I think there's a few missing pieces of information (in this discussion, probably could be looked up by some intrepid soul) Firstly what is the payout rate aga

I think there's a few missing pieces of information (in this discussion, probably could be looked up by some intrepid soul) 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.