Ulric B. and Evelyn L. Bray Social Sciences Seminar
Abstract: A basic test of fairness when we divide a "manna" Ω of private items between agents is the lowest welfare the rule guarantees to each agent, irrespective of others' preferences. Two familiar examples are: the equal Split Guarantee (the utility of 1/nΩ) when the manna is divisible and preferences are convex; and 1/n-th 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 Ω. 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 Ω 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 Ω) of a share they find acceptable, and the lowest bidder picks such a share. The resulting guarantee is between the minMax and Maxmin utilities.
Written with Anna Bogomolnaia and Richard Stong.