An elementary non-constructive proof of the non-emptiness of the core of the Housing Market of Shapley and Scarf

Shapley and Scarf, by using the theory of balanced games, prove, in a well-known paper of 1974 (Journal of Mathematical Economics, 1, 23-28), the non-emptiness of the core of the Housing Market. The present paper provides a non-constructive, simple and short proof that gives some intuition about how blocking can be done by players who have not traded.