On the Shapley–Scarf economy: the case of multiple types of indivisible goods

Abstract

We study a generalization of Shapley-Scarf's (1974) economy in which multiple types of indivisible goods are traded. We show that many of the distinctive results from the Shapley-Scarf economy do not carry over to this model, even if agents' preferences are strict and can be represented by additively separable utility functions. The core may be empty. The strict core, if nonempty, may be multi-valued, and might not coincide with the set of competitive allocations. Furthermore, there is no Pareto efficient, individually rational, and strategy-proof social choice rule. We also show that the core may be empty in the class of economies with a single type of indivisible good and agents consuming multiple units, even if no complementarity exists among the goods.

Random assignment of multiple indivisible objects

“…In a closely related context of multiple objects assignment problem with initial property rights, Konishi, Quint, and Wako (2001) show that there is no Pareto efficient, individually rational and strategyproof deterministic mechanism.…”

The coordinate-wise core for multiple-type housing markets is second-best incentive compatible

“…Moulin (1995) introduced multiple-type housing markets, but Konishi et al (2001) were the first to analyze the model. They demonstrate that when increasing the dimension of the model by adding other types of indivisible commodities, most of the positive results obtained for the one-dimensional case disappear: even for additively separable 2 preferences the core may be empty and no Pareto efficient, strategy-proof, and individually rational trading rule exists.…”

“…They demonstrate that when increasing the dimension of the model by adding other types of indivisible commodities, most of the positive results obtained for the one-dimensional case disappear: even for additively separable 2 preferences the core may be empty and no Pareto efficient, strategy-proof, and individually rational trading rule exists. For separable preferences, Konishi et al (2001) and Wako (2005) suggested an alternative solution to the core by first using separability to decompose a multiple-type housing market into "coordinate-wise submarkets" and second, determining the core in each submarket. Wako (2005) calls the resulting outcome the commodity-wise competitive allocation and shows that it is implementable in coalition-proof Nash equilibria, but not in strong Nash equilibria.…”

“…For multiple-type housing markets with more than one object type, i.e.,¯ ≥ 2, Konishi et al (2001) demonstrated that the core may be empty. Furthermore, if the core is not empty, then it may not be a singleton.…”

“…However, using the separability of preferences and the fact that a unique core allocation exists in each separate "marginal object type market," we now define the coordinate-wise core rule ϕ cc . Konishi et al (2001) mention the possibility to calculate the unique commodity-wise (strict) core for multiple-type housing markets with separable preferences. Wako (2005) introduced the coordinate-wise core under the name commodity-wise competitive allocations.…”