Re: Satoshi Dice -- Statistical Analysis

It's complicated by the fact that with 1 trial, there is a 0% chance that the house will take within 1.89 to 1.91 percent.

The chance is not 0%, but it is very small, perhaps <<0.001%.

Quote from: etotheipi on April 03, 2013, 04:15:51 PM

Markov chains are more useful when there's a relationship between the states of the system. In this case, it would be more like "losing 3 in a row changes your chances of winning the next one". Since we don't have that, you can use regular IID statistics.

It seems to me that SD could be modelled as a markov chain since the game is stochastic and has a markov property; that is, the outcome of trial B is not dependent on the outcome of trial A (I'm not sure if that's what you meant?). Although the probability of consecutive lessthan1 "successes" is very remote, it is still possible and a probability is associated with it. Given infinite number of trials, its bound to happen and there is no "losing 3 in a row changes your chance of winning the next one" for the markov model because the game has a markov property (or maybe it doesn't?). Maybe I'm misunderstanding the application of markov models, so please correct any errors I've made.

There's nothing stopping you from using Markov chains to model this problem and get a correct answer. But it would be like using calculus to compute the area of a square. It works, but there's simpler ways to do it.

I think this is somewhat a matter of taste. The MCMC method is more intuitive (at least to me, who is no math whiz), and it is easier to modify for input parameters for testing alternative hypotheses (satoshi dice does not work as claimed in some specific way that may be due to interactions between parameters). There is also a psychological factor in that you are more encouraged to go "outside the box" and mess around. And of course, you always get a nice approximation of the actual distribution, even if it is not normal. The tradeoff is cpu time and expenses. It is less limiting when you want to test other assumptions though, IMO.

edit: I thought of another way of putting it. MCMC slowly gives you an answer about the supposed 'square' you are actually measuring, while your method quickly gives an answer about perfect squares. Which is the better approach depends on how perfect the square is you are trying to measure.