EDIT: p.37-38 in Goodsell, 2023 gives a better proposal, which is to clip/truncate the utilities into the range [−t,t] and compare the expected clipped utilities in the limit as t→∞. This will still suffer from St Petersburg lottery problems, though.
Here’s an order that’s as strong as both expected utility and stochastic dominance, and overall seems promising to me:
tl;dr: For lotteries with finite utility payoffs (but possibly unbounded utility payoffs and infinite expected utility), we can take expectations through any subset with finite and well-defined expected utility, and then compare the resulting lotteries with stochastic dominance. We just need to find any pair of well-behaved “expected utility collapses” for which one lottery stochastically dominates the other. Allowing expected utility collapses over the infinite expected utilities can lead to A<A, so I rule that out.
In practice, you might just take one expectation over everything but the top X% and bottom Y% of each lottery, and compare those lotteries with stochastic dominance, for different values of X and Y. This allows you to focus on the tails of heavy-tailed distributions.
For a lottery X, a utility function U, and a countable (possibly finite and possibly empty) set of mutually exclusive non-empty measurable subsets of the measure space, P={Q1,Q2,…,Qn} (or basically a set of binary random variables whose sum is at most 1) and letting PC=(∪Q∈PQ)C be the complement of their union (so, for their indicator binary random variables, 1PC=1−∑Q∈P1Q), the expected utility collapse of X over P is:
XP=E[U(X)|Q] if Q, for Q∈P, and XP=X|PC, otherwise.
Or, in lottery notation, letting L(c) be the constant lottery with constant value c,
XP=P(PC)X|PC+∑Q∈PP(Q)L(E[U(X)|Q]).
In other words, we replace probability subsets of X with its expected utility over those subsets.
If furthermore, E[U(X)|Q] is well-defined and finite for each Q∈P, we call the expected utility collapse well-behaved.
Then, we define the order over lotteries as follows:
A<B if there exists well-behaved expected utility collapses AP1 and BP2 of A and B respectively such that BP2 strictly stochastically dominates AP1.
If you allow infinite actual utilities (including possibly infinities of different magnitudes), you could add a disjunctive or overriding condition to handle comparisons with those.
EDIT: p.37-38 in Goodsell, 2023 gives a better proposal, which is to clip/truncate the utilities into the range [−t,t] and compare the expected clipped utilities in the limit as t→∞. This will still suffer from St Petersburg lottery problems, though.
Here’s an order that’s as strong as both expected utility and stochastic dominance, and overall seems promising to me:
tl;dr: For lotteries with finite utility payoffs (but possibly unbounded utility payoffs and infinite expected utility), we can take expectations through any subset with finite and well-defined expected utility, and then compare the resulting lotteries with stochastic dominance. We just need to find any pair of well-behaved “expected utility collapses” for which one lottery stochastically dominates the other. Allowing expected utility collapses over the infinite expected utilities can lead to A<A, so I rule that out.
In practice, you might just take one expectation over everything but the top X% and bottom Y% of each lottery, and compare those lotteries with stochastic dominance, for different values of X and Y. This allows you to focus on the tails of heavy-tailed distributions.
For a lottery X, a utility function U, and a countable (possibly finite and possibly empty) set of mutually exclusive non-empty measurable subsets of the measure space, P={Q1,Q2,…,Qn} (or basically a set of binary random variables whose sum is at most 1) and letting PC=(∪Q∈PQ)C be the complement of their union (so, for their indicator binary random variables, 1PC=1−∑Q∈P1Q), the expected utility collapse of X over P is:
XP=E[U(X)|Q] if Q, for Q∈P, and XP=X|PC, otherwise.
Or, in lottery notation, letting L(c) be the constant lottery with constant value c,
XP=P(PC)X|PC+∑Q∈PP(Q)L(E[U(X)|Q]).
In other words, we replace probability subsets of X with its expected utility over those subsets.
If furthermore, E[U(X)|Q] is well-defined and finite for each Q∈P, we call the expected utility collapse well-behaved.
Then, we define the order over lotteries as follows:
A<B if there exists well-behaved expected utility collapses AP1 and BP2 of A and B respectively such that BP2 strictly stochastically dominates AP1.
If you allow infinite actual utilities (including possibly infinities of different magnitudes), you could add a disjunctive or overriding condition to handle comparisons with those.