Matching with Myopic and Farsighted Players

We study stable sets for marriage problems under the assumption that
players can be both myopic and farsighted. We introduce the new notion of
the myopic-farsighted stable set, which is based on the notion of a myopic-farsighted improving path. A myopic-farsighted stable set is the set of matchings such that there is no myopic-farsighted improving path from any matching in the set to another matching in the set (internal stability) and there
is a myopic-farsighted improving path from any matching outside the set to
some matching in the set (external stability). For the special cases where
all players are myopic and where all players are farsighted, our concept predicts
the set of matchings in the core. When all men are myopic and the top
choice of each man is a farsighted woman, we show that the singleton consisting
of the woman-optimal stable matching is a myopic-farsighted stable set.
The same result holds when all women are farsighted. We present examples
where this is the unique myopic-farsighted stable set as well as examples of
myopic-farsighted stable sets consisting of a core element different from the
woman-optimal matching or even of a non-core element.