The thing I originally said about (almost) uniqueness was maybe wrong. Oops! I edited, and it it is correct now.
To see that there might be many solutions under the weakest notion of egalitarianism, consider the case where there are three people, A, B, and C, with utility a, b, c, and each with probability 13. The constraints on utility are that a≤1, and that b2+c2≤1. The thing is that if we give a small enough weight to A, then almost everything we can do with B and C will be egalitarian, and anything on the Pareto frontier that gives both B and C positive utility will be able to be simultaneously egaltarian and utilitarian.
You can’t run into this problem with two people, or with everyone ending up with the same utility.
Here is a proof that we get existence and uniqueness if we also have the constraint that everyone ends up with the same utility. The construction in the main post gives existence, because everyone has utility 1.
For uniqueness, we may take some point that satisfies utilitarianism, egalitarianism, and gives everyone the same utility. WLOG, it gives everyone utility 1, and is utilitarian and egalitarian with respect to the weight vector that gives everyone weight 1. This point maximizes the expected utility with respect to your your probability distribution. It also maximizes the expected logarithm of utility. This is because it achieves an expected logarithm of 0, and the concavity of the logarithm says that the expectation of the logarithm is at most the logarithm of the expectation, which is at most the logarithm of 1, which is 0. Thus, this point is a Nash bargaining solution (i.e. the point that maximizes expected log utility), and since the Nash bargaining solution is unique, it must be unique.
Note this is only saying the utility everyone gets is unique. There still might be multiple different strategies to achieve that utility.
Sorry for the (possible) error! It might be that the original thing turns out to be correct, but It depends on details of how we define the tiered egalitarian solution.
Yeah, that is correct.
The thing I originally said about (almost) uniqueness was maybe wrong. Oops! I edited, and it it is correct now.
To see that there might be many solutions under the weakest notion of egalitarianism, consider the case where there are three people, A, B, and C, with utility a, b, c, and each with probability 13. The constraints on utility are that a≤1, and that b2+c2≤1. The thing is that if we give a small enough weight to A, then almost everything we can do with B and C will be egalitarian, and anything on the Pareto frontier that gives both B and C positive utility will be able to be simultaneously egaltarian and utilitarian.
You can’t run into this problem with two people, or with everyone ending up with the same utility.
Here is a proof that we get existence and uniqueness if we also have the constraint that everyone ends up with the same utility. The construction in the main post gives existence, because everyone has utility 1.
For uniqueness, we may take some point that satisfies utilitarianism, egalitarianism, and gives everyone the same utility. WLOG, it gives everyone utility 1, and is utilitarian and egalitarian with respect to the weight vector that gives everyone weight 1. This point maximizes the expected utility with respect to your your probability distribution. It also maximizes the expected logarithm of utility. This is because it achieves an expected logarithm of 0, and the concavity of the logarithm says that the expectation of the logarithm is at most the logarithm of the expectation, which is at most the logarithm of 1, which is 0. Thus, this point is a Nash bargaining solution (i.e. the point that maximizes expected log utility), and since the Nash bargaining solution is unique, it must be unique.
Note this is only saying the utility everyone gets is unique. There still might be multiple different strategies to achieve that utility.
Sorry for the (possible) error! It might be that the original thing turns out to be correct, but It depends on details of how we define the tiered egalitarian solution.