Happy Family of Stable Marriages

In this chapter, we study some aspects of the problem of stable marriage. There are two distinguished marriage plans: the fully transferable case, where money can be transferred between the participants, and the fully nontransferable case where each participant has its own rigid preference list regarding the other gender. We continue to discuss intermediate partial transferable cases. Partial transferable plans can be approached as either special cases of cooperative games using the notion of a core or as a generalization of the cyclical monotonicity property of the fully transferable case (fake promises). We introduce these two approaches and prove the existence of stable marriage for the fully transferable and nontransferable plans. The marriage problem is a special case of more general assignment problems, which has many application in mathematical economy and logistics, in particular, the assignment of employees to hiring firms. The fully cooperative marriage plan is also a special case of the celebrated problem of optimal mass transport, which is also known as Monge-Kantorovich theory. Optimal transport problem has countless applications in many fields of mathematics, physics, computer science and, of course, economy, transportation and traffic control.