I believe the entire point of the ergodicity question here is "If you apply this process n times, with n approaching infinity, obviously the result may depend on what point in the n-times iterated distribution you sample, but if you choose a volume of vanishingly small measure to exclude, can you make a single concrete statement about what the process is doing without taking an expected value over the different outcomes"
And the answer is yes - with probability approaching 1 as n increases (ie excluding a portion of the distribution whose measure decreases to 0), the random process matches a deterministic process which is described by "you lose 5% each round".
I should admit I'm being very generous to Peters here - I came to the conclusion that this is what he means only because the math of ergodicity (https://en.wikipedia.org/wiki/Ergodic_theory#Ergodic_theorem...) talks a lot about "except on a set of measure zero". He provides no explanation of how he moves from "the time average of values in a particular run of the process" (which is ergodicity) to "what does a typical process round do, with probability 1" (which is perhaps what someone computing a utility function cares about).
I asked a friend who is an econ professor "Why does this Peters guy explain this so poorly" and his response was more or less, yes, all of economics has been wondering that too since he first published his Nature Physics paper on this a decade ago.
This quote tells you all you need to know about the author's ability to understand things:
"my second criticism is more severe and I’m unable to resolve it: in maximizing the expectation value — an ensemble average over all possible outcomes of the gamble — expected utility theory implicitly assumes that individuals can interact with copies of themselves, effectively in parallel universes (the other members of the ensemble)."
And the answer is yes - with probability approaching 1 as n increases (ie excluding a portion of the distribution whose measure decreases to 0), the random process matches a deterministic process which is described by "you lose 5% each round".