A mixed manna contains goods (that everyone likes), bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others.
If all items are goods and utility functions are homothetic, concave (and monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash product of utilities: hence it is welfarist (determined utility-wise by the feasible set of profiles), single-valued and easy to compute.
We generalize the Gale-Eisenberg Theorem to a mixed manna. The Competitive division is still welfarist and related to the product of utilities or disutilities. If the zero utility profile (before any manna) is Pareto dominated, the competitive profile is unique and still maximizes the product of utilities. If the zero profile is unfeasible, the competitive profiles are the critical points of the product of disutilities on the efficiency frontier, and multiplicity is pervasive. In particular the task of dividing a mixed manna is either good news for everyone, or bad news for everyone.
We refine our results in the practically important case of linear preferences, where the axiomatic comparison between the division of goods and that of bads is especially sharp. When we divide goods and the manna improves, everyone weakly benefits under the competitive rule; but no reasonable rule to divide bads can be similarly Resource Monotonic. Also, the much larger set of Non Envious and Efficient divisions of bads can be disconnected so that it will admit no continuous selection.
当效用是可加的时，我们在之前的论文（Bogomolnaia et al.“Dividing Goods or Bads under Additive Utilities”）中发现了许多相似之处，但也发现了我们在划分（私人）商品或坏品时，熟悉的竞争规则（收入相等）的行为中令人惊讶的差异。在这两种情况下，该规则都会在效率边界上挑选出效用（或非效用）产品的临界点，但如果我们分享商品，则只有一个临界点，而在坏的情况下，可能会有许多个临界点。我们将这种分析扩展到混合项目的公平划分：每个项目可以被一些参与者视为好的，而被其他人视为坏的，具有相应的正或负的边际效用。我们发现，混合物品的划分可以归结为一个全好问题或全坏问题的变体，无论是规范上还是计算上：特别是划分非一次性物品的任务必须对每个人都是好消息，或者对每个人都是坏消息。如果至少有一个可行的效用配置文件是积极的，竞争规则挑选唯一的最大值的产品（积极的）效用。如果没有可行的效用曲线是正的，这个规则会在有效边界上选择所有非效用产品的临界点。
