Decisions in Economics and Finance https://doi.org/10.1007/s10203-018-0213-4 On the extension of binary relations in economic and game theories Athanasios Andrikopoulos 1 Received: 23 June 2017 / Accepted: 14 September 2018 © Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2018 Abstract Szpilrajn’s extension theorem on binary relations and its strengthening by Dushnik and Miller are fundamental in economic and game theories. Szpilrajn’s result entails that each partial order extends to a linear order. Dushnik and Miller use Szpilrajn’s theorem to show that each partial order has a realizer. Since then, many authors utilize Szpilrajn’s theorem and the well-ordering principle to prove more general theorems on extending binary relations. The original extension theorems of Szpilrajn, Dushnik- Miller and Moulin-Weymark are called: Szpilrajn extension theorem, Dushnik-Miller extension theorem and Moulin-Weymark’s Pareto extension theorem respectively. The generalizations of these theorems are called: Szpilrajn-type extension theorem, Dushnik-Miller-type extension theorem and Moulin-Weymark’s Pareto-type extension theorem respectively. The presented results generalize well-known extension theorems in the literature. Keywords Extension theorems · Consistent binary relations · Intersection of binary relations · Realizer JEL Classification C60 · C70 · D00 · D60 · D71 1 Introduction The Szpilrajn’s extension theorem shows that any transitive and asymmetric binary relation has an asymmetric, transitive and total extension (see Szpilrajn 1930). Arrow (1963, page 64), Hansson (1968) and Fishburn (1973) prove on the basis of the original Szpilrajn’s extension theorem that the result remains true if asymmetry is replaced with reflexivity, that is, any quasi-ordering has an ordering extension. While the property of being a quasi-ordering is sufficient for the existence of an ordering B Athanasios Andrikopoulos aandriko@ceid.upatras.gr 1 Department of Computer Engineering and Informatics, University of Patras, Patras, Greece 123