MichaelStJules comments on Impossibility results for unbounded utilities

[MichaelStJules]

Here are some ways to get more strict inequalities (less incomparability or equivalence):

1. Require $q=1-p$ to handle some more cases with both positive and negative expected infinities, but I’m not sure that the results would always be intuitive. There might be other relationships between $p$ and $q$ that that depend on the particular lotteries that work better. You could test the lim infs under multiple relationships, $q=1-f\left(p\right)$ for different $f$ from a specific set.

2. Replace the strict inequality condition with

$$\underset{(p,q)\to (0,1),0<p<q<1}{lim}\text{sgn}({\int }_p^q{Q}_{U\left(B\right)}\left(t\right)-Q_{U\left(A\right)}\left(t\right)dt)=1.$$

Equivalently, there are $p_0,q_0,0<p_0<q_0<1$ such that ${\int }_p^q{Q}_{U\left(B\right)}\left(t\right)-Q_{U\left(A\right)}\left(t\right)>0$ for all $p,q,0<p<p_0<q_0<q<1$.

$A<B$ would mean that the integral for $A$ never catches up with that for $B$ in the limit.