They're simply calculating expectation incorrectly. Expectation is a probability-weighted sum over every attempt. While the game of 50/40 is based on a probability-weighted product over every attempt.
Basically "average expectation" for that game is 0.95 (and they give a false name to it).
Tl;dr: it's got nothing to do with ergodicity.
More importantly, they quote "Peters coin game" which has no results in Google whatsoever.
The expectation of a single flip is 'positive' ( > 1 ), the expectation of n flips is 'negative' ( < 1).
My layman reading of this is that this 'appears' to be a violation of martingale theory because this 'should' be a sub-martingale but martingale theory requires finite variance so I suspect the variance of this must go infinite as n tends to infinity?
Earning 50% and then losing 40% is equivalent to losing 10%. The naïve calculation you're arguing against would yield a gain of 10%. kgwgk didn't make that mistake.
I agree that expectation may not be the most meaningful number here, in the long run is almost sure that we lose everything but there is a zero probability of infinite gains that makes up for it!
For any threshold (say $1, assuming you start with $1mn) and any confidence level (say 99%) there is a number of flips n such that you have over 99% probability of holding less than $1 after playing n times. On the other hand, the expected value of your wealth at that time is 1.05^n million dollars.
In fact, an even stronger statement is possible: If you start with $1MM, eventually you will fall below $1 and never go above $1 again no matter how long you keep playing.
And all of that not just on a fraction of sequences, but with probability 1.
Basically "average expectation" for that game is 0.95 (and they give a false name to it).
Tl;dr: it's got nothing to do with ergodicity.
More importantly, they quote "Peters coin game" which has no results in Google whatsoever.