# Competitive division of a mixed manna

## TL;DR

This paper extends the Gale-Eisenberg Theorem to the division of mixed manna containing goods and bads, analyzing the properties of competitive equilibria and their relation to utility maximization, uniqueness, and multiplicity.

## Contribution

It generalizes the Gale-Eisenberg Theorem to mixed manna, characterizing competitive divisions and their properties in complex scenarios with goods and bads.

## Key findings

- Competitive equilibrium maximizes the Nash product of utilities for goods.
- When the zero utility profile is Pareto dominated, the competitive profile is unique.
- Multiplicity of competitive profiles occurs when the zero profile is unfeasible.

## Abstract

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.