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I don't quite understand. Going with the worked out example in the post you link to:
To account for the compounding nature of losses, we can use the geometric expectation of total wealth instead of the arithmetic expectation of gains/losses when evaluating the alternatives. This is the mathematically correct thing to do, although I leave out the proof. If you want to dig into the gory details, see The Kelly Capital Growth Investment Criterion; MacLean, Thorp, & Ziemba; World Scientific Publishing; 2011. Yes, it’s that Thorp.
- If we replace the worn out part for $5,000, we are guaranteed to end up with a reduced wealth of $45,000.
If we wait and see, there are two things that could happen:
- With 90 % probability, our wealth will be unchanged at $50,000.
- With 10 % probability, our wealth will be reduced to $10,000.
The geometric expectation of these two outcomes is
50,0000.9×10,0000.1≈43,000.50,0000.9×10,0000.1≈43,000.
Okay, so we're optimizing for geometric expectation now.
Meanwhile, the same post:
[...] misconception 3: the Kelly criterion does not require logarithmic utility, it only requires trying to long-term maximise something that grows geometrically.
So we're ... not optimising for logarithmic utility. But isn't optimizing the geometric expectation equivalent to optimizing the log of utility? Last time I checked, E[ln(X)] = ln(G(X)) where G(X) is the geometric expectation of X (mind you I only used chatgpt to confirm my intuition but it could be wrong)
Which is fine actually. I do care more about my geometric expected wealth than I do about my expected wealth, but that would have been a much shorter post