First-Choice Maximal and First-Choice Stable School Choice Mechanisms

School Choice: Nash Implementation of Stable Matchings through Rank-Priority Mechanisms

“…Hence, any mechanism in their class can be described as a "twophase" mechanism where the first phase consists of the first n = π ia (n) rank-priority steps of the Boston mechanism. Then, from the observation in the previous paragraph, we obtain Theorem 2 in Dur et al (2016b): any mechanism in the class considered by Dur et al (2016b) Nash implements the set of stable matchings.…”

“…In particular, we obtain Nash implementation for the adaptive Boston or immediate acceptance with skips mechanism (Alcalde, 1996, Harless, 2015, and Dur, 2015. Dur et al (2016b) consider the class of mechanisms that (1) maximize the number of students matched to their reported first choices and (2) yield a matching in which no student forms a blocking pair with his first choice. They show that the set of students that receive their first choice under each of these mechanisms always coincides with the set of students that receive their first choice under the Boston or immediate acceptance mechanism (Dur et al, 2016b, Lemma 1).…”

School choice: Nash implementation of stable matchings through rank-priority mechanisms

“…. 10 Since (4) is a condition on π, we will interchangeably refer to the quasi-monotonicity of π and ϕ π . Note that quasimonotonicity in fact only imposes restrictions on the rank-priority pairs that appear in π before position π(n), i.e., the position in which priority n appears for the first time.…”

“…In particular, we obtain Nash implementation for the adaptive Boston or immediate acceptance with skips mechanism (Alcalde, 1996, Harless, 2015, and Dur, 2015. Dur et al (2016b) consider the class of mechanisms that (1) maximize the number of students matched to their reported first choices and (2) yield a matching in which no student forms a blocking pair with his first choice. They show that the set of students that receive their first choice under each of these mechanisms always coincides with the set of students that receive their first choice under the Boston or immediate acceptance mechanism (Dur et al, 2016b, Lemma 1).…”

“…Dur et al (2016b) consider the class of mechanisms that (1) maximize the number of students matched to their reported first choices and (2) yield a matching in which no student forms a blocking pair with his first choice. They show that the set of students that receive their first choice under each of these mechanisms always coincides with the set of students that receive their first choice under the Boston or immediate acceptance mechanism (Dur et al, 2016b, Lemma 1). Hence, any mechanism in their class can be described as a "twophase" mechanism where the first phase consists of the first n = π ia (n) rank-priority steps of the Boston mechanism.…”