Dividing goods and bads under additive utilities.

When utilities are additive, we uncovered in our previous paper (Bogomolnaia et al. "Dividing Goods or Bads under Additive Utilities") many similarities but also surprising differences in the behavior of the familiar Competitive rule (with equal incomes), when we divide (private) goods or bads. The rule picks in both cases the critical points of the product of utilities (or disutilities) on the efficiency frontier, but there is only one such point if we share goods, while there can be exponentially many in the case of bads. We extend this analysis to the fair division of mixed items: each item can be viewed by some participants as a good and by others as a bad, with corresponding positive or negative marginal utilities. We find that the division of mixed items boils down, normatively as well as computationally, to a variant of an all goods problem, or of an all bads problem: in particular the task of dividing the non disposable items must be either good news for everyone, or bad news for everyone. If at least one feasible utility profile is positive, the Competitive rule picks the unique maximum of the product of (positive) utilities. If no feasible utility profile is positive, this rule picks all critical points of the product of disutilities on the efficient frontier.