What is the math behind a casino's "max bet" and bankroll management?

[APCHI]

Hello everyone, I'm interested in understanding the math behind casino bankroll management, for example :

for example, there's a big site that has a crash game with a 1% house edge, and they auto-cashout users at a win of 0.75% of their bankroll , so this is the max win.

example:

lets say their bankroll is 100$ and I bet 0.01$, Its not possible for me to win in a single round more than 0.75$ .



 how did they decide that the game mentioned must have a max win of 0.75%? what is the formula?

Will appreciate every answer, thanks and have a good day

[1715088682]

1715088682

[ralle14]

I remember some casinos use the kelly criterion to determine the max bet/profit while others uses a fixed percentage of the bankroll.

I don't think there's a must as some of them are willing to take more risks but I guess variance plays a big part since whales could easily put a dent on their bankroll.

I'm not good with explanations but this thread might help with the formula.

[Kyraishi]

Most sites that I know of use some sort of static max bet (in terms of BTC) that they know is sustainable. These are generally closed to outside investors and they probably use the Kelly criterion to determine the safe upper limit.

Others like JD which accepts bankroll investments will have a dynamic max bet that is generally a fixed percentage of their bankroll, but generally in line with the Kelly criterion.

Don't think there are any other major formulae involved other than this.

[shield132]

Here is a good quote from old Yolodice's medium article.

What is the profit limit of a bet?
Limit on profit of a single bet is
p_{\rm max} = \min\left\{ 0.005 B_{\rm eff}, 0.025 B_{\rm d} \right\}pmax​=min{0.005Beff​,0.025Bd​}
where BB is the effective bankroll (aka leveraged bankroll, including leverage loans), B_{\rm d}Bd​ is deposited bankroll. The player cannot place a bet that would result in profit larger than p_{\rm max}pmax​.
This limit ensures optimal bankroll growth rate while minimizing the variation in the bankroll.
TL;DR; A single bet cannot win more than 0.5% of effective bankroll or 2.5% of bankroll, whichever is lower.

[RHavar]

But the tldr is that a casino has to manage a who set of trade-offs. A non-comprehensive list:

* Bigger bets result in bigger EV
* Bigger max-profits result in higher risks
* There are risks that aren't represented (e.g. the site gets hacked and the hacker pretends to be a gambler) so higher limits make it easier to abuse
* Bigger bets/max-profits allow the game to appeal to more people (i.e. whales and newly-created whales)
* Big bets are great for marketing (who doesn't like a whale show!)

for example, there's a big site that has a crash game with a 1% house edge, and they auto-cashout users at a win of 0.75% of their bankroll , so this is the max win.

I assume you're talking about bustabit? If so, it's going to be a bit different due to the multiplayer aspect. Namely it needs to have a per-round max-profit (to protect the casino) and then a per-bet max-profit (so a single whale can't win almost all the per-round max-profit and ruin the fun for everyone).

--

But if you're interested in bankroll management, the goto resource would be:
https://en.wikipedia.org/wiki/Kelly_criterion

although its a bit intimidating at first

[slaman29]

Great replies and thanks for sharing everyone. I think most casinos doing some form of Kelly have really got it right, but RHavar mentions something that most casinos also don't really share which is hacks, which have resulted in some casinos losing and lowering drastically their max bets or max payouts. Which causes bigger players not to come or older players to leave (as it restricts martingale for example).

Something else I believe happens also is individual accounts also have their own max limits increased or decreased, depending on their pattern of play. For example, in Sportsbooks they do this depending on player ratio of win/loss versus expected win/loss.

I believe for a casino this works like:
1. if a player bets more and plays more on high house edge games with low max payout, they increase the wager limits.
2. if a player chooses highest payouts a lot, or plays low edge games, they lower limits.

I have no proof a casino actually does that but if Sportsbooks do it, I see no reason casinos implement the same. If, for example, you have a whale who loves betting millions on 1.1x dice, why would you stop them from betting a lot every day. Good advertisement for casino and still relatively low bankroll adjustment for his big bets even if he wins a lot.

[RealMalatesta]

It is basically just a way for the casinos to not lose all of their money very quickly, while trying to make as much profit as they can. This is literally the simplest way that anyone can explain it to you, that sentence is the whole deal. If you have 100 btc as your bankroll as a casino, if you allow someone to bet something that profits 100 btc, then if they win you are out of money and casino is over, but if you only allow 0.0000001 btc to be max profit then you are going to both not get a lot of people, and you are not going to profit at all.

So, that means you need to basically manage to have some middleground in this, both make a bit of money but also do not risk all of your money at the same time. These are all the simplest ways that max bet could be explained, and that is how they are calculating it, only finding the perfect balance.

[RHavar]

I'm also going to go out on a limb and say the Kelly criterion isn't nearly as important as people think. Although it's important (or a casino operator) to really understand it and the whole concept of "expected bankroll growth" (which the kelly maximizes) and how that can even be negative (!?!!).

But the reason it's not as important as it might seem, is because the casino isn't the one picking bet sizes. They can just put limits, which will and can drive players to other casinos. For a small and bankroll-constrained casino, it's probably not too unreasonable to risk up to twice the kelly optimal (at the expensive of _stupid_ variance). A big established casino probably doesn't even need or want the stress of running a full kelly (which is actually pretty extreme).

And you also can't be complacent, I remember when the (late) moneypot went to 3.33x kelly, I wrote them a simulation showing they were negative-expected-bankroll growth and a whale would most likely bankrupt them. They ignored me and thought it was too "theoretical". Couple months later they were actually bankrupt...

[YOSHIE]

Will appreciate every answer, thanks and have a good day

in Crash game in my opinion, doesn't require any skill or barrier to do it, in Crash bets only need to cash and double as much as possible, before the accident occurs, after the rocket takes off to aim for double profits.

In the Crash game to do multiplier odds is almost impossible for someone to make predictions, unlike casino games, sports and jokers, As far as I know, every gambling site has its own level of "house edge" in determining the player's win, in the Crash game the percentage can be said, it doesn't take any skill or barrier to do it, it's purely a matter of will and long lasting before the rocket falls, I can say Crash's game is based on luck.

[pawanjain]

I don't know how the casinos make this possible but if I were a casino site owner I would simply implement a code in the website which does the below.

I would get the bet amount from the user and get the bet result through the provably fair algorithm.
Upon receiving the winning bet I would calculate the total winning amount.
If the winning amount is greater than 0.75% of bankroll then I would keep an auto cashout where current bet result amount = 0.75% of bankroll
All of this would be happening very quickly.

[jaberwock]

It is basically just a way for the casinos to not lose all of their money very quickly, while trying to make as much profit as they can. This is literally the simplest way that anyone can explain it to you, that sentence is the whole deal. If you have 100 btc as your bankroll as a casino, if you allow someone to bet something that profits 100 btc, then if they win you are out of money and casino is over, but if you only allow 0.0000001 btc to be max profit then you are going to both not get a lot of people, and you are not going to profit at all.

So, that means you need to basically manage to have some middleground in this, both make a bit of money but also do not risk all of your money at the same time. These are all the simplest ways that max bet could be explained, and that is how they are calculating it, only finding the perfect balance.

It's complicated and much more complex than it seems from the outside. Casinos will most likely be challenged by a potentially bigger fish than the casino itself at some point. It's known to everyone that the bigger small shall swallow the smaller one. But here, the casino manages to eek out the bigger fish because they have a house edge in favor and have the ability to set max bet such that the casino doesn't lose quickly.

There isn't a set rule of maths or percentage the casinos will allocate towards the max bet. It all comes down to how the casino operates and maybe how much cash flow it has.

[barbara44]

how did they decide that the game mentioned must have a max win of 0.75%? what is the formula?

I haven't checked, so apology if someone has already answered this. But let me try and explain from my own understanding of the max bet. It depends on two things.

1- The Bankroll of the casino: If the casino has let's say $1million bankroll then usually the max bet will be something like $10 or so because the casino has to ensure that it is capable of affording a large number of losses, just in case.

2- Maybe the risk casino is able to take: Some casinos might have a smaller bankroll but allow larger max bets just because they have the appetite to risk more and allow players to gamble big.

I remember there used to be a casino who allowed max bet so big that once a player won big, they had to agree on a deal where the casino will pay the player in installments each month. Not sure how it worked out though.

[nakamura12]

I haven't checked, so apology if someone has already answered this. But let me try and explain from my own understanding of the max bet. It depends on two things.

1- The Bankroll of the casino: If the casino has let's say $1million bankroll then usually the max bet will be something like $10 or so because the casino has to ensure that it is capable of affording a large number of losses, just in case.

2- Maybe the risk casino is able to take: Some casinos might have a smaller bankroll but allow larger max bets just because they have the appetite to risk more and allow players to gamble big.

I remember there used to be a casino who allowed max bet so big that once a player won big, they had to agree on a deal where the casino will pay the player in installments each month. Not sure how it worked out though.

I think 1 is a lot more used of the casino wants less risky. The number 2 is a bit risky if a player won when he/she won but the bankroll is not enough to cover the payout and installment each month is how they will pay until it is fully paid. I don't think it is a problem if the casino have lots of gamblers using their service and if someone made a max bet can be paid even if it's a huge amount to pay.

[South Park]

I'm also going to go out on a limb and say the Kelly criterion isn't nearly as important as people think. Although it's important (or a casino operator) to really understand it and the whole concept of "expected bankroll growth" (which the kelly maximizes) and how that can even be negative (!?!!).

But the reason it's not as important as it might seem, is because the casino isn't the one picking bet sizes. They can just put limits, which will and can drive players to other casinos. For a small and bankroll-constrained casino, it's probably not too unreasonable to risk up to twice the kelly optimal (at the expensive of _stupid_ variance). A big established casino probably doesn't even need or want the stress of running a full kelly (which is actually pretty extreme).

And you also can't be complacent, I remember when the (late) moneypot went to 3.33x kelly, I wrote them a simulation showing they were negative-expected-bankroll growth and a whale would most likely bankrupt them. They ignored me and thought it was too "theoretical". Couple months later they were actually bankrupt...

And this is the problem of trying to maximize the profits at all costs, there is a concept called risk of bankruptcy, there are some equations to calculate this on stocks but I have seen the same concept applied to traders, gamblers and casinos and the only acceptable number for that number is zero, because anything above it means that sooner or later enough unfortunate events will happen that will lead you in that direction.