“…. ., we update these sets and matchings 25 The tracking of reference matchings has no counterpart in earlier formulations of the deferred acceptance algorithms of, among many others, Gale and Shapley (1962), Roth (1984), Adachi (2000), Fleiner (2003), Echenique and Oviedo (2004), Hatfield and Milgrom (2005), Echenique and Oviedo (2006), Echenique and Yenmez (2007), Ostrovsky (2008), Kojima (2010), andBando (2014). In these papers, there is no need to track reference matchings and the deferred acceptance algorithm terminates when there are no more rejections and no new offers.…”