  A well known result states that stability criterion for matchings in
two-sided markets doesn't ensure uniqueness. This opens the door for a moral
question with regard to the optimal stable matching from a social point of
view. Here, a new notion of social optimality is proposed. Its novelty is its
ability to take into account the possibility of agents to leave the matching
after it has already been established. To formalize this real-life scenario,
this work includes a well-defined probability model and a social cost function
that maintain the general guidelines of leaving-agents situations. Finally,
efficient algorithms to optimize this function are developed either under
stability constraint or without it.
