now I am confused. if 2x is leading to 0 bank toll growth then 3x is even worse and should lead to bank roll = investors losses. I dont see a reason to allow 3x then. please correct me if I am wrong

2x Kelly was the risk in the worst-case scenario, where all bets in a round target the same multiplier and would reach the round's profit limit. The intention behind that is to provide high rollers with a good experience on bustabit when several of them are playing at the same time, even if it means accepting suboptimal bets.

Even so, investors were never exposed to risk that carries an expectation of negative bankroll growth. When I suspended the commission, the worst-case risk dropped to 1.5x Kelly, ensuring that their expected bankroll growth was exclusively positive.



That is only the case for an individual investor that (ab)uses the offsite investment system to risk more than he actually has.

The offsite system is meant to allow investors to lower their counterparty risk and free up liquidity by not depositing their entire investment. I strongly recommend to only use it for that purpose. If you use the offsite system properly or don't use it at all, you never have an expectation of negative growth.

No, it was 2x Kelly with the commission and is now 1.5x Kelly without it.

Without the commission, investors' EV for each bet is 1 % (the house edge) and the risk per round is 1.5 % of the bankroll.

Oh, I see what you mean! Yeah, that's correct.

No. Thanks for sharing.

Well, the bankroll can never go negative - the house only ever risks a percentage of it, so at worst it approaches zero. Maybe you mean the site profit goes negative, but that's not necessarily the case either. It's possible for the house to over-leverage and still make a profit by getting lucky.

Thanks to Ryan's linked article I think I understand it now. The point I wasn't understanding is that "expected value" isn't the same as "expected bankroll growth".

Specifically, the expected value of each bet is 1% of the amount wagered, whether the house is over-leveraged or not. It's always positive so long as the house edge is positive.

Whereas the expected bankroll growth is more to do with the factor by which the bankroll grows with each bet.

For instance, suppose the house is risking 50% of its bankroll with every bet, that the payout is 2x, and the chance of the player winning is 49.5%. That's a standard 1% house edge bet, but massively over-leveraged.

Each time the house wins a bet like that, it adds 50% to its bankroll, or in other words it multiplies the bankroll by 1.5
Each time the house loses a bet like that, it loses 50% of its bankroll, or in other words it multiplies the bankroll by 0.5

In the long run, the house wins 0.505 times per bet, and loses 0.495 times per bet.

We can calculate the expected bankroll growth factor per bet as (growth when winning)^(probability of winning) * (growth when losing)^(probability of losing).

In this case, that's:

  1.5^0.505 * 0.5^0.495 = 0.8708

In other words, when the house is risking 50% of its bankroll every roll with a 1% house edge on a 49.5% bet, the house expects to multiply its bankroll by 0.8708 each roll, on average.

If we change the maximum bet to allow players to only win 2% of the house bankroll each bet, the multiplies change to 1.02 for a win and 0.98 for a loss, and so the expected bankroll growth factor changes to:

  1.02^0.505 * 0.98^0.495 = 1

In other words risking 2% of the bankroll with a 1% edge leads to a static bankroll.


What really made me see it, is imagine the case whether the house risks 50% of its bankroll each time, and starts with 100 units.

If the player wins their first bet, the bankroll is down to 50 units. Then if the player loses a bet, the bankroll only goes up to 75.
On the other hand, if the player loses their first bet, the bankroll goes up to 150 untis, but then if the player wins, the bankroll again goes down to 75.

So each time the player makes two bets where one wins and one loses, the bankroll drops 25%. In the long run, the number of wins and losses is going to be about the same (assuming 50% betting, and no house edge, let's say). We can pair each winning bet with a losing bet, and see that the house is going to lose 25% about n/2 times, where n is the number of bets that were made. There's a tiny chance that the player does exceptionally badly and ends up massively down, but by far the most likely outcome is that the house loses almost all of its bankroll.

I thought you had already lost all your money on just-dice!

"To protect investors, the most a single player can win in one game is 0.75 % of the bankroll, in line with the Kelly criterion."




You guys definitely defrauded investors, lied, and abused your powers.  Full scam accusation coming soon.

Stop asking me to leave to try and hide.  We had the opportunity to handle this in a civil manner, but RHavar, being the scumbag he is, tried to discredit me by calling me an idiot and encouraged everyone to look away.

When millions of dollars are at stake and you are both profiting off of it, the details need to be fully investigated and combed through, not just swept under the rug like you guys have attempted to do.

You are both dishonest and shameful human beings.

You would think with millions of dollars at stake, you would take it more serious and actually make sure you have all the facts stated clearly and run the proper models so that what you claim is true.

You did not do this.  There were much simpler solutions to keep investors protected, but you both refused to admit to mistakes and greedily kept accepting higher bets while soliciting for a larger bankroll.

You two are dishonest and shameful human beings.

please dont forget the HE of a casino or the edge a player could have in case he uses KC