Dividing goods and bads under additive utilities

Understand

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.