A note on the strong core of a market with indivisible goods (Q1057768)

\textit{L. S. Shapley} and \textit{H. E. Scarf} [ibid. 1, 23-37 (1974; Zbl 0281.90014)] studied a market where each agent i owns one indivisible good \(w_ i\), hence resources are \(w=(w_ 1,w_ 2,...,w_ n)\). Each agent has preferences among \(w_ i\), but does not desire more than one good. An allocation \(x=(x_ 1,x_ 2,...,x_ n)\) is a permutation of w. A competitive allocation is defined as usual with a price vector: each agent obtains the good which is best in his budget. An allocation is in the strong core if no coalition can weakly improve by only trading among themselves (at least one agent gets better off and not necessarily all). It is shown in this note that the strong core (which may be empty) is a subset of the competitive allocations, using the method of ''top trading cycles'' as introduced in the above-mentioned paper. An example shows that (normal) core allocations need not be competitive.