Fairness, risk preferences and independence: Impossibility theorems

The most widely used economic models of social preferences are specified only for certain outcomes. There are two obvious methods of extending them to lotteries. If we do so by expected utility theory, so that the independence axiom is satisfied, our results imply that the resulting preferences do not exhibit ex ante fairness. If we do so by replacing certain outcomes with their expected utilities for each individual, so that individual risk preferences are preserved, then ex ante fairness may be preserved, but neither the independence axiom nor ex post fairness is satisfied. Both ex ante and ex post fairness can be satisfied but then the individual does not have well defined preferences over own lotteries.