Guarantees in Fair Division: general or monotone preferences.

A basic test of fairness when we divide a manna of private items between n agents is the lowest welfare the rule guarantees to each agent, irrespective of others preferences. Two familiar examples are: the Equal Split Guarantee when the manna is divisible and preferences are convex; and 1/n of the utility of a heterogenous non atomic cake, if utilities are additive. The minMax utility of an agent is that of her best share in the worst possible n-partition of. It is weakly below her Maxmin utility, that of her worst share in the best possible n-partition. The Maxmin guarantee is not feasible, even with two agents, if non convex preferences are allowed. The minMax guarantee is feasible in the very general class of problems where the manna is non atomic and utilities are continuous, but not necessarily additive or monotonic. The proof uses advanced algebraic topology techniques. And the minMax guarantee is implemented by the n-person version of Divide and Choose due to Kuhn (1967). When utilities are co-monotone (a larger part of the manna is weakly better for everyone, or weakly worse for everyone) better guarantees than minMax are feasible. In our Bid & Choose rules, agents bid the smallest size (according to some benchmark measure of the manna) of a share they find acceptable, and the lowest bidder picks such a share. The resulting guarantee is between the minMax and Maxmin utilities. 1