They may express their preferences in various ways: Several considerations may be important in designing algorithms for house allocation.
Probably the simplest algorithm for house allocation is serial dictatorship: the agents are ordered in some arbitrary order (e.g. by seniority), and each agent in turn picks the best remaining house by his/her preferences.
[3] Abdulkadiroglu and Sönmez[1] consider an extended setting in which some agents already own a house while some others are house-less.
Ergin[4] considers rules that are also consistent, that is, their predictions do not depend of the order in which the assignments are realized.
In computer science and operations research, the primary efficiency requirement is maximizing the sum of utilities.
Algorithmic problems related to fairness of the matching have been studied in several contexts.