Abstract
We build an abstract model, closely related to the stable marriage problem and motivated by Hungarian college admissions. We study different stability notions and show that an extension of the lattice property of stable marriages holds in these more general settings, even if the choice function on one side is not path independent. We lean on Tarski’s fixed point theorem and the substitutability property of choice functions. The main virtue of the work is that it exhibits practical, interesting examples, where non-path independent choice functions play a role, and proves various stability-related results.
Highlights
In this paper, we study different generalizations of Gale and Shapley’s marriage and college admissions model
An observation attributed to Conway generalizes man and woman optimality. It states that stable marriages form a complete lattice for the partial order defined by the men
We describe the stability notion used in the Hungarian college admission scheme
Summary
We study different generalizations of Gale and Shapley’s marriage and college admissions model. A well-known result of Blair in [3] generalizes the case of strict preferences, by proving that if both sides of the matching market has path-independent substitutable choice functions, stable solutions form a lattice under a natural partial order. We study four kinds of stabilities: dominating stability, three-stability (which is defined by a three-partition of the contract set), four-stability (which comes from a four-partition of the contract set) and score-stability This last notion is generalized to so-called “loser-free” choice functions, allowing us to work with more flexible models describing diverse market situations, like company-worker admissions, with no strict preference ordering on the company’s side.
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