Stability in matching markets: literary review from static matching to dynamic matching

This elaborate aims to collect and review the young but quite large literature on dynamic matching, which originates from the one developed in the 60s on (static) matching markets, which in turn led two of its main contributors, the mathematician Llyod Shapley and the economist Alvin Roth, to win the Nobel Prize for Economy in 2012. Some concepts and models from applied mathematics are also presented: it is interesting to draw comparisons and to observe that several problems, such as optimality in assignment and matching, have been addressed also in several disciplines besides matching theory. Both static matching and dynamic matching chapter cover two-sided and one-sided frameworks in which sidepayments are not present. The dynamic matching framework entails facing several challenges in defining stability and related concepts, even the slightest nuances in definitions and assumptions can lead to problems in describing behaviours and conjectures consistently.