The set of stable payoffs in assignment games is often large. We seek to refine this set in the spirit of the Nash (1953) program, where an idealized (or “cooperative”) (the work is joint with Mihai Manea.) solution is also supported by a non-cooperative mechanism whose Nash equilibria (possibly refined as in Nash (1953)) yield outcomes that exactly mirror the idealized solution. These dual perspectives jointly reinforce and validate one another.