Suppose that you have inconsistent but monotone credences: if p entails q then P(Q)≥P(p). Level Set Integrals (LSI) provide a way of evaluating expected utilities that escapes Dutch Books and avoids domination failure: if E(f)≥E(g) then g cannot dominate f.
Sadly, by running this simple script, I’ve just discovered that LSI need not escape multi-shot domination failure. Suppose you have these monotonic credences for a coin toss:
Heads: 1/4
Tails: 1/4
Heads or Tails: 1
Neither: 0.
Suppose you’ll first be offered a choice between these two wagers:
- A: $1 if Heads and $1 if Tails
- A′: $3 if Heads and $0 if Tails
and then second you will be offered a choice between these two wagers:
- B: $1 if Heads and $1 if Tails
- B′: $0 if Heads and $3 if Tails.
You will first choose A over A′ and then B over B′. This is true regardless of whether you use the independent multi-shot decision procedure where you ignore previous wagers or the cumulative method where you compare the expected utilities of the sum of all the unresolved wagers. The net result of your choices is getting $2 no matter what. But if you chose A′ over A and B′ over B, you’d have received $3 no matter what.
Stepping back, classical expected utility with consistent credences has the following nice property: When you are offered a sequence of choices between wagers, with the offers not known in advance to you but also not dependent on your choices, and you choose by classical expected utility, you won’t choose a sequence of wagers dominated by another sequence you could have chosen.
Level Set Integrals with in my above case (and I strongly suspect more generally, but I don’t have a theorem yet) do not have this property.
I wonder how much it matters that one does not have this property. The property is one that involves choosing sequentially without knowing what choices are coming in the future, but the choices available in the future are not dependent on the choices already made. This seems a pretty gerrymandered situation.
If you do know what choices will be available in the future, it is easy to avoid being dominated: you figure out what sequence of choices has the overall best Level Set Integral, and you stick to that sequence.
