I am confused about something. You write that a preference ordering $L⪯M$ is geometrically rational if$G_{U\sim P}{E}_{O\sim L}U\left(O\right)\le G_{U\sim P}{E}_{O\sim M}U\left(O\right)$.

This is compared to VNM rationality which favours $L⪯M$ if and only if $E_{O\sim L}U\left(O\right)\le E_{O\sim M}U\left(O\right)$. 

Why, in the the definition of geometric rationality, do we have both the geometric average and the arithmetic average? Why not just say "an ordering is geometrically rational if it favours $L⪯M$ if and only if $G_{O\sim L}U\left(O\right)\le G_{O\sim M}U\left(O\right)$ " ?

As I understand it, this is what Kelly betting does. It doesn't favour lotteries over either outcome, but it does reject the VNM continuity axiom, rather than the independence axiom.

I think this is a good idea, thanks for implementing!

Very minor but the link lesswrong.com/moderation#rejected-comments just goes to the same page as lesswrong.com/moderation#rejected-posts (the written address is correct but the hyperlink goes to the wrong page)

The link to Harsanyi's paper doesn't work for me. Here is a link that does, if anyone is looking for one: 

https://hceconomics.uchicago.edu/sites/default/files/pdf/events/Harsanyi_1955_JPE_v63_n4.pdf 

The infinite non-uniform discrete case is not much more difficult. If $X$ is a finite or countably infinite set, $f:X\to R^{\ge 0}$ assigns a nonnegative value to each $x\in X$, and $P$ is a probability distribution on $Y$, then $E_{x\sim P}f\left(x\right)={\sum }_{x\in X}P\left(x\right)f\left(x\right)$

Very minor, but shouldn't this read "$P$ is a probability distribution on $X$" not $Y$?

Thanks for writing this. I think that the arguments in parts III and IV are particularly compelling and well-written.

Thanks for writing this. I wanted to write something about how Deutsch performs a bit a of motte-and-bailey argument (motte:'there are some problems in physics which are hard to solve using the dynamical laws approach'. bailey:'these problems can be solved using constructor theory specifically, rather than other approaches'). Your comment does a good job of making this case. In the end I didn't include it, as the piece was already too long. I just wrote the sentence 

Pointing out problems in the dynamical laws approach to physics and trying to find solutions is useful, even if constructor theory turns out not to be the best solution to them. 

and left it at that.