Upon reflection, I now suspect that the Impact $I\left(a_i\right)$ is analogous to Shapley Value. In particular, the post could be reformulated using Shapley values and would attain similar results. I'm not sure whether Impact-scarcity of Shapley values holds, but the examples from the post suggest that it does.

(thanks to TurnTrout for pointing this out!)

Answering questions one-by-one:

• I played fast and loose with IEU in the intro section. I think it can be consistently defined in the Bayesian game sense of "expected utility given your type", where the games in the intro section are interpreted as each player having constant type. In the Bayesian Network section, this is explicitly the definition (in particular, player i's IEU varies as a function of their type).
• Upon reading the Wiki page, it seems like Shapley value and Impact share a lot of common properties? I'm not sure of any exact relationship, but I'll look into connections in the future.
• I think what's going on is that the "causal order" of $\omega$ and $a_i$ is switched, which makes $a_i$ "look as though" it controls the value of $\omega$. In terms of game theory the distinction is (I think) definitional; I include it because Impact has to explicitly consider this dynamic.
• In retrospect: yep, that's conditional expectation! My fault for the unnecessary notation. I introduced it to capture the idea of a vector space projection on random variables and didn't see the connection to pre-existing notation.

We conjecture this? We've only proven limiting cases so far, (constant-sum, and strongly suspected for common-payoff), but we're still working on formulating a more general claim.