Before revision:  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.
After revision:  It is weakly below her Maxmin utility , that of her worst share in her best partition -- cannot be guaranteed, even for two agents. If for all agents more manna is better than less (or less is better than more), 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.