so y_m is actually [...] which clutters up the notation so much that I don't want to write it down anymore.

Yes.

We also get into trouble with taking the expectation, the observations x{k+1:n} are only considered in modeling the actions of the future agents, but not now. What is M(yx<k,yx_k:n) even supposed to mean, where do the x's come from?

Oops, you are right. The sum should have been over x_{k:n}, not just over x_k.

So let's torture some indices: [...]

Yes, that is a cleaner and actually correct version what I was trying to describe. Thanks.