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The Nobel Prize in Economic Sciences 2012 and Matching Theory

The Nobel Prize in Economic Sciences 2012 was awarded jointly to A. E. Roth and L. S. Shapley “for the theory of stable allocations and the practice of market design." The reason why it was awarded to A. E. Roth and L. S. Shapley is two-fold: their extremely valuable efforts in applying scientific findings to very important real life problems such as kidney exchange and student placement problems, and their contribution to the theory of stable matchings. In this mini survey, we will first present the theory of stable matchings starting from the basics such as the Gale-Shapley Algorithm, and then discuss some variations encountered in various contexts. Two important applications, namely student placement and kidney exchange problems, will be given special consideration. The main focus of the survey will be the role of graph theory in the study of stable matchings. In particular, the links between stable matchings and the problem of finding an inclusion-wise maximal matching of minimum size will be explored. As a natural consequence of this link, the field of graph classes which became increasingly important, will be presented and illustrated with examples from matching theory.