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.
Maybe a reference to this paper [1] ?
[1] The time resolution of the St. Petersburg paradox, by Ole Peters
As he says in the Twitter thread anyway, they are coining this name.
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?
For 2 flips there are 4 equiprobable cases with gains +125%, -10%, -10%, -64% and the expectation is 10.25%. The expected gain for n flips is 1.05^n-1
If you earn 10% and then lose 10% you are worse off than before. Start with 10: (11-1.1 < 10).
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!
Why do you say almost sure to lose everything?
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.
It's just a regular sub-martingale with 'positive' expectation as you say.