Unless I'm very mistaken the example they give is not a dynamic system, so asking whether it's ergodic doesn't make sense. For it to be a dynamic system it would need to have an invariant measure, which I can't even begin to figure out from their description.
The process of simply flipping coins, even a biased coin, does correspond to a dynamic system where the space is all infinite sequences of heads and tails, the invariant measure is the joint probability, and the operator is the left-shift (i.e. if you sample 1 series of coin flips and throw away the first result you still end up with a sample of the same distribution).
But this doesn't translate at all to their proposed scenario of starting everyone at 1 and letting the results diverge from there.
The process of simply flipping coins, even a biased coin, does correspond to a dynamic system where the space is all infinite sequences of heads and tails, the invariant measure is the joint probability, and the operator is the left-shift (i.e. if you sample 1 series of coin flips and throw away the first result you still end up with a sample of the same distribution).
But this doesn't translate at all to their proposed scenario of starting everyone at 1 and letting the results diverge from there.