Topic: Just-Dice.com : Invest in 1% House Edge Dice Game - page 124. (Read 435457 times)

Thanks organofcorti. If variance = entropy for Gaussian distributions, then I think we use the central limit theorem to justify a bunch of discrete Bernoulli processes morphing into a process with a Gaussian PDF. 

I have a vague memory of variance and entropy having a monotonous  relationship for some continuous distribution, and I think that for gaussian distributions variance == entropy, and stdev  = sqrt(variance). I wouldn't have thought the relationship would hold for a discrete distribution though.

Can anyone help me justify why stdev would be the square root of entropy?

Your visualization shows him pretty much riding the 2 sigma line most of the way, no?

Remember, dabs, vegas rule #1: never talk a sale. Do you know what I mean?

Changing the topic a bit, but check out this graph of Nakowa's winnings over time:




Bugpowder had made some very nice histograms via simulation over 100,000 trials that show where on a PDF Nakowa's gambling profits were when he stopped (see p. 106 of this thread).  I wanted to devise an analytic way to do the same thing *over time* and I think I may have.  But I need some of the experts here to confirm or refute  

The variance of a *single* bet is

  V = B^2 * p * (1-p)

But the rule-of-thumb "variances add" does not apply to individual dice rolls: individual dice rolls are Bernoulli processes which do not have Gaussian PDFs.  (If you assume that variances add, then Nakowa's win shown above would be > 4 sigma, which would be really disconcerting!)

Borrowing some concepts from information theory, the entropy of a Bernoulli process is defined as:

  H(p) = -p log_2 (p) - (1-p) log_2 (1-p)

and entropy *does* add.  The entropy cloud above was calculated by accumulating the entropy from Bet #1 to Bet n:

  H_accumulated = H(.495)*B1^2 + H(.495)*B2^2 + ... + H(.495)*Bn^2

Here is my question: if I take the square root of H_accumulated at Bet n, it seems to give me standard deviation at that particular point in the bet sequence.  My PDF matches Bugpowder's results almost exactly.  

Can anyone help me justify why stdev would be the square root of entropy?

I think if you do it that way, it is exactly the same as if "offsite" reserves weren't implemented at all, and investor B just invested 10 BTC at 10% risk.  After losing his first bet he would have 9 BTC left, risk 10% of it, ie. 0.9 BTC, as in your working above.  The point of the "offsite" reserve is that it doesn't shrink with losses.  He deposited 10 BTC, claims to have 90 BTC more at home, and we take him on his word.


And that is how it is meant to work.  "offsite" investment is a way for player B to invest his whole 100 BTC without having to trust us with more than 10 BTC of it at a time.  It should be exactly equivalent to investor A investing the whole 100 BTC at once.


We don't care whether he really has the other 90 BTC or not.  If he loses the first 10 BTC without topping it up, he gets divested.

The only downside I can see to this is that it can cause sudden changes in the available max profit.  To mitigate that, we limit the amount of leverage investors are allowed access to.

Anyways, I've stopped begging a long time ago. The ones who "invested" in my "gambling security" did it willingly and contacted me directly. No one ever contributed to the "Group Bet" I started even earlier.

I have a slight suggestion for dooglus in the way he implements the lucky number generation. Use the nonce twice. Once as concatenated input to the "secret", and second as concatenated input to the "message".

OK, so I'm announcing it.  At midnight UTC Friday I'll change max profit to 0.5%.  (half Kelly).

That's in 25 hours.  I'll have the site spam this news to the chat periodically too.

1) "Investors don't want to gamble" I beg to differ. Like someone said, this is a startup casino using startup money which is highly volatile. That's gambling. Just with a 1% edge. So I reject that premise. Plus it's real simple-- most investors on the site gamble some too.

2) We use the Kelly Criterion to determine the max profit, which is 1% of your investment (or .25% now). But that's the max. In reality, we're almost never risking full kelly, except on the occasional times when someone makes a bet that pays full max profit (usually a whale, but sometimes somebody going for a longshot, betting little for a big payoff). So when someone bets 0.008 when max profit is 80BTC, that wager is actually 0.0001x Kelly. Hardly relevant. And individual satoshi, even less. Look at the awesome Kelly chart someone uploaded a couple pages ago. Look how little volatility 0.0001x Kelly contributes. Thus, no need for a minimum, maximum is already taken care of rationally. Done.

EDIT: Keep in mind that casinos have minimums for several other reasons that are irrelevant here: segregate high-rollers from us poor folk, there are fixed costs to having dealers, it's physically difficult and risky to have the dealer have to handle wildly different denominations on one table (imagine having stacks or 1 cent chips next to a stack of $10,000 or $500,000 chips. That's a recipe for disaster!). Again, none of this is relevant at JD!

I would say adjusting max profit up to 0.5% is ok to do anytime when Nak isn't playing or waiting on a deposit to arrrive in the middle of a session as it only really affects him anyway, so don't worry about giving too much notice.

I would probably leave it at 0.25% though as it seems to be doing quite well and with investment going back up the max bet will soon be 100btc. Don't feel pressured into raising it back up a bit if it proves that 0.25% is working well.

While I work on the change, I am happy to put the max profit back up to 0.5%.  Half Kelly seems like a reasonable compromise.

How do I do this in a way that doesn't piss anyone off?  Changes without notice are "out of order".  Where do I announce it?  I don't have any way of contacting most of the investors.

Do I just post here "max profit will be adjusted to 0.5% of bankroll at (midnight tonight, UTC)" and leave it at that?  How much notice does it need?  That's only 2 hours away, and so probably not long enough.

What about "midnight pacific"?  That's 9 hours away.

Thoughts?  How much notice do people need?  And where's a good place to put such annoucements?

I made the point about the robbing. I'm glad you took it in the spirit in which it was intended. I was worried it would come off... I dunno, mean or tasteless.