Balanced Exchange in a Multi-Unit Shapley-Scarf Market

Abstract

We study markets in which each agent is endowed with multiple units of an indivisible and agent-specific good. Monetary compensations are not possible. An outcome of a market is given by a circulation which consists of a balanced exchange of goods. Agents only have (responsive) preferences over the bundles they receive. We prove that for general capacity configurations there is no circulation rule that satisfies individual rationality, Pareto-efficiency, and strategy-proofness. We characterize the (so-called irreducible) capacity configurations for which the three properties are compatible, and show that in this case the Circulation Top Trading Cycle (cTTC) rule is the unique rule that satisfies all three properties. We also explore the incentive and efficiency properties of the cTTC rule for general capacity configurations and provide a characterization of the rule for lexicographic preferences. Next, we introduce and study the family of so-called Segmented Trading Cycle (STC) rules. These rules are obtained by first distributing agents' endowments over a number of different smaller markets (the market segments), then applying the standard Top Trading Cycle algorithm within each market segment separately, and finally lumping together the induced circulations. We show that STC rules are individually rational, strategy-proof, and nonbossy. Even though STC rules do not satisfy group-strategy-proofness in general, they do satisfy weaker notions of group-strategy-proofness. For irreducible capacity configurations the family of STC rules collapses to the cTTC rule which then is also group-strategy-proof. Finally, we characterize one particularly interesting STC rule by means of top unanimity and self-enforcing group-strategy-proofness.