Competitive Division of a Mixed Manna

Abstract

A mixed manna contains goods (that everyone likes) and bads (that everyone dislikes), \nas well as items that are goods to some agents, but bads or satiated to others. \nIf all items are goods and utility functions are homogeneous of degree 1 and concave \n(and monotone), the competitive division maximizes the Nash product of utilities \n(Gale–Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles), \nunique, continuous, and easy to compute. \nWe show that the competitive division of a mixed manna is still welfarist. If the zero \nutility profile is Pareto dominated, the competitive profile is strictly positive and still \nuniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance, \nif all items are bads), the competitive profiles are strictly negative and are the \ncritical points of the product of disutilities on the efficiency frontier. The latter allows \nfor multiple competitive utility profiles, from which no single-valued selection can be \ncontinuous or resource monotonic. \nThus the implementation of competitive fairness under linear preferences in interactive \nplatforms like SPLIDDIT will be more difficult when the manna contains bads \nthat overwhelm the goods.

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