On the fair division of a random object

Ann likes oranges and dislikes apples; Bob likes apples and dislikes oranges. Tomorrow they will receive one fruit that will be an orange or an apple with equal probability 0.5. Giving to each half of that fruit is fair for each realisation of the fruit; but 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 object, 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 realised 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 that use 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 designer, these rules perform in the same range as the best fair rule having full knowledge of this distribution.