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!

So, a market of Kelly betters re-distributes money according to Bayesian updates.

So take the following with a (large) grain of salt before I can recheck my reasoning, but:

Everything you've written (as I currently understand it) also applies for many other betting strategies. eg if everyone was betting (the same constant) fractional Kelly.

Specifically the market will clear at the same price (weighted average probability) and "everyone who put money on the winning side picks up a fraction of money proportional to the fraction they originally contributed to that side".

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:

So take the following with a (large) grain of salt before I can recheck my reasoning, but:

Everything you've written (as I currently understand it) also applies for many other betting strategies. eg if everyone was betting (the same constant) fractional Kelly.

Specifically the market will clear at the same price (weighted average probability) and "everyone who put money on the winning side picks up a fraction of money proportional to the fraction they originally contributed to that side".