Competitive division of a mixed manna

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 homogeneous of degree one, concave (and monotone), the competitive division maximizes the Nash product of utilities (Gale-Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles), unique, continuous and easy to compute. We show that the competitive division of a mixed manna is still welfarist. If the zero utility profile is Pareto dominated, the competitive profile is strictly positive and still uniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance if all items are bads), the competitive profiles are strictly negative, and are the critical points of the product of disutilities on the efficiency frontier. The latter allows for multiple competitive utility profiles, from which no single-valued selection can be continuous or Resource Monotonic. Thus the implementation of competitive fairness under linear preferences in interactive platforms like SPLIDDIT will be more difficult when the manna contains bads that overwhelm the goods.