121).

n = (3 * 3 ) / (4 * 0.01 * 0.01)
n = 9 / 0.0004
n = 22,500

In short, as long as we bet 500 bitcoins 22,500 times, we are guaranteed that the error will be no more than 1%. But the house edge is 1%, so this just states profit will be between 0% and 2%. (house edge +/- 1% is 0% to 2%.). That doesn't help us.

To get +/- 0.1%  or 0.9% < profit < 1.1%, we need to set E to be 0.001 not 0.01:

n = (3 * 3 ) / (4 * 0.001 * 0.001)
n = 9 / 0.000004
n = 2,250,000

There we go. How convenient. As you can see, we have just rolled over 2.4 million bets at 500 max bet. Therefore we arrive at the following connundrum:

1. 2,250,000 bets at 500 BTC is enough to guarantee variance within 0.9% < profit < 1.1%.
2. Actual sample size is a minimum of 2,400,000 because not all bets were made at max kelly bet.
3. Actual site profit is less than 0.2%.

0.2% < 0.9% < profit < 1.1%

This is a serious problem.

I am not merely suggesting something is wrong, I am proving it.

If Dooglus is interested in hiring me as a consultant I will help him fix this problem. Then again, the solution is obvious, but I think Dooglus needs someone to tell him. And no I will not advise anyone for free. You get what you pay for in life. That does not mean I am greedy it means I want Dooglus to listen to me, pay attention to what I say, and do it, or I will not waste my time. If he cannot value my advice then it has no value to him. It's that simple. That being said my rates are exceedingly cheap.

Chat soon~
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September 22, 2013, 03:59:57 PM
 #1910

@usagi

That being said my rates are exceedingly cheap.

what is cheap? as I want to open an online casino I would be interested in Your opinion. I know what is wrong with JD. lets say I think I know and as I want to be sure I really would be interested in Your opinion. so how much  Huh

cheers
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September 22, 2013, 04:02:46 PM
 #1911

Charlotte is doing some serious betting right now.  Cheesy

And the profit is falling... Sad
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September 22, 2013, 04:05:02 PM
 #1912

@usagi

That being said my rates are exceedingly cheap.

what is cheap? as I want to open an online casino I would be interested in Your opinion. I know what is wrong with JD. lets say I think I know and as I want to be sure I really would be interested in Your opinion. so how much  Huh

cheers
I think the answer is to simply put in a hard cap for the max profit.  So it can be 1% of the bankroll up until X.  X can be 50 BTC or 100 BTC or whatever.  This would decrease variance significantly. At the same time, Just-Dice would still have the highest max profit of any site.  I mean, when put nearest competitors have a max profit of 20BTC and we were over 550 BTC, something is probably wrong. If the OP cannot stay invested in his own site, that means something is wrong.  There is a reason other sites do not have a similar system in play.

I will take Devil's advocate - if we were up double the expected (instead of on 0.15% of expected), due to the same high variance conditions, noone would be complaining.  However, such high variance and risk of ruin is not compatible with long-term sustainability. 
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September 22, 2013, 04:07:55 PM
 #1913

@usagi

That being said my rates are exceedingly cheap.

what is cheap? as I want to open an online casino I would be interested in Your opinion. I know what is wrong with JD. lets say I think I know and as I want to be sure I really would be interested in Your opinion. so how much  Huh

cheers
I think the answer is to simply put in a hard cap for the max profit.  So it can be 1% of the bankroll up until X.  X can be 50 BTC or 100 BTC or whatever.  This would decrease variance significantly. At the same time, Just-Dice would still have the highest max profit of any site.  I mean, when put nearest competitors have a max profit of 20BTC and we were over 550 BTC, something is maybe wrong.
Then simply multi-accounting would bypass the limit.
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September 22, 2013, 04:10:56 PM
 #1914

how is it possible?
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September 22, 2013, 04:15:08 PM
 #1915

@usagi

That being said my rates are exceedingly cheap.

what is cheap? as I want to open an online casino I would be interested in Your opinion. I know what is wrong with JD. lets say I think I know and as I want to be sure I really would be interested in Your opinion. so how much  Huh

cheers
I think the answer is to simply put in a hard cap for the max profit.  So it can be 1% of the bankroll up until X.  X can be 50 BTC or 100 BTC or whatever.  This would decrease variance significantly. At the same time, Just-Dice would still have the highest max profit of any site.  I mean, when put nearest competitors have a max profit of 20BTC and we were over 550 BTC, something is maybe wrong.
Then simply multi-accounting would bypass the limit.
I think the issues is the cap is so high that martingales are highly probably to work for those with extremely large bankrolls.  A more sane limit would decrease the effectiveness of that strategy.  We just need to be a better deal than the competition (which as I said has a max profit of 20 BTC per bet).  Having a straight 1% max profit is extreme.  

I would also be interested in determining if the results we have seen are statisically probable but I do not have the statistics background to do so confidently.  I do not think Dooglus is cheating or the server is compromised (meaning is playing under alt names are giving out server seed).  In that case, if the results turn out to be statistically improbable, is there any explanation as to why beyond a compromised server seed?
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September 22, 2013, 04:16:04 PM
 #1916

Wagered = 1,295,557.00504856   
Profit = -135.25142283

Expected profit = 12.95 k btc / Profit = - 135 btc.

Seems legit.
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September 22, 2013, 04:17:27 PM
 #1917

The Kelly Criterion derives from a model that differs from JD's real situation in a couple of important ways.

First, in the Kelly model, the player with the edge controls the betting.  Since he has an edge, he keeps on betting.  In contrast, with JD the house has the edge but must wait passively for the whales to bet.  Since the whales don't have an edge, they can and should stop when they're ahead.

Second, the Kelly model runs on "bet time", where the unit of time is one bet.  The Kelly Criterion maximizes the return over the number of bets.  In contrast, JD runs on "calendar time".  Investors count their return in percent per day or month or year, and count their opportunity cost the same way.

Because of these differences, it does not follow that setting the maximum bet based on the Kelly Criterion will maximize JD's return in calendar time.  

The maximum bet policy has been questioned before, and it's always been answered by an appeal to the Kelly Criterion, or to simulations based on the Kelly model.  I'm suggesting that the model doesn't match the reality, so it's time for a fresh look.


You are correct. But the problem is much much deeper than this. Let me begin by asking the simple question; since the KC maximizes profit over number of bets, and since variance decreases with bet size, how many max bets would we need to make in order to decrease variance to get, say, 0.9% < profit < 1.1% assuming all bets were max bets? Going with a set max bet size, say 500 BTC, guarantees we will find a sample size more than sufficient to limit profit in this way. Let's further simplify by going after the RNG and not the house edge.

So now we have simplified the problem into determining how many coin flips we need to make to show whether or not a coin is fair. Which is actually a well known problem. If we calculate this number, and determine that actually, just-dice has "flipped the coin" more times, we have then proven that just-dice is not a fair coin. It does not matter that we are not using the actual formulas for just-dice's statistics; our results are a superset of theirs. In short, if just-dice's numbers are within the sample size we require it may or may not be fair (we won't know) but if their numbers lie outside of ours we have proven that they are unfair. This proves using actual just-dice statistical formulas will merely create numbers x and y such that our figures bracket them as such; 0.9% < x < profit < y < 1.1%.

I'll even draw a picture. We will end up with a number (sample size) which will appear in one of the following places: A, B or C:
Code:
0 ...======================================================================... infinity
        (A)       JUST-DICE-STATS    (B)   OUR-SIMPLE-STATS         (C)
   

If our number shows up as A or B, we will not know which one it is (since we are calculating a simplified version of the statistics). In the case of A and B all we know is that just dice has not yet achieved the sample size we require to limit profit to 0.9% < profit < 1.1%. If, however, just dice has a sample size which falls at (C) -- which is greater than what we require -- we have guaranteed that profit should be limited to 0.9% < profit < 1.1%.

The formula for required sample size is (Z*Z)/(4*E*E), where E is the desired error (ex. 0.01 for 1%) and Z is how many standard deviations you want. 3 standard deviations gives a 99.7% level of confidence, which is less frequently broken than 1/300. A quick glance at a chart which shows how likely you are to die from various causes shows that it is far more probable that you will die by falling down (1:246) if you don't first die from committing suicide (1:121).

n = (3 * 3 ) / (4 * 0.01 * 0.01)
n = 9 / 0.0004
n = 22,500

In short, as long as we bet 500 bitcoins 22,500 times, we are guaranteed that the error will be no more than 1%. But the house edge is 1%, so this just states profit will be between 0% and 2%. (house edge +/- 1% is 0% to 2%.). That doesn't help us.

To get +/- 0.1%  or 0.9% < profit < 1.1%, we need to set E to be 0.001 not 0.01:

n = (3 * 3 ) / (4 * 0.001 * 0.001)
n = 9 / 0.000004
n = 2,250,000

There we go. How convenient. As you can see, we have just rolled over 2.4 million bets at 500 max bet. Therefore we arrive at the following connundrum:

1. 2,250,000 bets at 500 BTC is enough to guarantee variance within 0.9% < profit < 1.1%.
2. Actual sample size is a minimum of 2,400,000 because not all bets were made at max kelly bet.
3. Actual site profit is less than 0.2%.

0.2% < 0.9% < profit < 1.1%

This is a serious problem.

I am not merely suggesting something is wrong, I am proving it.

If Dooglus is interested in hiring me as a consultant I will help him fix this problem. Then again, the solution is obvious, but I think Dooglus needs someone to tell him. And no I will not advise anyone for free. You get what you pay for in life. That does not mean I am greedy it means I want Dooglus to listen to me, pay attention to what I say, and do it, or I will not waste my time. If he cannot value my advice then it has no value to him. It's that simple. That being said my rates are exceedingly cheap.

Chat soon~
You seem to be suggesting the RNG is not random and there is a pattern that can be ascertained?
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September 22, 2013, 04:21:50 PM
 #1918

The Kelly Criterion derives from a model that differs from JD's real situation in a couple of important ways.

First, in the Kelly model, the player with the edge controls the betting.  Since he has an edge, he keeps on betting.  In contrast, with JD the house has the edge but must wait passively for the whales to bet.  Since the whales don't have an edge, they can and should stop when they're ahead.

Second, the Kelly model runs on "bet time", where the unit of time is one bet.  The Kelly Criterion maximizes the return over the number of bets.  In contrast, JD runs on "calendar time".  Investors count their return in percent per day or month or year, and count their opportunity cost the same way.

Because of these differences, it does not follow that setting the maximum bet based on the Kelly Criterion will maximize JD's return in calendar time.  

The maximum bet policy has been questioned before, and it's always been answered by an appeal to the Kelly Criterion, or to simulations based on the Kelly model.  I'm suggesting that the model doesn't match the reality, so it's time for a fresh look.


You are correct. But the problem is much much deeper than this. Let me begin by asking the simple question; since the KC maximizes profit over number of bets, and since variance decreases with bet size, how many max bets would we need to make in order to decrease variance to get, say, 0.9% < profit < 1.1% assuming all bets were max bets? Going with a set max bet size, say 500 BTC, guarantees we will find a sample size more than sufficient to limit profit in this way. Let's further simplify by going after the RNG and not the house edge.

So now we have simplified the problem into determining how many coin flips we need to make to show whether or not a coin is fair. Which is actually a well known problem. If we calculate this number, and determine that actually, just-dice has "flipped the coin" more times, we have then proven that just-dice is not a fair coin. It does not matter that we are not using the actual formulas for just-dice's statistics; our results are a superset of theirs. In short, if just-dice's numbers are within the sample size we require it may or may not be fair (we won't know) but if their numbers lie outside of ours we have proven that they are unfair. This proves using actual just-dice statistical formulas will merely create numbers x and y such that our figures bracket them as such; 0.9% < x < profit < y < 1.1%.

I'll even draw a picture. We will end up with a number (sample size) which will appear in one of the following places: A, B or C:
Code:
0 ...======================================================================... infinity
        (A)       JUST-DICE-STATS    (B)   OUR-SIMPLE-STATS         (C)
   

If our number shows up as A or B, we will not know which one it is (since we are calculating a simplified version of the statistics). In the case of A and B all we know is that just dice has not yet achieved the sample size we require to limit profit to 0.9% < profit < 1.1%. If, however, just dice has a sample size which falls at (C) -- which is greater than what we require -- we have guaranteed that profit should be limited to 0.9% < profit < 1.1%.

The formula for required sample size is (Z*Z)/(4*E*E), where E is the desired error (ex. 0.01 for 1%) and Z is how many standard deviations you want. 3 standard deviations gives a 99.7% level of confidence, which is less frequently broken than 1/300. A quick glance at a chart which shows how likely you are to die from various causes shows that it is far more probable that you will die by falling down (1:246) if you don't first die from committing suicide (1:121).

n = (3 * 3 ) / (4 * 0.01 * 0.01)
n = 9 / 0.0004
n = 22,500

In short, as long as we bet 500 bitcoins 22,500 times, we are guaranteed that the error will be no more than 1%. But the house edge is 1%, so this just states profit will be between 0% and 2%. (house edge +/- 1% is 0% to 2%.). That doesn't help us.

To get +/- 0.1%  or 0.9% < profit < 1.1%, we need to set E to be 0.001 not 0.01:

n = (3 * 3 ) / (4 * 0.001 * 0.001)
n = 9 / 0.000004
n = 2,250,000

There we go. How convenient. As you can see, we have just rolled over 2.4 million bets at 500 max bet. Therefore we arrive at the following connundrum:

1. 2,250,000 bets at 500 BTC is enough to guarantee variance within 0.9% < profit < 1.1%.
2. Actual sample size is a minimum of 2,400,000 because not all bets were made at max kelly bet.
3. Actual site profit is less than 0.2%.

0.2% < 0.9% < profit < 1.1%

This is a serious problem.

I am not merely suggesting something is wrong, I am proving it.

If Dooglus is interested in hiring me as a consultant I will help him fix this problem. Then again, the solution is obvious, but I think Dooglus needs someone to tell him. And no I will not advise anyone for free. You get what you pay for in life. That does not mean I am greedy it means I want Dooglus to listen to me, pay attention to what I say, and do it, or I will not waste my time. If he cannot value my advice then it has no value to him. It's that simple. That being said my rates are exceedingly cheap.

Chat soon~

The math is all wrong - or, rather, irrelevant.

The volume the site has had of large bets is nowhere near millions.  The millions of tiny bets become pretty much irrevelant compared to the much smaller number of large bets when looking at variance.

Try looking at what happens if you have 3 million bets at 1 satoshi and 1 bet at 500 BTC.  How likely is it that profits would be anywhere near expectation?  It's not just unlikely, it's impossible.

The real scenario isn't quite so extreme - but that extreme example clearly shows the futility of trying to do math that would only be valid if all bets were of the same size when we KNOW all bets aren't anything like the same size.

The site simply has to decide whether it's willing to take massive variance to maximise expected profits.  As there's no way to get hundreds of people playing at high stakes then the choices are either:

1.  Allow large max bets - getting the most possible expected profit but also taking on huge variance.
2.  Don't allow large bets - the smaller you reduce the max bet the lower variance becomes (as the max bet becomes progressively nearer to the mean bet and the long-term arrives faster).

I'm fine with #1 - as it currently is.  If you reduce max bets not only do you lose the few players who use them but also those who come to the site to watch them and one of the biggest factors differentiating J-D from other sites.
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September 22, 2013, 04:24:08 PM
 #1919

Quote
The formula for required sample size is (Z*Z)/(4*E*E), where E is the desired error (ex. 0.01 for 1%) and Z is how many standard deviations you want.
Don't know where you got this formula from, but I'll assume for now that it is correct.

Quote
To get +/- 0.1%  or 0.9% < profit < 1.1%, we need to set E to be 0.001 not 0.01:

n = (3 * 3 ) / (4 * 0.001 * 0.001)
n = 9 / 0.000004
n = 2,250,000

There we go. How convenient. As you can see, we have just rolled over 2.4 million bets at 500 max bet.
2.4 million bets at 500 max bet? That's 1.2 billion BTC wagered! 100 times the total amount in existence. Surely you mean 24000 bets at 500 max bet. Conveniently, a lot closer to your first example.

Quote
0.2% < 0.9% < profit < 1.1%

This is a serious problem.

Ooh, big letters *and* colours. Let me try:

Your math is wrong!

This is a serious problem.


Quote
If Dooglus is interested in hiring me as a consultant I will help him fix this problem. Then again, the solution is obvious, but I think Dooglus needs someone to tell him. And no I will not advise anyone for free. You get what you pay for in life. That does not mean I am greedy it means I want Dooglus to listen to me, pay attention to what I say, and do it, or I will not waste my time. If he cannot value my advice then it has no value to him. It's that simple. That being said my rates are exceedingly cheap.

If you are interested in hiring me as a mathematician I will help you fix this problem (actually I just did). Then again, the solution is obvious, but I think you need someone to tell you (I just did). And no I will no advise anyone for free (except I just did!). You get what you pay for in life (counter-quote: "the best things in life are free"). That does not mean I am greedy it means I want you to listen to me, pay attention to what I say, and do it, or I will not waste my time (I probably just did). If you cannot value my advice then it has no value to you. It's that simple. That being said my rates are exceedingly cheap (Can't beat free!).

edit: Deprived, stop beating me to the punch!
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September 22, 2013, 04:27:54 PM
 #1920

You are correct. But the problem is much much deeper than this. Let me begin by asking the simple question; since the KC maximizes profit over number of bets, and since variance decreases with bet size, how many max bets would we need to make in order to decrease variance to get, say, 0.9% < profit < 1.1% assuming all bets were max bets? Going with a set max bet size, say 500 BTC, guarantees we will find a sample size more than sufficient to limit profit in this way. Let's further simplify by going after the RNG and not the house edge.

Variance definitely does not decrease with bet size (assuming total amount wagered is kept constant). The standard deviation for a binomial process is calculated by:
s = sqrt( n * p * (1 - p) )
(s = standard deviation, n = number of samples, p = probability)
If every bet is the same (size b), we can obtain the standard deviation for the total profit (S):
S = b sqrt( n * p * (1 - p) )
If we increase bet-size (b), we decrease n (number of bets) proportionally. But since S is proportional to b and proportional to sqrt(n), we see that with increased bet-size, the standard deviation of the expected total profit goes up.


Which is exactly what he said. Lol.
If you increase bet size, you increase variance.
If you decrease bet size, you decrease variance.

=> Variance is decreasing with bet size.
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