Random objects arrive sequentially and must be divided among a fixed set of n agents upon arrival: the distribution \mu of the profile of utilities is IID across periods, but arbitrary across agents. We consider a class of online division rules that take, as an input, only the realized utility profile, and from \mu the individual expected utilities. They have no record from past realized utilities, and do not know either if and how many new objects will appear in the future. We call such rules prior-independent. A rule is fair if each agent , ex ante, expects at least 1/n-th of his utility for the object if it is a good, at most 1/n-th of his disutility for it if it is a bad. Among fair prior-independent rules to divide goods (bads)we uncover those collecting the largest (lowest) total expected (dis)utility . There is exactly one fair rule for bads that is optimal in this sense. However, for goods, the set of optimal fair rules is one dimensional. Both in the worst case and in the asymptotic sense, our optimal rules perform much better than the natural Proportional rule (for goods or for bads), and not much worse than the optimal fair prior-dependent rule that knows the full distribution \mu in addition to realized utilities .
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 .
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Random objects arrive sequentially and must be divided among a fixed set of n agents upon arrival: the distribution \mu of the profile of utilities is IID across periods, but arbitrary across agents. We consider a class of online division rules that take, as an input, only the realized utility profile, and from \mu the individual expected utilities. They have no record from past realized utilities, and do not know either if and how many new objects will appear in the future. We call such rules prior-independent. A rule is fair if each agent , ex ante, expects at least 1/n-th of his utility for the object if it is a good, at most 1/n-th of his disutility for it if it is a bad. Among fair prior-independent rules to divide goods (bads)we uncover those collecting the largest (lowest) total expected (dis)utility . There is exactly one fair rule for bads that is optimal in this sense. However, for goods, the set of optimal fair rules is one dimensional. Both in the worst case and
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
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asymptotic sense, our optimal rules perform much better than the natural Proportional rule (for goods or for bads), and not much worse than the optimal fair prior-dependent rule that knows the full distribution \mu in addition to realized utilities
same range as the best fair rule having full knowledge of this distribution