bustabit – The original crash game

[RHavar]

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

Yup, great explanation btw. I think your summary is a lot clearer than the original.

So now that we have a way of calculating the expected bankroll growth, we can introduce another term "the kelly".

If you plot the expected bankroll growth (EBG) against how much you are risking -- you will see the chart is sort of parabolic. And the point in which  EBG is maximized is known as the kelly.

Now a casino isn't really able to make every bet "a kelly" because that would require telling players how to bet, but what it can do is limit bets that get too risky. I'm not really sure there's a good robust way to do this, at it totally depends on the context.

For instance it's my unconfirmed belief that physical casinos employ very strict limits on the general tables because they know if someone wins "big" they will just walk, instead of turning over the money. (i.e. last casino I was in, had a $50 limit on a number in roulette, unless you were a high roller in which case they were happy taking over x10 that).

Bustabit also has some pretty unique constraints, as the limits can everyone. So it really needs to balance the idea of having a "hard cap" risk amount it accepts per game, while also being able to accept as much from a single player as possible. The system it uses of having a 1x kelly per-player and a hard-cap of a 2x kelly (or currently 1.5x) per-game seems pretty reasonable.

The other interesting thing about 2x kelly, is that for a binary bet (win all, or lose all) there is 0-EBG (but positive +EV). But actually bustabit (from the houses perspective) isn't a binary bet, it can lose some and win some. So assuming I calculated everything right (which very well might not be the case, as I can't come up with an analytical solution) it's actually quite common for a 2x kelly to be positive expected bankroll growth! (and the worst case is a 2x kelly is 0 EBG)

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[quickmaffs]

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

Yup, great explanation btw. I think your summary is a lot clearer than the original.

So now that we have a way of calculating the expected bankroll growth, we can introduce another term "the kelly".

If you plot the expected bankroll growth (EBG) against how much you are risking -- you will see the chart is sort of parabolic. And the point in which  EBG is maximized is known as the kelly.

Now a casino isn't really able to make every bet "a kelly" because that would require telling players how to bet, but what it can do is limit bets that get too risky. I'm not really sure there's a good robust way to do this, at it totally depends on the context.

For instance it's my unconfirmed belief that physical casinos employ very strict limits on the general tables because they know if someone wins "big" they will just walk, instead of turning over the money. (i.e. last casino I was in, had a $50 limit on a number in roulette, unless you were a high roller in which case they were happy taking over x10 that).

Bustabit also has some pretty unique constraints, as the limits can everyone. So it really needs to balance the idea of having a "hard cap" risk amount it accepts per game, while also being able to accept as much from a single player as possible. The system it uses of having a 1x kelly per-player and a hard-cap of a 2x kelly (or currently 1.5x) per-game seems pretty reasonable.

The other interesting thing about 2x kelly, is that for a binary bet (win all, or lose all) there is 0-EBG (but positive +EV). But actually bustabit (from the houses perspective) isn't a binary bet, it can lose some and win some. So assuming I calculated everything right (which very well might not be the case, as I can't come up with an analytical solution) it's actually quite common for a 2x kelly to be positive expected bankroll growth! (and the worst case is a 2x kelly is 0 EBG)

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.

[Luxo42]

The other interesting thing about 2x kelly, is that for a binary bet (win all, or lose all) there is 0-EBG (but positive +EV). But actually bustabit (from the houses perspective) isn't a binary bet, it can lose some and win some. So assuming I calculated everything right (which very well might not be the case, as I can't come up with an analytical solution) it's actually quite common for a 2x kelly to be positive expected bankroll growth! (and the worst case is a 2x kelly is 0 EBG)

2x kelly is not 0-ebg for any binary (two-option) bet. It's only for 50% bet. ADDED later: and one of multipliers is 0.
For bets with low probability 0-EBG point is very slightly less than 2x kelly. Wolfram alpha
Something like up to 1.9933 Kelly
For bets with high probability 0-EBG point is more than 2x kelly. Wolfram alpha

[JackpotRacer]

The other interesting thing about 2x kelly, is that for a binary bet (win all, or lose all) there is 0-EBG (but positive +EV). But actually bustabit (from the houses perspective) isn't a binary bet, it can lose some and win some. So assuming I calculated everything right (which very well might not be the case, as I can't come up with an analytical solution) it's actually quite common for a 2x kelly to be positive expected bankroll growth! (and the worst case is a 2x kelly is 0 EBG)

2x kelly is not 0-ebg for any binary (two-option) bet. It's only for 50% bet.
For bets with low probability 0-EBG point is very slightly less than 2x kelly. Wolfram alpha
Something like up to 1.9933 Kelly
For bets with high probability 0-EBG point is more than 2x kelly. Wolfram alpha

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

[Luxo42]

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

HE=1% (look at ev - it's 0.99)

[quickmaffs]

The other interesting thing about 2x kelly, is that for a binary bet (win all, or lose all) there is 0-EBG (but positive +EV). But actually bustabit (from the houses perspective) isn't a binary bet, it can lose some and win some. So assuming I calculated everything right (which very well might not be the case, as I can't come up with an analytical solution) it's actually quite common for a 2x kelly to be positive expected bankroll growth! (and the worst case is a 2x kelly is 0 EBG)

2x kelly is not 0-ebg for any binary (two-option) bet. It's only for 50% bet.
For bets with low probability 0-EBG point is very slightly less than 2x kelly. Wolfram alpha
Something like up to 1.9933 Kelly
For bets with high probability 0-EBG point is more than 2x kelly. Wolfram alpha

https://i.imgur.com/8t8k8IP.png

"In general, all investors have only been exposed to +EV and positive expected bankroll growth."

RHavar

[RHavar]

The other interesting thing about 2x kelly, is that for a binary bet (win all, or lose all) there is 0-EBG (but positive +EV). But actually bustabit (from the houses perspective) isn't a binary bet, it can lose some and win some. So assuming I calculated everything right (which very well might not be the case, as I can't come up with an analytical solution) it's actually quite common for a 2x kelly to be positive expected bankroll growth! (and the worst case is a 2x kelly is 0 EBG)

2x kelly is not 0-ebg for any binary (two-option) bet. It's only for 50% bet.
For bets with low probability 0-EBG point is very slightly less than 2x kelly. Wolfram alpha
Something like up to 1.9933 Kelly
For bets with high probability 0-EBG point is more than 2x kelly. Wolfram alpha

Oh wow, brilliant work. I always thought 2x kelly = 0 EBG on a binary bet, but your formula looks correct and it shows that 2x kelly = 0EBG only for the specific case of 2x bet. This actually seems like it's rather good news for investors.

So currently bustabit restricts the per-game limit to 1.5x kelly, but let's assume it used 2x -- the worst behavior could be triggered by a whale max-betting on 200 accounts which would set the forced-point to 1.01x (and stop the game server accepting any new bets). In this kind of insane case, it would only be very slightly negative EBG. But in a more common case of people aiming at higher multiples (especially lottery-style bets), a 2x kelly still leads to very healthy +EBG.

What would be a really cool solution, is that that you dynamically adjusted the limits based on the persons bet multiple. However, I can't really see how to do that cleanly in bustabit without removing the manual-cashout, which would totally suck.

Probably the best thing for bustabit would to never go above 1.99x at the most extreme. That should guarantee that all games are +EBG. Although honestly, I'm not entirely sure that's the best idea. As EBG is only one of the many things that bustabit needs to control for, another thing for instance is controlling variance. Because investing isn't provably fair, if a whale came and won 80% of the bankroll it would look pretty bad even though the site is doing everything correctly.

[JackpotRacer]

Because investing isn't provably fair, if a whale came and won 80% of the bankroll it would look pretty bad even though the site is doing everything correctly.

who will  finally bring the Provably Fair option for investors?

[devans]

who will  finally bring the Provably Fair option for investors?

For dice games, bustadice goes pretty far. I've seen some interesting ideas to make it entirely provably fair for investors, but unfortunately none of them were really feasible.

[Luxo42]

So currently bustabit restricts the per-game limit to 1.5x kelly, but let's assume it used 2x -- the worst behavior could be triggered by a whale max-betting on 200 accounts which would set the forced-point to 1.01x (and stop the game server accepting any new bets). In this kind of insane case, it would only be very slightly negative EBG. But in a more common case of people aiming at higher multiples (especially lottery-style bets), a 2x kelly still leads to very healthy +EBG.

Vice versa. The worst case for casino is low-value-betting on high multipliers, until forced cashout. According to my calculations, max-profit = bankroll * house_edge * (2-house-edge) would be EBG 0+ for any cases.

[RHavar]

Vice versa. The worst case for casino is low-value-betting on high multipliers, until forced cashout. According to my calculations, max-profit = bankroll * house_edge * (2-house-edge) would be EBG 0+ for any cases.

You sure about that? It looks to me that the worst case is for the casino is low-value betting on high multipliers from the casinos perspective, but this is generated by a player betting a large amount on a low multiplier. (i.e. if a player is betting 1 BTC to try win 10 BTC,  from the casinos perspective this is a 10 BTC bet trying to win 1 BTC)

[RGBKey]

who will  finally bring the Provably Fair option for investors?

For dice games, bustadice goes pretty far. I've seen some interesting ideas to make it entirely provably fair for investors, but unfortunately none of them were really feasible.

I'm happy with the solution that bustadice uses, but I'm still in search for the pot of gold at the end of the rainbow that would be a fully trustless zero-knowledge on-demand RNG. Something where nobody has to trust any number of people and could be used to make games fair for both players and investors.

[Luxo42]

Vice versa. The worst case for casino is low-value-betting on high multipliers, until forced cashout. According to my calculations, max-profit = bankroll * house_edge * (2-house-edge) would be EBG 0+ for any cases.

You sure about that? It looks to me that the worst case is for the casino is low-value betting on high multipliers from the casinos perspective, but this is generated by a player betting a large amount on a low multiplier. (i.e. if a player is betting 1 BTC to try win 10 BTC,  from the casinos perspective this is a 10 BTC bet trying to win 1 BTC)

Rechecked again. Sure.
Player: low-value high-multiplier low-win-probability
Casino: high-value low-multiplier high-win-probability
0-EBG at <2 Kelly

Maybe, it's really counterintuitive.
But, for example, casino (HE=1%) may lose 5% of bankroll per bet with high probability (slightly less than 99%). But player's bet is much higher than the whole bankroll. And that's is EBG+ for casino.

[RHavar]

Rechecked again. Sure.

Touché, I checked again and you do indeed seem right. I'm both pretty embarrassed about my math and impressed by yours. Well done

[Luxo42]

In fact, it does not matter whether the maximum profit, kelly criterion, fees and others things was set correctly or not.

What if some lucky player won a lot at casino? Investors lost money.
Investor can not prove that a lucky player is a casino. And that is not just luck, but scam.
Casino can not prove that a lucky player is not casino. And that is not scam.
So such investment is entirely based on trust to the owner.

[Luxo42]

What would be a really cool solution, is that that you dynamically adjusted the limits based on the persons bet multiple. However, I can't really see how to do that cleanly in bustabit without removing the manual-cashout, which would totally suck.

I think you shouldn't pay attention to this. 0-EBG point moves from 2xKelly very slow and only in extreme cases.
max profit = 1xKelly or 2xKelly minus HE looks good solution for me.

[devans]

Now that investors are comfortably back in profit at approximately 324 BTC, I have reinstated the wager commission of 0.25 %.

Consequently, the bet limit and the maximum profit per player have been lowered to 0.75 % of the bankroll again and the maximum profit per game has been lowered to 1.125 % of the bankroll.

[dooglus]

the maximum profit per game has been lowered to 1.125 % of the bankroll

inb4 "dishonest and shameful"...

[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? Whether they are lying about it is only "revealed" when their onsite gets to 0, so why would A's onsite go to 0 whereas B's onsite doesn't - even before it is known whether they are lying or not? Surely the site can not predict the future of whether A or B will turn out to have been lying, and then set A to -EBG and B to +EBG based on predicting the future. Whether they are at +EBG or -EBG depends solely on their onsite and offsite, obviously...

[DarkStar_]

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? Whether they are lying about it is only "revealed" when their onsite gets to 0, so why would A's onsite go to 0 whereas B's onsite doesn't - even before it is known whether they are lying or not? Surely the site can not predict the future of whether A or B will turn out to have been lying, and then set A to -EBG and B to +EBG based on predicting the future. Whether they are at +EBG or -EBG depends solely on their onsite and offsite, obviously...

When you leverage, the expectation is that you are willing to move those off-site coins on site as you lose, so that you do not get margin called. You are at a 2x Kelly maximum (current bustabit rules), which allows for +EBG. That person can expect their bankroll to increase.

A 50 BTC + 50 BTC fake off-site gives a 4x Kelly, assuming no off-site is 2x Kelly. Once your investment loses 50 BTC, you are left with nothing. It doesn't matter what the site thinks about your investment. By treating 1 coin as 2 coins, you are increasing your exposure. If everything invested with 2x leverage on bustabit, the max win would technically be 2.5% of the actual investment.