Saturday, October 28, 2017

Two-sided matching with indifferences by Aytek Erdil and Haluk Ergin

Aytek Erdil and Haluk Ergin have a new paper in JET:

Two-sided matching with indifferences
Journal of Economic Theory, 171, 268-292, September 2017

"Most of the two-sided matching literature maintains the assumption that agents are never indifferent between any two members of the opposite side. In practice, however, ties in preferences arise naturally and are widespread. Market design needs to handle ties carefully, because in the presence of indifferences, stability no longer implies Pareto efficiency, and the deferred acceptance algorithm cannot be applied to produce a Pareto efficient or a worker-optimal stable matching.

"We allow ties in preference rankings and show that the Pareto dominance relation on stable matchings can be captured by two simple operations which involve rematching of workers and firms via cycles or chains. Likewise, the Pareto relation defined via workers' welfare can also be broken down to two similar procedures which preserve stability. Using these structural results we design fast algorithms to compute a Pareto efficient and stable matching, and a worker-optimal stable matching."

It is a followup to their earlier important paper

Erdil, Aytek; Ergin, Haluk, "What's the matter with tie-breaking? Improving efficiency in school choice," American Economic Review, 98, 3, 669-689, JUN 2008

In several school choice districts in the United States, the student proposing deferred acceptance algorithm is applied after indifferences in priority orders are broken in some exogenous way. Although such a tie-breaking procedure preserves stability, it adversely affects the welfare of the students since it introduces artificial stability constraints. Our main finding is a polynomial-time algorithm for the computation of a student-optimal stable matching when priorities are weak. The idea behind our construction relies on a new notion which we call a stable improvement cycle. We also investigate the strategic properties of the student-optimal stable mechanism.

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