Stable Matching: Gale and Shapley Algorithm

讲解下该问题 ( from Algorithm Design written by Jon and Eva)

Origin:

 双方根据自己的喜好配对

Formulating the Problem:

如果以applicant 和 company 举例，company 会招很多applicant, 并且company 的slot 比applicant多. 所以此时我们简化问题。

bare-bones version: 双方都只能和对方其中一个配对。 

preferences:  each member in M ranks each member in W,  no ties

Definition 

Matching S: a set of ordered pairs, with the property that each member of M and each member of W appears in at most one pair in S

Perfect matching S:  each member of M and each member of W appears in exactly one pair in S

   简单来说，matching S 可以只包含部分元素（如果两边个数不等）， 但stable matching 需要两边数目一样。

Stable Matching:   (1) perfect matching (2) no instability with respect to S   首先一定要perfect, 不然没法判断有没有stable 的pair 

   一个matching中可以有多个stable matching (书中最简单的例子，每边2个，只要左边的人选的都是第一志愿，他们肯定不会分开；同理，还可以是右边所有人)

Stable Matching

1.  stable matching exists for every set of preference list  ( 一定存在)