You wrote:

“ The reason why your ensemble model goes to zero with more trials is because you are keeping the number of individuals constant while increasing the trials.”

Expectation has the notion of probability in it. It is the mean of the distribution of sum of pn x xn, as explained in Round One article. The average of a number of realizations is therefore NOT the expectation unless pn is known. This is a rudimentary mistake some make in probability.

In order to calculate pn, this can only be done in the frequency interpretation context. Therefore a sufficient number of repetitions is required. (See A. Papoulis reference)

Sufficient samples requirement for expectation then also applies to the number of trials (repetitions), not only to the number of individuals. This is exactly where the problem lies with these claims of non-ergodicity for this gamble.

If you increase the number of individuals, then you need to also increase the number of trials for sufficient samples. Otherwise pn is not known and expectation is not known.

However, it would be better to try to do the simulations yourself. I just repeated the simulation for 100000 individuals and 520 trials. Yes, you are right the expectation diverges. But when I increase the trails to 5200, it drops to 0 by 1000 trials.

Without skin-in-the-game with these simulations it is hard to understand what is happening.

I would prefer to see an article with simulations, rather than vague comments.

Thank you.