  The stable matching problem is a prototype model in economics and social
sciences where agents act selfishly to optimize their own satisfaction, subject
to mutually conflicting constraints. A stable matching is a pairing of adjacent
vertices in a graph such that no unpaired vertices prefer each other to their
partners under the matching. The problem of finding stable matchings is known
as stable marriage problem (on bipartite graphs) or as stable roommates problem
(on the complete graph). It is well-known that not all instances on
non-bipartite graphs admit a stable matching. Here we present numerical results
for the probability that a graph with $n$ vertices and random preference
relations admits a stable matching. In particular we find that this probability
decays algebraically on graphs with connectivity $\Theta(n)$ and exponentially
on regular grids. On finite connectivity Erd\"os-R\'enyi graphs the probability
converges to a value larger than zero. Based on the numerical results and some
heuristic reasoning we formulate five conjectures on the asymptotic properties
of random stable matchings.
