Competitive Division of a Mixed Manna

Abstract

A mixed manna contains goods (that everyone likes) and 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 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), 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.

Equitable Allocations of Indivisible Goods

“…However, there are subtle technical di erences between the two settings. In the context of (approximate) envy-freeness, this contrast has been noted in several works (Peterson and Su, 2009;Bogomolnaia et al, 2018Bogomolnaia et al, , 2017Brânzei and Sandomirskiy, 2019). To take one example, it is known that an allocation of goods that is both envy-free up to one good and Pareto optimal can be found by allocating goods so that the product of the agents' utilities-the Nash social welfare-is maximized .…”

“…Several papers study a model with mixed items, wherein an item can be a good for one agent and a chore for another. Bogomolnaia et al (2017) examine this model for divisible items and show that unlike the goods-only case, the set of competitive utility pro les (Varian, 1974;Eisenberg and Gale, 1959) can be multivalued; for the chores-only problem, the multiplicity can be exponential in the number of agents/items (Bogomolnaia et al, 2018). Segal-Halevi (2018a) and Meunier and Zerbib (2019) consider a generalization of the cake-cutting problem to the mixed utilities setting, and study envy-free divisions with connected pieces.…”

Competitive Equilibria with a Constant Number of Chores

“…While most of the work in fair division focuses on goods, there are a few works for the case of bads, e.g., (Azrieli & Shmaya, 2014;Brams & Taylor, 1996;Robertson & Webb, 1998;Su, 1999). The study of competitive division with a mixed manna was initiated by Bogomolnaia et al (2017). They establish equilibrium existence and show further properties, e.g., that multiple, disconnected equilibria may exist, and polynomial-time computation is possible if there are either two agents or two items with linear utility functions (Bogomolnaia, Moulin, Sandomirskiy, & Yanovskaia, 2019).…”

“…Instance Types. In (Bogomolnaia et al, 2017), the authors show that every fair division instance with mixed manna falls into one of three types: positive, negative, or null. The type roughly indicates whether there is a 'surplus' of goods or bads.…”

“…In a seminal contribution (Bogomolnaia, Moulin, Sandomirskiy, & Yanovskaia, 2017) the existence of competitive equilibria under general conditions for instances with mixed manna was established. Moreover, even for mixed manna, CEEI retain their attractive fairness properties.…”