Stable matchings in three-sided systems with cyclic preferences Original Research Article

We consider generalizations of the Gale–Shapley (Amer. Math. Monthly 69 (1962) 9) Stable Marriage Problem to three-sided families. Alkan (Math. Social Sci. 16 (1988) 207) gave an example which shows that in this case stable matchings do not always exist. Here we provide a simpler example demonstrating this fact. Danilov (NATO Advanced Research Workshop on Mathematical Theory of Allocation of Discrete Resources: Equilibria, Matching, Mechanisms, Sabanci University, Istanbul, Turkey, 16–19 December 2001; Math. Social Sci. 46(2) (2003) 145) proved that stable matchings always exist for the special case of certain acyclic preferences and he raised the problem for another special case involving cyclic preferences. Here we show that the answer is still negative by constructing a three-sided system with lexicographically cyclic preferences for which no stable matching exists. Finally, we also consider possible generalizations to image-sided families, for image.