“…The lattice property of competitive equilibrium payoffs and prices is intensively investigated in various directions, like matching models, e.g., Shapley and Shubik (1971), Demange and Gale (1985), Hatfield and Milgrom (2005) The rural hospital theorem of stable matchings and competitive equilibria has been established in different models, like matching models, e.g., Roth (1986), Demange and Gale (1985), Hatfield and Milgrom (2005), Hatfield and Kojima (2010), and Kojima et al (2018), and models of trading networks, e.g., Hatfield and Kominers (2012), Fleiner et al (2018), and Schlegel (2019). It shows the invariance of the sets of matched agents in different stable matchings and competitive equilibria.…”