  We consider priority-based matching problems with limited farsightedness. We
show that, once agents are sufficiently farsighted, the matching obtained from
the Top Trading Cycles (TTC) algorithm becomes stable: a singleton set
consisting of the TTC matching is a horizon-$k$ vNM stable set if the degree of
farsightedness is greater than three times the number of agents in the largest
cycle of the TTC. On the contrary, the matching obtained from the Deferred
Acceptance (DA) algorithm may not belong to any horizon-$k$ vNM stable set for
$k$ large enough.
