If we’re choosing between two states of the world, and one state is better for at least one person and worse for no one (in terms of expected utility as defined above), then our ethical system should always prefer that state.
It’s also worth mentioning that this Pareto efficiency assumption, applied to expected utilities over uncertain options and not just actual utilities over deterministic options, rules out (terminal) other-centered preferences for utility levels to be distributed more equally (or less equally) ex post, as well as ex post prioritarian and ex post egalitarian social welfare functions.
You would be indifferent between these two options over the utility levels of Alice and Bob (well, if you sweeten either an arbitrarily low amount, you should prefer the sweetened one; continuity gives you indifference without sweetening):
50% chance of 1 for Alice and 0 for Bob, and 50% chance of 0 for Alice and 1 for Bob.
50% chance of 1 for each and 50% chance of 0 for each.
But an ex post egalitarian might prefer 2, since the utilities are more equal in each definite outcome.
Between the two following options, an ex post prioritarian or ex post egalitarian might strictly prefer the first, as well, even though the expected utilities are the same (using the numbers as the same utilities used to for expected utilities):
50% chance of −1 for Alice and a 50% chance of 1 for Alice.
100% chance of 0 for Alice.
HAT’s assumptions together also rule out preferences for or against ex ante equality and so ex ante priotarianism and ex ante egalitarianism, i.e., you should be indifferent between the following two options, even though the first seems more fair ex ante:
50% chance of 1 for Alice and 0 for Bob, and 50% chance of 0 for Alice and 1 for Bob.
It’s also worth mentioning that this Pareto efficiency assumption, applied to expected utilities over uncertain options and not just actual utilities over deterministic options, rules out (terminal) other-centered preferences for utility levels to be distributed more equally (or less equally) ex post, as well as ex post prioritarian and ex post egalitarian social welfare functions.
You would be indifferent between these two options over the utility levels of Alice and Bob (well, if you sweeten either an arbitrarily low amount, you should prefer the sweetened one; continuity gives you indifference without sweetening):
50% chance of 1 for Alice and 0 for Bob, and 50% chance of 0 for Alice and 1 for Bob.
50% chance of 1 for each and 50% chance of 0 for each.
But an ex post egalitarian might prefer 2, since the utilities are more equal in each definite outcome.
Between the two following options, an ex post prioritarian or ex post egalitarian might strictly prefer the first, as well, even though the expected utilities are the same (using the numbers as the same utilities used to for expected utilities):
50% chance of −1 for Alice and a 50% chance of 1 for Alice.
100% chance of 0 for Alice.
HAT’s assumptions together also rule out preferences for or against ex ante equality and so ex ante priotarianism and ex ante egalitarianism, i.e., you should be indifferent between the following two options, even though the first seems more fair ex ante:
50% chance of 1 for Alice and 0 for Bob, and 50% chance of 0 for Alice and 1 for Bob.
100% chance of 1 for Alice and 0 for Bob.