  This paper addresses two deficiencies of models in the area of matching-based
market design. The first arises from the recent realization that the most
prominent solution that uses cardinal utilities, namely the Hylland-Zeckhauser
(HZ) mechanism, is intractable; computation of even an approximate equilibrium
is PPAD-complete. The second is the extreme paucity of models that use cardinal
utilities. Our paper addresses both these issues by proposing
Nash-bargaining-based matching market models. Since the Nash bargaining
solution is captured by a convex program, efficiency follows. In addition, it
possesses several desirable game-theoretic properties. Our approach yields a
rich collection of models: for one-sided as well as two-sided markets, for
Fisher as well as Arrow-Debreu settings, and for a wide range of utility
functions, all the way from linear to Leontief. We give very fast
implementations for these models using Frank-Wolfe and Cutting Plane
algorithms. These help solve large instances with several thousand agents and
goods in a matter of minutes on a PC, even for a one-sided matching market
under piecewise-linear concave utility functions and a two-sided matching
market under linear utility functions. In contrast, using HZ, going beyond even
$n = 10$ is prohibitive. Several new ideas were needed, beyond the standard
methods, to obtain these implementations. In particular, we present several
lower bounding schemes, which not only help improve the convergence of our
solution methods but also shed light on fairness properties of the
Nash-bargaining-based models.
