Robust Minimal Instability of the Top Trading Cycles Mechanism

Abstract

In the context of priority-based resource allocation, we formulate methods to compare assignments in terms of their stability as binary relations (on the set of possible assignments) that depend on the preference and the priority profile. We introduce three basic properties, stability preferred, separability, and consistency, that a reasonable stability comparison should satisfy. We show that, for any stability comparison satisfying the three properties, the top trading cycles (TTC) mechanism is minimally unstable among efficient and strategy-proof mechanisms in one-to-one matching. An important consequence is the robustness of a recent result by Abdulkadiroglu et al. (2019), which uses a particular stability comparison method where an assignment is more stable than another assignment if the set of blocking pairs in the former assignment is a subset of the set of blocking pairs in the latter assignment. Our unifying approach covers basically all natural comparison methods and it includes many cardinal stability comparison methods as special cases. * Battal Dogan acknowledges financial support from the British Academy/Leverhulme Trust (SRG1819\190133). Lars Ehlers acknowledges financial support from the SSHRC (Canada) and the FRQSC (Québec). We thank participants at several seminars and conferences for valuable comments and suggestions.

Minimally Unstable Pareto Improvements over Deferred Acceptance

“…Yet, Abdulkadiroglu et al (2020) has shown that if we fix a set of agents and a set of schools with unit capacities, then TTC is minimally unstable among efficient and strategy-proof mechanisms. In our companion paper Dogan and Ehlers (2020b), we show that TTC is also minimally unstable among efficient and strategy-proof mechanisms for any of our stability comparisons if we fix a set of agents and a set of schools with unit capacities.…”

“…8 Not all conclusions have to be sensitive, however. InDogan and Ehlers (2020b), we show that a result byAbdulkadiroglu et al (2020), which relies on a particular stability comparison method, is, in fact, robust to the choice of the stability comparison method.…”

Minimally unstable Pareto improvements over deferred acceptance

“…Yet, Abdulkadiroğlu et al (2020) has shown that if we fix a set of agents and a set of schools with unit capacities, then TTC is minimally unstable among efficient and strategy‐proof mechanisms. In our companion paper Doğan and Ehlers (2020b), we show that TTC is also minimally unstable among efficient and strategy‐proof mechanisms for any of our stability comparisons if we fix a set of agents and a set of schools with unit capacities.…”

“…88 Not all conclusions have to be sensitive, however. In Doğan and Ehlers (2020b), we show that a result by Abdulkadiroğlu et al (2020), which relies on a particular stability comparison method, is, in fact, robust to the choice of the stability comparison method. …”

Criteria to compare mechanisms that partially satisfy a property: an axiomatic study

“…When mechanisms do not perfectly satisfy a property of interest, another approach consists in comparing them using a criterion formalizing "by how much" each solution violates the property. In the case of stability, which requires the absence of blocking pairs, Combe et al (2017), Abdulkadiroglu et al (2019), Dogan and Ehlers (2020b) and Bonkoungou and Nesterov (2020) compare mechanisms by measuring, in different ways, the number of blocking pairs or the number of players participating to a blocking pair in each profile. Dogan and Ehlers (2020a) axiomatically characterize criteria for stability comparisons based on axioms specific to this property.…”