No problem! Glad it was helpful. I think your fix makes sense.
I’m not quite sure what the error was in the original proof of Lemma 3; I think it may be how I converted to and interpreted the vector representation.
Yeah, I figured maybe it was because the dummy variable ℓ was being used in the EV to sum over outcomes, while the vector l was being used to represent the probabilities associated with those outcomes. Because ℓ and l are similar it’s easy to conflate their meanings, and if you apply ϕ to the wrong one by accident that has the same effect as applying ϕ−1 to the other one. In any case though, the main result seems unaffected.
No problem! Glad it was helpful. I think your fix makes sense.
Yeah, I figured maybe it was because the dummy variable ℓ was being used in the EV to sum over outcomes, while the vector l was being used to represent the probabilities associated with those outcomes. Because ℓ and l are similar it’s easy to conflate their meanings, and if you apply ϕ to the wrong one by accident that has the same effect as applying ϕ−1 to the other one. In any case though, the main result seems unaffected.
Cheers!