I also looked into this after that discussion. At the time I thought that this might have been something special about Kelly, but when I did some calculations afterwards I found that I couldn't get this to work in the other direction. I haven't fully parsed what you mean by:

(And since payoffs of the bet-against-yourself strategy are exactly identical to Kelly betting payoffs, a bunch of Kelly bets at house odds rearrange money in exactly the same way as this.)

But this is clearly equivalent to how hypotheses redistribute weight during Bayesian updates!

I'm not sure what you mean here. What is "this" in "looked into this" --
Critch's theorem? What is "the other direction"?
It seems obvious to me that the market will clear at the same price if everyone
is using the same fractional Kelly, but if people are using different Kelly
fractions, the weighted sum would be correspondingly skewed, right? Anyway,
that's not really important here...
The important thing for the connection to Critch's theorem is: the total wealth
gets adjusted like Bayes' Law. Other betting strategies may not have this
property; for example, fractional Kelly means losers lose less, and winners win
less. This doesn't limit us to exactly Kelly (for example, the
bet-against-yourself strategy in the post also has the desired property);
however, all such strategies must be equivalent to Kelly in terms of the payoffs
(otherwise, they wouldn't be equivalent to Bayes in terms of the updates!).
For example, if everyone uses fractional Kelly with the same fraction, then on
the first round of betting, the market clears with all the right prices, since
everyone is just scaling down how much they bet. However, the subsequent
decisions will then get messed up, because the everyone has the wrong weights
(weights changed less than they should).

I also looked into this after that discussion. At the time I thought that this might have been something special about Kelly, but when I did some calculations afterwards I found that I couldn't get this to work in the other direction. I haven't fully parsed what you mean by:

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