Robust stability in matching markets

Abstract

In a matching problem between students and schools, a mechanism is said to be robustly stable if it is stable, strategy-proof, and immune to a combined manipulation, where a student first misreports her preferences and then blocks the matching that is produced by the mechanism. We find that even when school priorities are publicly known and only students can behave strategically, there is a priority structure for which no robustly stable mechanism exists. Our main result shows that there exists a robustly stable mechanism if and only if the priority structure of schools is acyclic (Ergin 2002), and in that case, the student-optimal stable mechanism is the unique robustly stable mechanism.

Group robust stability in matching markets

“…We say that a mechanism is group robustly stable if it is non-manipulable via a combined manipulation by any group of students whereby each member of a manipulating group is at least weakly better off, and at least one member is strictly better off. Then, in contrast to the main theorem of Kojima (2011), which shows the existence of a robustly stable mechanism under acyclic priority structures (Ergin, 2002) (henceforth, the acyclic priority structure condition refers to Ergin, 2002 unless otherwise cited), our first result demonstrates that there is no group robustly stable mechanism even under acyclic priority structures. Given this impossibility result even under the demanding acyclicity condition, 2 we define a weak version of group robust stability, called weak group robust stability, and seek a condition under which it is achieved.…”

“…Yet, there has to be some rematching student in the group by the definition of combined manipulation. Kojima (2011) shows that there is no robustly stable mechanism, and since the group robust stability notion is a generalization of robust stability, the non-existence result also holds for group robust stability. Fact 1.…”

“…Kojima (2011) says that a mechanism is robustly stable if it is stable, strategy-proof, 5 and non-manipulable via combined manipulations, first misreporting preferences and then blocking (rematching) the produced matching, by individual students. However, he only considers the individual students' incentives.…”

“…In the current study, we extend the robust stability notion of Kojima (2011) to any group of students (including individual ones) in standard matching markets without interdependent values. We say that a mechanism is group robustly stable if it is non-manipulable via a combined manipulation by any group of students whereby each member of a manipulating group is at least weakly better off, and at least one member is strictly better off.…”

“…Chakraborty and Citanna (2009), then, extend the stability notion of them to any group of agents in the same environment. On the other hand, for standard matching markets without interdependent values, Kojima (2011) adopts the strong stability notion of Chakraborty et al (2010) and introduces the concept of robust stability requiring robustness against a combined manipulation by individual students.…”