The Unbalanced Matching in a Director Market

I construct an intertemporal searching model (take it or leave it offer) in a frictional directorship market to explain the unbalanced matching between the director and the firm. In this model, potential candidates for outside directors and firms have heterogeneous (also, well ordered) quality levels. Also, both parties have strictly ordered preferences over the quality of counterpart from high levels to low levels. A candidate considers his quality ranking to compare the value of accepting a favorite offer at present to the value of waiting for successful matching with a better offer in the future. My model suggests that that highly qualified candidates would be likely to be matched with bad (not too bad) firms. The best candidate could go to the 150th ranked firm over 250 firms under the uniform distribution for the quality of the firm, and the 140th ranked firm under the extreme value distribution