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  This section focuses on the algorithm developed by David Gale and Lloyd Shapley in 1962. The algorithm, called the Stable matching problem, is meant to answer the question: "Could one design a college admissions process, or a job recruiting process, that was

self-enforcing?“ This algorithm can also be illustrated by matching groups of people into “couples” so that they all are matched with the best possible individual to lead to a stable relationship environment. The remainder of the section goes on to discuss implementation and results of the algorithm and related proofs which will be discussed after this summary. /As aforementioned, the motivation for the problem is to develop a college admissions/job recruitment process that self-enforced and avoided people “jumping ship” to other opportunities.

  Here is a sketch of the algorithm as given by the text:
  Initially all m E M and w E W are free
  While there is a man m who is free and hasn’t proposed to
  every woman
   Choose such a man m
   Let w be the highest-ranked woman in m’s preference list
   to whom m has not yet proposed
   If w is free then
   (m, w) become engaged
   Else w is currently engaged to m’
      If w prefers m’ to m then
     m remains free
      Else w prefers m to m’
  (m,w) become engaged
  m' becomes free
      Endif
  Endif

Endwhile Return the set S of engaged pairs