Shapley-Scarf Markets with Objective Indifferences
Will Sandholtz, Andrew Tai
TL;DR
This paper introduces objective indifferences in Shapley-Scarf markets, where all agents share the same indifference classes over object types, and analyzes Top Trading Cycles with fixed tie-breaking ($\text{TTC}_{\succ}$) in this domain. It proves that TTC$_{\succ}$ is Pareto efficient, core-selecting, and group strategy-proof on the objective-indifferences domain, and that these properties fail on larger domains, with objective indifferences providing a maximal setting for PE and CS, and a symmetric-maximal setting for GSP. The results offer a precise boundary for when TTC remains desirable in the presence of indifferences and suggest careful domain specification (e.g., how to partition objects into indifference classes) in practical applications like housing or school choice. The work also clarifies the relationship between core structure (essentially single-valued under objective indifferences) and TTC outcomes, and points to future directions in constrained indifferences and axiomatic characterizations.
Abstract
Top trading cycles with fixed tie-breaking (TTC) has been suggested to deal with indifferences in object allocation problems. Unfortunately, under general indifferences, TTC is neither Pareto efficient nor group strategy-proof. Furthermore, it may not select an allocation in the core of the market, even when the core is non-empty. However, when indifferences are agreed upon by all agents (``objective indifferences''), TTC maintains Pareto efficiency, group strategy-proofness, and core selection. Further, we characterize objective indifferences as the most general setting where TTC maintains these properties.