Thanks! I think you’re right. I think I actually should have defined ≻ϕ differently, because writing it out, it isn’t what I want. Having written out a small example, intuitively, L≻ϕM should hold iff ϕ(L)≻ϕ(M), which will also induce u(ϕ(oi)) as we want.
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. Probably it’s more natural to represent Eℓ∼ϕ−1(L)[u(ℓ)] as u⊤(Pϕ−1l)=(u⊤Pϕ−1)l, which makes your insight obvious.
The post is edited and the issues should now be fixed.
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.
Thanks! I think you’re right. I think I actually should have defined ≻ϕ differently, because writing it out, it isn’t what I want. Having written out a small example, intuitively, L≻ϕM should hold iff ϕ(L)≻ϕ(M), which will also induce u(ϕ(oi)) as we want.
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. Probably it’s more natural to represent Eℓ∼ϕ−1(L)[u(ℓ)] as u⊤(Pϕ−1l)=(u⊤Pϕ−1)l, which makes your insight obvious.
The post is edited and the issues should now be fixed.
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!