Source: have worked with a situation which required me analyzing the process of getting betting software through regulatory approval.
Edit to add: Here's the relevant regulations in Nevada. https://gaming.nv.gov/modules/showdocument.aspx?documentid=4...
The thing to search for in the regulations is the "Theoretical hold percentage", which is the expectation for the house expressed as a percent. The fact that they call it "theoretical" is probably enough to demonstrate that it isn't a mandated rate, but the theoretical average rate.
You can set the odds of a fully random process, just take an rng returning a uniform distribution between 1 and 100 and set the bar at whichever threshold you want, or do the same with dice, the odds of any given face is 1/faces, so get however many faced dice you need then set the faces (or a threshold again for numbered faces) e.g. on a 20-faced dice you set the threshold at 3+ and you’ve got a 90% payout rate.
This is why they are programmed to increase payout odds if there was no payout for long time, and this is why some players wait for such machines. (In modern casinos machines are pooled, so it’s actually more complicated)
In your example, there is no guarantee that 90% of rolls will be a 3 or above. Statistically, over the long term, that will be the case, but I wouldn't expect the payout laws to be written in such a way that "eventually" you'll "probabilistically" hit your payout targets.
More likely it's something like "you must pay out X over time period Y". You can't 100% guarantee that with a random process. It's possible (and not even all that unlikely) that, over the span of a day (week, month, whatever), you'd end up with only 89.9% of rolls being a 3 or above, and that would violate the payout laws.
Firstly what is the payout rate against, I could imagine it being:
- Plays - Time - Stake
Secondly, how is compliance defined:
- Convergence in expected value - Degree of breach of threshold
On the second one, it's kind of interesting if the degree of breach is 'never breach threshold', because when a machine is 'first turned on' the first play must always be a win (crudely speaking), if the second play has a sufficiently large stake (such that a loss would break the payout rate threshold) then that probably has to be a win too, etc until people can't keep exponentially raising the stages... This is a boot-strapping problem really but still.
If compliance is in expected value then I think any of the rates can be made to work
For a rate in terms of plays or stake then you can just have a binomial variable with some probability. In this case each player/play would be exposed to the same probability; each play would be independent.
For a rate in terms of time the plays would not be independent because the duration of previous plays and payout would affect what payout with subsequent plays would be, so you would need a dynamic adjustment of odds over the time domain, but this can still converge.
For a threshold like you describe then at such a point you would breach the threshold you may need to payout in a situation where you would have not paid out when converging in expectation.
I think what you are saying is that this shunting of probabilities would make the process not random.
However, I think the probability that a 'shunt' occurs could itself be modelled as a random variable dependent on the play history. Assuming some random properties to the sequence of plays (which is a reasonable modelling assumption), this would still be random.
It wouldn't be 'fair' in the sense of giving each player the same probabilities or for plays to be independent, but it would still be random just a distribution weirder than a bernoulli one.
Also FWIW:
> I wouldn't expect the payout laws to be written in such a way that "eventually"
That is exactly how I'd expect them to be written, seems the most natural way to do things, with the same ultimate benefit.
This is exactly how they are written. Except, there is no specific payout target (this game must payout 92% of wagers), there is a mandated minimum, any game which exceeds the minimum is lawful. Note the word "theoretical" in each of these:
New Jersey Admin Code 13:69E-1.28A: (a) Except as otherwise provided in this section, each slot machine game which requires a wager shall have a theoretical return to player (RTP) equal to or greater than 83 percent.
Mississippi Title 13, Chapter 12, Rule 12.5: Rule 12.5 Minimum Standards For Gaming Devices. All gaming devices submitted for approval:v(a) Must be electronic in design and operation and must be controlled by a microprocessor or the equivalent. Microcontrollers are allowed.v(b) Must theoretically pay out a mathematically demonstrable percentage of all amounts wagered, which must not be less than eighty percent (80%) or greater than one hundred percentv(100%) for each wager available for play on the device
Pennsylvania 58 Pa. Code § 461a.7: (a) A slot machine may not be set to pay out less than the theoretical payout percentage, which may not be less than 85%, calculated using the lowest possible wager that could be played for any single play, or equal or exceed 100%, calculated using the highest eligible wager available.