On the fair division of a random object
Résumé
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.