When we talk about "growth", we're talking about multipliers, not deltas.

For example, if a player's bankroll grows from 10 to 11 while the house's bankroll shrinks from 10 to 9, the player's bankroll growth is 1.1x while the house's is 0.9x.

It doesn't make sense to sum those growth multipliers.

"Negative expected growth" means an expected multiplier of less than 1.

I kind of ran out of steam at that point. I can't imagine that "real leverage" is better than "offsite investing" in every case. Shouldn't there be a trade-off?

Back to the example:

A has 10 onsite and 90 offsite.
C has 10 onsite with 10x "real" leverage.

Suppose a whale plays for a long period with 50% chance of winning each bet. All bets are the maximum, aiming to win 1% of the bankroll. The payout when he wins is 1.98x (1% house edge), and he wins and loses the same amount of bets (as he is expected to do, long term). Let's call the effective bankroll "B".

50% of the time the player wins 1% of the bankroll: B/100. When this happens, A's effective bankroll is multiplied by 99/100, and C's is multiplied by 90/100.

The other 50% of the time the player loses his stake. He's aiming to profit by B/100, with a payout multiplier of 1.98x, so he's risking and losing B/98. When this happens, A's effective bankroll is multiplied by 99/98, and C's is multiplied by 108/98.

Since the whale wins and loses the same number of bets, we can pair these bets. Each pair consists of one win and one loss.

For each pair, A's effective bankroll is multiplied by 99/100 and 99/98, for a net growth factor of 99/100 * 99/98 = 9801/9800 ~= 1.0001x

And C's effective bankroll is multiplied by 90/100 and 108/98, for a net growth factor of 90/100 * 108/98 = 9720 9800 ~= 0.9918x

And so we see that my intuition was correct, and that "real" leverage is worse than this "offsite investment" thing. The "offsite investment" has a (small but) positive expected bankroll growth whereas the "real leverage" expects to lose almost 1% of the bankroll for every pair of (1 win + 1 lose) bets.

So you have nothing but semantics and quibbling over the meaning of the word "growth"? Do you perhaps have something substantial to say about the argument I made?

Let's define this properly. Let B_0 be the initial bankroll and B_N the bankroll after a sequence of N bets. Then BG is a random variable which, when given said sequence of N bets, takes the value B_0 * log(B_N B_0) N. Likewise for PG.

Yes, EBG<1 for 10x leverage.
But leverage multiplier should be limited!

For max win = 1KK max leverage = 2x.
For max win = 0.5KK max leverage = 4x.
Investor with max leverage is EBG 0+ always.
When player is whale, regular investor's EBG+, max leverage investor's EBG-zero.
But when player is not whale, both EBG are positive. And regular investor's EBG is lower than max leverage investor's EBG. So, leverage works.

Therefore, "real leverage investment" properly works in all cases. But "offsite investement" not. Or am I wrong?

I do not exactlly understand what is meant by a a lot of bankrolls and virtual bankroll.

My maths are next:
Investor 1: 5 x1
Investor 2: 1 x2
Investor 3: 1 x3
Bankroll = 5*1+1*2+1*3 = 10
Bankroll's shares: investor 1 - 50%, investor 2 - 20%, investor 3 - 30%.
Casino'a wins/losses divides among investors according to their shares.

I'm definitely for being able to choose the max kelly risk. The only thing keeping me from investing into the site is the crazy amount of potential volatility when a whale/whales bets for max profit (as many have seen with that multi account mega whale). There are huge potential swings in the bankroll that can occur at 1.5x kelly and even at 1x kelly. I would personally like to invest at 0.5x kelly and I'm sure many investors probably feel the same.

Also I think a per game max profit of 0.5x, 1x, and 1.5x kelly (current) should be sufficient.

Yes your % share of overall bankroll would change according to the kelly you set. The max profit would also be a variable % of overall bankroll (wouldn't really make a difference to players as they only see the max profit value).

In the hypothetical case that true "leverage" investing is offered, how would the dilution fee work?

Would someone investing 1 BTC at 2x kelly be charged the same dilution fee as someone investing 1 BTC at 1x kelly? Or would they be charged the same amount?

Would people be allowed to change their kelly setting at will, without incurring dilution fees? Or would it work much like the "offsite" feature does at the moment?

Allowing people to reduce their kelly multiplier for free re-enables 'whale dodging" behavior, which the dilution fee was designed to prevent.

I guess the most reasonable way to work it would be to charge dilution fees on ((onsite + offsite) * kelly), since that is the effective contribution to the bankroll.

Of course it was because your argument is towards the former owner and the current owner of the site whom he sold the site too so whatever you had found any evidence about on the old system was transitioned over to the new owner and now they need to fess up about it if not confronted to in the first place.

Usually the bankroll isn't being fully utilized, i.e. players aren't going for maximum profit in every round. As an investor, however, you ideally want your risk to be the Kelly criterion for optimal growth.

If players are only attempting to win 10% of the maximum profit in "slow" rounds, an investor can optimize his own bankroll growth by leveraging to 10x, meaning he will be risking the Kelly criterion again. Then, when a high roller starts playing and fully utilizing the maximum profit, the investor would stop leveraging again, thereby dodging the whale.

This is bad for the casino because it leads to a poor experience for the high roller and is arguably unfair towards the investors that aren't micromanaging their investment. Giving investing and indirectly divesting a cost disincentives "whale dodging".