  Consider the complete bipartite graph on $n+n$ vertices where the edges are
equipped with i.i.d. exponential costs. A matching of the vertices is stable if
it does not contain any pair of vertices where the connecting edge is cheaper
than both matching costs. There exists a unique stable matching obtained by
iteratively pairing vertices with small edge costs. We show that the total cost
$C_{n,n}$ of this matching is of order $\log n$ with bounded variance, and that
$C_{n,n}-\log n$ converges to a Gumbel distribution. We also show that the
typical cost of an edge in the matching is of order $1/n$, with an explicit
density on this scale, and analyze the rank of a typical edge. These results
parallel those of Aldous for the minimal cost matching in the same setting. We
then consider the sensitivity of the matching and the matching cost to
perturbations of the underlying edge costs. The matching itself is shown to be
robust in the sense that two matchings based on largely identical edge costs
will have a substantial overlap. The matching cost however is shown to be noise
sensitive, as a result of the fact that the most expensive edges will with high
probability be replaced after resampling. Our proofs also apply to the complete
(unipartite) graph and the results in this case are qualitatively similar.
