Thanks for the detailed reply! I want to push back on the claim that the EE agent and the log-utility maximizer make different choices
You say that in the additive case, the EE agent is exactly risk-neutral while the log-utility maximizer remains risk-averse. But consider what a log-utility maximizer actually does when facing a long sequence of independent additive bets.
For a single additive bet with payoff x, the contribution to log wealth is ln(W + x) - ln(W) ≈ x/W for small x relative to W. So the log-utility maximizer treats each bet approximately linearly. And as the number of independent additive bets grows, each individual bet becomes smaller relative to total wealth (because wealth is growing via the accumulated positive-EV bets), making the linear approximation increasingly exact.
In the limit of infinitely many independent additive bets, the log-utility maximizer’s per-bet behavior converges to exact risk-neutrality — which is exactly what EE prescribes. So in the regime where the EE prescription is most cleanly motivated (many repeated bets, which is the whole setup for time-average reasoning), the two agents converge.
Where they diverge is for large additive bets relative to current wealth. But here I think the log-utility maximizer is actually more faithful to the “maximize time-average growth rate” objective than the EE agent is. If you’re risk-neutral about an additive bet that could cost you half your wealth, you’re exposing yourself to ruin risk, which destroys your time-average growth rate. The identity mapping tells you the additive process is ergodic, but being ergodic doesn’t mean you should be risk-neutral about large bets within it — not if your goal is to maximize the growth rate of your own single trajectory.
So it seems to me that either (a) the bets are small relative to wealth, and the log-utility maximizer behaves (approximately, converging to exactly) like the EE agent, or (b) the bets are large relative to wealth, and the log-utility maximizer is arguably more correct about maximizing time-average growth. In neither case do the two agents clearly diverge in a way that supports the independence violation argument.
What am I missing?
If I’m right then again it doesn’t seem like the EE agent is really violating independence, given that its behaviour is replicated by a vnm agent that just values its final log wealth
But I don’t think EE would give different answers! EE would give the answer that would maximise the time-averaged geometric growth rate. That would be the same answer that would maximise the expected final log wealth after a long series of identical decisions.