Quote from: usagi on September 22, 2013, 03:43:15 PM

Quote from: Oleander on September 22, 2013, 01:09:09 AM

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.