bustabit – The original crash game

[suchmoon]

I'm surprised nobody has called JD "dishonest and shameful"... or maybe they have and I just missed the drama.

Fun fact: not a single person has ever been "margin called" on JD. Probably because the bankroll is so large compared to the size of bets that happen that there's never a significant percentage downturn in the bankroll.

Even if you're at 25x offsite, the bankroll needs to draw down 4% before anyone is margin called. The current bankroll is 19.6 million, so the site would need to lose 784,000 CLAMs to cause the riskiest investor to get margin called. That's 8 max-profit bets. (JD max profit is 0.5% of the bankroll).

It's not impossible, but it does seem pretty unlikely.

All it takes is a whale betting 9 million CLAM at 99%

Slightly different question - is there a reason why "leverage" or "offsite" on BAB and JD is not percentage-based but a fixed amount? I.e. if I have 5 coins onsite, I might want to set offsite to 100% of onsite instead of 5 coins and that way offsite could go up or down. This would make the whole EBG- theory of the last few pages more in line with reality.

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[dooglus]

All it takes is a whale betting 9 million CLAM at 99%

Or more realistically betting 10k CLAM at 10%, and hitting 8 times more than expected.

Slightly different question - is there a reason why "leverage" or "offsite" on BAB and JD is not percentage-based but a fixed amount? I.e. if I have 5 coins onsite, I might want to set offsite to 100% of onsite instead of 5 coins and that way offsite could go up or down. This would make the whole EBG- theory of the last few pages more in line with reality.

I designed it to allow investors to reduce their counterparty risk. The amount of value they have offsite that they are holding back is closer to a constant than a multiple of their onsite amount. When their onsite amount doubles due to a whale losing, that doesn't change how much they own that isn't deposited.

And BAB pretty much copied JD's design for the 'offsite' stuff.

Another reason is purely practical. If all the investors have the same leverage, we don't need to calculate each investor's bankroll size after every bet. All we need to store is what percentage of the bankroll is theirs. If you have 10% of the bankroll before someone bets, you still have 10% of the bankroll after they bet. All I need to update is the size of the bankroll. If you are using 2x leverage and I'm using 3x, my share of the bankroll will increase and yours will decrease each time a player loses a bet. That adds additional complexity that I didn't want to deal with.

Following on from my post earlier about "real leverage" vs. the fixed offsite investment, maybe it's interesting to compare how going 10x using offsite investment compares with using an actual 10x leverage:

Suppose A uses offsite investment, starts with 10 BTC and declares 90 BTC "offsite". We can call this "10x offsite". He has an effective bankroll of 100 BTC and risks 1% of it per bet. So he's risking 1 BTC on the first bet.

And suppose C uses actual 10x leverage (not currently possible on bustabit, but let's assume). He starts with 10 BTC, and risks 10% of it per bet. So he's also risking 1 BTC on the first bet.

The first bet has the same effect on both investors. They win or lose 1 BTC.

If the house wins:
  * A now has an effective bankroll of 11 BTC onsite + 90 BTC offsite for a total of 101 BTC, and risks 1.01 BTC on the next bet.
  * C now has a bankroll of 11 BTC and risks 10% of it, or 1.1 BTC on the next bet.

We can see that C (using actual 10x leverage) has increased his risk per bet by 10%, while A (using 10x offsite) has only increased his risk per bet by 1%.

If the house wins again:
  * A wins 1.01 BTC, has 12.01 + 90 = 102.01 effective, and risks 1.0201 BTC on the next bet.
  * C wins 1.1 BTC, has 12.1 BTC, and risks 1.21 BTC on the next bet.

After 10 wins:
  * A has profited by 100 * 1.01**10 - 100 = 10.4622 BTC (104.62% profit)
  * B has profited by 10 * 1.1**10 - 10 = 15.9374 BTC (159.37% profit)

After 50 wins:
  * A has 7.4x'ed his 10 BTC (((100 * 1.01**50) - 90) / 10 = 7.4)
  * B has 117.4x'ed his 10 BTC ((10 * 1.1**50) / 10 = 117.39085287969579)

So in the best case, the 'real leverage' investor does a lot better than the fixed 'offsite' investor.

In the worse case, where the player wins all his max bets, the 'real leverage' investor also clearly does a lot better than the fixed 'offsite' investor. The fixed 'offsite' investor always (until he goes bust) has at least 90 BTC invested (his offsite amount), and so risks at least 0.9 BTC every bet. Since he started with 10 BTC, he will be bust after just 11 or 12 winning max bets, whereas the 'real leverage' investor will still have 0.515 BTC left even if the player wins 50 max bets in a row.

This likely means that the "real leverage" player does worse than the fixed offsite investor in the much more common case where the player wins and loses more equally.

[suchmoon]

Another reason is purely practical. If all the investors have the same leverage, we don't need to calculate each investor's bankroll size after every bet. All we need to store is what percentage of the bankroll is theirs. If you have 10% of the bankroll before someone bets, you still have 10% of the bankroll after they bet. All I need to update is the size of the bankroll. If you are using 2x leverage and I'm using 3x, my share of the bankroll will increase and yours will decrease each time a player loses a bet. That adds additional complexity that I didn't want to deal with.

Makes sense, thanks.

This likely means that the "real leverage" player does worse than the fixed offsite investor in the much more common case where the player wins and loses more equally.

Shouldn't "real leverage" still do better in that scenario if the players lose slightly more than they win as expected due to house edge?

[01010100b]

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.

This statement is false, just think about it. Suppose there are two investors, A and B, and each have 50 btc onsite and 50 btc offsite. A is lying about it but B isn't. Why would A have -EBG but B have +EBG?

I've been attempting to find a good answer to your question, and came up with a surprising realization. Your "A" guy would appear to be running a 2x kelly risk, and so we would say he has zero expected bankroll growth. We know that if the percentage you risk is twice the house edge then your expected growth is zero.

BUT... he is only risking 2x on the first bet. After that his offsite stays constant. If the house loses money, A starts risking more than 2x kelly, and if the house wins money, A finds himself risking less than 2x kelly. A isn't really using "leverage" at all. Leverage would be where he is constantly risking 2%. But this "offsite" feature means he is always risking 1% of (onsite + 50) - which is quite a different thing.

In conclusion, we can't accurately say that if you go 2x using offsite investment then you can expect 0 bankroll growth. That's not true, because kelly only talks about what happens if you are risking the same percentage of your actual bankroll on every bet, and A isn't...

Yes, that's exactly why I'm saying that it would be best if they just stopped referring to it as leverage, because the system doesn't function as one.

[01010100b]

However I think he made the same mistake I struggled with too, which was thinking that if the bankroll is -EBG that would allow a player to be +EBG.  Fortunately (for investors) this isn't the case, so there's no real abuse avenue.

Can you check where I'm going wrong with this? Here's how I looked at it:

Let BG and PG be two random variables, BG is the house's bankroll growth and PG is the player's bankroll growth (in absolute numbers). Then:

BG + PG = 0 (money only moves back and forth between the house and the player, so a loss for one is a gain for the other and vice versa)
E[BG + PG] = E[0] (taking the expected value of both sides)
E[BG + PG] = 0 (expected value of a constant is that constant)
E[BG] + E[PG] = 0 (by linearity of expected value)
E[BG] = -E[PG]

Which leads to the player's EBG being the opposite of the house's EBG, so if the house is at -EBG then the player is at +EBG and vice versa. Perhaps by EBG is meant the median bankroll growth and not the mean bankroll growth? In which case the linearity condition above wouldn't apply.

[Luxo42]

Investment with fixed offsite bankrolls looks very bad unlike with fixed multiplier / leverage

It is not clear for me how such investment system will behave in some cases. I would like to see the source code or investment math of BaB and JD (if that's possible and appropriate for dooglus and devans)

And here is a case.
Two investors - A is onsite and B is offsite. 1xKK.
Very bad day for casino. Investor B is in pre-margin-call position.
Investor A: 8 onsite, 0 offsite
Investor B: 1 onsite,  991 offsite
Bankroll: 1000, max profit: 10. Player bet and win.
And thas is disaster. Max profit more than onsite bankroll. It's more than casino's "cash"! Casino can not pay to player. What EBG are you talking about?
Of course, this is rare case. But math should works always everywhere. if it is not, then the whole investement system is working incorrectly.

Looks like there is no such problems at leverage investment system.

Maybe, would be better to create separate topic?

[Luxo42]

However I think he made the same mistake I struggled with too, which was thinking that if the bankroll is -EBG that would allow a player to be +EBG.  Fortunately (for investors) this isn't the case, so there's no real abuse avenue.

Can you check where I'm going wrong with this? Here's how I looked at it:

Let BG and PG be two random variables, BG is the house's bankroll growth and PG is the player's bankroll growth (in absolute numbers). Then:

BG + PG = 0 (money only moves back and forth between the house and the player, so a loss for one is a gain for the other and vice versa)
E[BG + PG] = E[0] (taking the expected value of both sides)
E[BG + PG] = 0 (expected value of a constant is that constant)
E[BG] + E[PG] = 0 (by linearity of expected value)
E[BG] = -E[PG]

Which leads to the player's EBG being the opposite of the house's EBG, so if the house is at -EBG then the player is at +EBG and vice versa. Perhaps by EBG is meant the median bankroll growth and not the mean bankroll growth? In which case the linearity condition above wouldn't apply.

No. That is for expected value only. EVcasino + EVplayer = 0.

Try EBG * EPG = 1
For example, EBG = 1.05 (+5%)
So, EPG = 1/1.05 = 0.952 (-4.76%)
But i'm not sure it would work.

[01010100b]

However I think he made the same mistake I struggled with too, which was thinking that if the bankroll is -EBG that would allow a player to be +EBG.  Fortunately (for investors) this isn't the case, so there's no real abuse avenue.

Can you check where I'm going wrong with this? Here's how I looked at it:

Let BG and PG be two random variables, BG is the house's bankroll growth and PG is the player's bankroll growth (in absolute numbers). Then:

BG + PG = 0 (money only moves back and forth between the house and the player, so a loss for one is a gain for the other and vice versa)
E[BG + PG] = E[0] (taking the expected value of both sides)
E[BG + PG] = 0 (expected value of a constant is that constant)
E[BG] + E[PG] = 0 (by linearity of expected value)
E[BG] = -E[PG]

Which leads to the player's EBG being the opposite of the house's EBG, so if the house is at -EBG then the player is at +EBG and vice versa. Perhaps by EBG is meant the median bankroll growth and not the mean bankroll growth? In which case the linearity condition above wouldn't apply.

No. That is for expected value only.

Yes, the expected value of the bankroll growth in absolute numbers. BG and PG are random variables.

[Luxo42]

Yes, the expected value of the bankroll growth in absolute numbers. BG and PG are random variables.

"Expected value of the bankroll growth in absolute numbers" - crazy confusing wording.
There are "expected value" and "expected growth".

[01010100b]

Yes, the expected value of the bankroll growth in absolute numbers. BG and PG are random variables.

"Expected value of the bankroll growth in absolute numbers" - crazy confusing wording.
There are "expected value" and "expected growth".

Do you know what a random variable is? Do you know what the expected value of a random variable is? I'm not talking about the expected value of the bet, I'm talking about the expected value of the bankroll growth.

[01010100b]

It's at least definitely possible for a player to be +EBG yet -EV, the edge case is simple. Suppose the casino bets 100% of its bankroll on each bet, then even though it may have +EV, at some point it's going to lose. So the player will have, with probability one, gained money and hence is at +EBG - though he may require a deep bankroll.

[quickmaffs]

It's at least definitely possible for a player to be +EBG yet -EV, the edge case is simple. Suppose the casino bets 100% of its bankroll on each bet, then even though it may have +EV, at some point it's going to lose. So the player will have, with probability one, gained money and hence is at +EBG - though he may require a deep bankroll.

Yes, that's the angry whale with tons of money scenario with over aggressive kelly criterion applied. It was present in Bustabit for a big stretch there all while devans and RHavar profited in excess of a million dollars.

The thing is though that these models assume that there are no investments or divestments in between.  

Because the whale is betting at a negative house edge, there should be opportunities for you, as an investor, to pull the rug and divest (not to be confused with deleveraging onsite/offsite) your earnings with profit.  This then means that you're not investing anymore, but gambling at a positive house edge.  

If, however, you keep your money locked in the investment forever, you make zero at 0EBG and lose it all at -EBG.  This is one of the reasons that the predatory system which punishes investors for divesting is just plain wrong.

While taking commission, devans makes his highest profit when the bankroll is its highest; the higher the bankroll, the higher the max profit and amount of wagering can be.  That's why it is dishonest and shameful to keep soliciting for a bigger bankroll and to try and pitch Bustabit as a stable investment (lie) with only positive expected bankroll growth (lie) at 1x kelly (lie).  

By far, the easiest solution was to cap max profit per game/round at 1% (or 0.75% with commission) with no dilution.  You'd think they'd figure it out by now.  The excuse that it was to protect the players is absolute bullshit.  It's a 1% house edge game strictly against the house with no bonuses at all.  

[dooglus]

Can you check where I'm going wrong with this? Here's how I looked at it:

Let BG and PG be two random variables, BG is the house's bankroll growth and PG is the player's bankroll growth (in absolute numbers). Then:

BG + PG = 0 (money only moves back and forth between the house and the player, so a loss for one is a gain for the other and vice versa)

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.

[01010100b]

Can you check where I'm going wrong with this? Here's how I looked at it:

Let BG and PG be two random variables, BG is the house's bankroll growth and PG is the player's bankroll growth (in absolute numbers). Then:

BG + PG = 0 (money only moves back and forth between the house and the player, so a loss for one is a gain for the other and vice versa)

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

Obviously, and that is indeed the reason why I'm not summing them. I said "in absolute numbers" multiple times now. Anyway, I'm pretty sure my argument is correct, especially since the edge case trivially shows that it is indeed possible for a player to be -EV and +EBG.

[dooglus]

This likely means that the "real leverage" player does worse than the fixed offsite investor in the much more common case where the player wins and loses more equally.

Shouldn't "real leverage" still do better in that scenario if the players lose slightly more than they win as expected due to house edge?

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.

[dooglus]

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

Obviously, and that is indeed the reason why I'm not summing them. I said "in absolute numbers" multiple times now.

These "absolute numbers" you're talking about are the expected profits, not the expected growth factors. They are different things.

Expected profits are "absolute numbers" - you can add them. Expected growths are multipliers - you can multiply them.

[RHavar]

I can't imagine that "real leverage" is better than "offsite investing" in every case. Shouldn't there be a trade-off?

Agree. I think they're largely orthogonal. Although "offsite investing" can be used to (poorly) emulate "real leverage" and vice versa, but I think they largely serve different goals. There's definitely no reason you couldn't combine both

[01010100b]

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

Obviously, and that is indeed the reason why I'm not summing them. I said "in absolute numbers" multiple times now.

These "absolute numbers" you're talking about are the expected profits, not the expected growth factors. They are different things.

Expected profits are "absolute numbers" - you can add them. Expected growths are multipliers - you can multiply them.

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.

[suchmoon]

This likely means that the "real leverage" player does worse than the fixed offsite investor in the much more common case where the player wins and loses more equally.

Shouldn't "real leverage" still do better in that scenario if the players lose slightly more than they win as expected due to house edge?

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.

But isn't "B" changing as well and affecting the investors differently, as well as the max bet? I'm really bad at this abstract math thing so I tried to fill out a spreadsheet:



Maybe I messed something up but I can figure out what exactly.

Edit: I did mess it up. My previous calculation didn't do the leverage correctly. I have updated the screenshot however I still don't get the bankroll loss for C.

Edit2: Never mind. I think I get it now. Will pour myself a couple more and try to actually understand it.

[Luxo42]

This likely means that the "real leverage" player does worse than the fixed offsite investor in the much more common case where the player wins and loses more equally.

Shouldn't "real leverage" still do better in that scenario if the players lose slightly more than they win as expected due to house edge?

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.

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?