A mixed manna contains goods (that everyone likes) and bads (that everyone dislikes), well as items that are goods to some agents, but bads or satiated to others. all items are goods and utility functions are homogeneous of degree 1 and 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), , continuous, and easy to compute. show that the competitive division of a mixed manna is still welfarist. If the zero profile is Pareto dominated, the competitive profile is strictly positive and still maximizes the product of utilities. If the zero profile is unfeasible (for instance, all items are bads), the competitive profiles are strictly negative and are the points of the product of disutilities on the efficiency frontier. The latter allows multiple competitive utility profiles, from which no single-valued selection can be or resource monotonic. the implementation of competitive fairness under linear preferences in interactive like SPLIDDIT will be more difficult when the manna contains bads overwhelm the goods.
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Bogomolnaia et al. (2017) studied this question.