  Ann likes oranges much more than apples; Bob likes apples much more than
oranges. Tomorrow they will receive one fruit that will be an orange or an
apple with equal probability. Giving one half to each agent is fair for each
realization of the fruit. However, agreeing that whatever fruit appears will go
to the agent who likes it more gives a higher expected utility to each agent
and is fair in the average sense: in expectation, each agent prefers his
allocation to the equal division of the fruit, i.e., he gets a fair share.
  We turn this familiar observation into an economic design problem: upon
drawing a random object (the fruit), we learn the realized utility of each
agent and can compare it to the mean of his distribution of utilities; no other
statistical information about the distribution is available. We fully
characterize the division rules using only this sparse information in the most
efficient possible way, while giving everyone a fair share. Although the
probability distribution of individual utilities is arbitrary and mostly
unknown to the manager, these rules perform in the same range as the best rule
when the manager has full access to this distribution.
